p-ADIC L-FUNCTIONS AND EULER SYSTEMS: A TALE IN TWO TRILOGIES

MASSIMO BERTOLINI, FRANCESC CASTELLA, HENRI DARMON SAMIT DASGUPTA, KARTIK PRASANNA, VICTOR ROTGER

Abstract. This article surveys six different special value formulae for p-adic L-functions, stressing their common features and their eventual arithmetic applications via the theory of “Euler systems”.

Contents

Introduction 1 1. Classical examples 3 1.1. Circular units 3 1.2. Elliptic units 8 1.3. Heegner points 15 2. Euler systems of Garrett-Rankin-Selberg type 19 2.1. Beilinson-Kato elements 20 2.2. Beilinson-Flach elements 25 2.3. Gross-Kudla-Schoen cycles 29 Conclusion 35 References 36

Introduction

This article surveys six different special value formulae for p-adic L-functions, stressing their common features and their eventual arithmetic applications via the theory of “Euler systems”. The most classical instances are:

(1) Leopoldt’s formula for the value at s = 1 of the Kubota-Leopoldt p-adic L-function in terms of p-adic logarithms of circular units;

(2) Katz’s p-adic Kronecker limit formula for values of the two variable p-adic Lfunction of a quadratic imaginary field at finite order characters in terms of p-adic logarithms of associated elliptic units.

They are reviewed in Sections 1.1 and 1.2 respectively. Section 1.3 describes the more recent formula of [BDP] and explains why it is a direct generalisation of the formulae of Leopoldt and Katz in the setting where special units are replaced by Heegner points. The three parallel treatments in Chapter 1 suggest that both elliptic and circular units might be viewed as degenerate cases of the Euler system of Heegner points, obtained by successively replacing the cusp forms and ordinary CM points that arise in the latter setting by Eisenstein series and cusps.

The second part of this survey attempts to view the Euler system introduced by Kato [Kato] in a similar way, as the “most degenerate instance” of a broader class of examples. Referred to as “Euler systems of Garrett-Rankin-Selberg type” because of the role played by the formulae of Rankin-Selberg and Garrett in relating them to special values of Lfunctions, these examples consist of

(1) Kato’s original Euler system of (p-adic families of) Beilinson elements in the second K-group of modular curves, whose global objects are indexed by pairs {u_{1}, u_{2}} of modular units. Their connection to L-values follows from Rankin’s method applied to a cusp form and the pair of weight two Eisenstein series corresponding to the logarithmic derivatives of u_{1} and u_{2}.

(2) The Euler system of Beilinson-Flach elements in the first K-group of a product of two modular curves.

(3) The Euler system of generalised Gross-Kudla-Schoen diagonal cycles, whose connection with L-values arises from the formula of Garrett for the central critical value of the convolution L-series attached to a triple of newforms.

The global cohomology classes in (3) are indexed by triples (f, g, h) of cusp forms and take values in the tensor product of the three p-adic representations attached to f, g and h. Example (1) (resp. (2)) can in some sense be viewed as a degenerate instance of (3) in which g and h (resp. h only) are replaced by Eisenstein series.

Ever since the seminal work of Kolyvagin [Ko], there have been many proposals for axiomatizing and classifying the Euler systems that should arise in nature (cf. [Ru], [PR3], [Cz1], [Cz2], [MR], …) with the goal of understanding them more conceptually and systematizing the process whereby arithmetic information is coaxed from their behaviour. The present survey is less ambitious, focussing instead on six settings where the associated global objects have been constructed unconditionally, attempting to organise them coherently, and suggesting that they arise from two rather than six fundamentally distinct classes of examples. Like the ten plagues of Egypt in the Jewish Passover Haggadah, Euler Systems can surely be counted in many ways. The authors believe (and certainly hope) that their “two trilogies” are but the first instalments of a richer story in which higher-dimensional cycles on Shimura varieties and p-adic families of automorphic forms are destined to play an important role.

Acknowledgements: The authors thank Minhyong Kim for encouraging them to write this survey.

p-ADIC L-FUNCTIONS AND EULER SYSTEMS 3

1. Classical examples

1.1. Circular units. Let χ : (ℤ/Nℤ)^{×} ⟶ ℂ^{×} be a primitive, non-trivial even Dirichlet character of conductor N. The values of the Dirichlet L-function L(s, χ) at the negative odd integers belong to the field ℚ_{χ} ⊂ ℚ̄ generated by the values of χ. This can be seen by realising L(1 − k, χ) for even k ≥ 2 as the constant term of the holomorphic Eisenstein series

(1)  E_{k,χ}(q) := L(1 − k, χ) + 2 ∑_{n=1}^{∞} σ_{k−1,χ}(n)q^{n},  σ_{k−1,χ}(n) := ∑_{d|n} χ(d)d^{k−1}

of weight k, level N and character χ, and invoking the q-expansion principle to argue that the constant term in (1) inherits the rationality properties of the coefficients σ_{k−1,χ}(n).

If p is any prime (possibly dividing N), the ordinary p-stabilisation

(2)  E_{k,χ}^{(p)}(q) := E_{k,χ}(q) − χ(p)p^{k−1}E_{k,χ}(q^{p})

has Fourier expansion given by

(3)  E_{k,χ}^{(p)}(q) = L_{p}(1 − k, χ) + 2 ∑_{n=1}^{∞} σ_{k−1,χ}^{(p)}(n)q^{n},

where

(4)  L_{p}(1 − k, χ) = (1 − χ(p)p^{k−1})L(1 − k, χ),  σ_{k−1,χ}^{(p)}(n) = ∑_{p∤d|n}χ(d)d^{k−1}.

For each n ≥ 1, the function on ℤ^{≥1} sending k to the n-th Fourier coefficient σ_{k−1,χ}^{(p)}(n) extends to a p-adic analytic function of k ∈ (ℤ/(p − 1)ℤ) × ℤ_{p}. The article [Se] explains why the constant term of (3) inherits the same property. The resulting extension to ℤ/(p − 1)ℤ × ℤ_{p} of L_{p}(s, χ), defined a priori as a function on the negative odd integers, is the Kubota-Leopoldt p-adic L-function attached to χ. The elegant construction of L_{p}(s, χ) arising from this circle of ideas (and its subsequent extension to totally real fields) was one of the original motivations for the theory of p-adic modular forms initiated in [Se].

The collection of eigenforms in (3) is a prototypical example of a p-adic family of modular forms, whose specialisations at even integers k ≤ 0, while not classical, continue to admit a geometric interpretation as p-adic modular forms^{1} of weight k and level N_{0}, the primeto-p part of N. When k = 0, these are just rigid analytic functions on the ordinary locus A ⊂ X_{1}(N_{0})(ℂ_{p}) obtained by deleting from X_{1}(N_{0})(ℂ_{p}) all the residue discs attached to supersingular elliptic curves in characteristic p. In particular, the special value L_{p}(1, χ) can be interpreted as the the value at the cusp ∞ of such a rigid analytic function, namely the weight 0 Eisenstein series

(5)  E_{0,χ}^{(p)}(q) = L_{p}(1, χ) + 2 ∑_{n=1}^{∞} (∑_{p∤d|n}χ(d)d^{−1}) q^{n}.

1they are even overconvergent, but this stronger property will not be exploited here.

An independent expression for this function can be derived in terms of the Siegel units g_{a} ∈ O_{Y_{1}(N)}^{×} attached to a fixed choice of primitive N-th root of unity ζ and a parameter 1 ≤ a ≤ N − 1, whose q-expansions are given by

(6)  g_{a}(q) = q^{1/12}(1 − ζ^{a}) ∏_{n>0}(1 − q^{n}ζ^{a})(1 − q^{n}ζ^{−a}).

More precisely, let Φ be the canonical lift of Frobenius on A which sends the point corresponding to the moduli of the pair (E, t) ∈ A to the moduli of the pair (E/C, t + C) where C ⊂ E(ℂ_{p}) is the canonical subgroup of order p in E. The rigid analytic function

(7)  g_{a}^{(p)} := Φ^{∗}(g_{a})g_{a}^{−p} = g_{pa}(q^{p})g_{a}(q)^{−p}

maps the ordinary locus A to the residue disc of 1 in ℂ_{p}, and therefore its p-adic logarithm log_{p} g_{a}^{(p)} is a rigid analytic function on A, with q-expansion given by

log_{p} g_{a}^{(p)} = log_{p} (\frac{1 − ζ^{ap}}{(1 − ζ^{a})^{p}} ) + p ∑_{n=1}^{∞} (∑_{p∤d|n} \frac{ζ^{ad} + ζ^{−ad}}{d} ) q^{n}.

Letting

(8)  𝔤(χ) = ∑_{a=1}^{N−1} χ(a)ζ^{a}

denote the Gauss sum attached to χ, a direct computation shows that the rigid analytic function on A given by

(9)  h_{χ}^{(p)} := \frac{1}{p𝔤(χ^{−1})} × ∑_{a=1}^{N−1} χ^{−1}(a) log_{p} g_{a}^{(p)}

has q-expansion equal to

(10)  h_{χ}^{(p)}(q) = −\frac{(1 − χ(p)p^{−1})}{𝔤(χ^{−1})} ∑_{a=1}^{N−1} χ^{−1}(a) log_{p}(1 − ζ^{a}) + 2 ∑_{n=1}^{∞} (∑_{p∤d|n}χ(d)d^{−1}) q^{n}.

Theorem 1.1 (Leopoldt). Let χ be a non-trivial even primitive Dirichlet character of conductor N. Then

L_{p}(1, χ) = −\frac{(1 − χ(p)p^{−1})}{𝔤(χ^{−1})} ∑_{a=1}^{N−1} χ^{−1}(a) log_{p}(1 − ζ^{a}).

Proof. Comparing q-expansions in (5) and (10) shows that the difference E_{0,χ}^{(p)} − h_{χ}^{(p)} is constant on the residue disc of a cusp, and hence on all of A since it is rigid analytic on this domain. In fact,

(11)  E_{0,χ}^{(p)} = h_{χ}^{(p)},

since both these p-adic modular functions have nebentype character χ ≠ 1. Leopoldt’s formula follows by equating the constant terms in the q-expansions in (5) and (10). For more details on this “modular” proof of Leopoldt’s formula, see [Katz, §10.2]. □

Remark: Recall the customary notations in which E_{k}(ψ, χ) denotes the Eisenstein series attached to a pair (ψ, χ) of Dirichlet characters, having the q-expansion

E_{k}(ψ, χ) = δ_{ψ=1}L(1 − k, ψχ) + 2 ∑_{n=1}^{∞} (∑_{d|n} ψ(n/d)χ(d)d^{k−1}) q^{n},

and let χ_{p} be the Dirichlet character of modulus Np which agrees with χ on (ℤ/Npℤ)^{×}, so that E_{k}(1, χ) = E_{k,χ} and E_{k}(1, χ_{p}) = E_{k,χ}^{(p)}. The rigid analytic function E_{0,χ}^{(p)} = h_{χ}^{(p)} which is a key actor in the proof of Leopoldt’s formula above is a Coleman primitive of the weight two Eisenstein series E_{2}(χ_{p}, 1), a non-ordinary modular form of critical slope since its p-th Fourier coefficient is equal to p. The pattern whereby special values of p-adic L-series outside the range of classical interpolation arise as values of Coleman primitives of p-adic modular forms at distinguished points of the modular curve (namely cusps, or ordinary CM points) will recur in Sections 1.2 and 1.3.

The expressions of the form (1 − ζ^{a}) (when N is composite) and \frac{1−ζ^{a}}{1−ζ^{b}} (when N is prime) that occur in Leopoldt’s formula are called circular units. These explicit units play an important role in the arithmetic of the cyclotomic field ℚ(ζ). Letting F_{χ} denote the field cut out by χ viewed as a Galois character, and ℤ_{χ} the ring generated by its values, the expression

u_{χ} := ∏_{a=1}^{N−1}(1 − ζ^{a})^{χ^{−1}(a)} ∈ (O_{F_{χ}}^{×} ⊗ ℤ_{χ})^{χ}

is a distinguished unit in F_{χ} (or rather, a ℤ_{χ}-linear combination of such) which lies in the χ-eigenspace for the natural action of the absolute Galois group G_{ℚ} of ℚ (in which the second factor ℤ_{χ} in the tensor product is fixed by this group).

A notable feature of the unit u_{χ} is that it is essentially a “universal norm” over the tower of cyclotomic fields whose n-th layer is F_{χ,n} = F_{χ}(μ_{p^{n}}). More precisely, after fixing a sequence (ζ = ζ_{N}, ζ_{Np}, ζ_{Np^{2}}, …, ζ_{Np^{n}}, …) of primitive Np^{n}-th roots of unity which are compatible under the p-power maps, and setting

u_{χ,n} = ∏_{a=1}^{N−1}(1 − ζ_{Np^{n}}^{a})^{χ^{−1}(a)} ∈ (O_{F_{χ,n}}^{×} ⊗ ℤ_{χ})^{χ},

we find that

Norm_{F_{χ,n}}^{F_{χ,n+1}}(u_{χ,n+1}) = \{ \begin{matrix}u_{χ,n} & if n ≥ 1, \\ u_{χ} ⊗ (χ(p) − 1) & if n = 0. \end{matrix}

After viewing χ as a ℂ_{p}-valued character, let ℤ_{p,χ} be the ring generated over ℤ_{p} by the values of χ (endowed with the trivial G_{ℚ}-action) and let ℤ_{p,χ}(χ) be the free module of rank one over ℤ_{p,χ} on which G_{ℚ} acts via the character χ. The symbols ℚ_{p,χ} and ℚ_{p,χ}(χ) are likewise given the obvious meaning. The images

κ_{χ,n} := δu_{χ,n} ∈ H^{1}(F_{χ,n}, ℤ_{p,χ}(1))^{χ} = H^{1}(F_{n}, ℤ_{p,χ}(1)(χ^{−1}))

under the connecting homomorphism δ : (F_{χ,n}^{×} ⊗ ℤ_{χ})^{χ} ⟶ H^{1}(F_{χ,n}, ℤ_{p,χ}(1))^{χ} of Kummer theory can be patched together in a canonical element κ_{χ,∞} := (κ_{χ,n})_{n≥0} belonging to

(12)  \underleftarrow{lim}_{n} H^{1}(F_{n}, ℤ_{p,χ}(1)(χ^{−1})) = \underleftarrow{lim}_{n} H^{1}(ℚ, ℤ_{p}[G_{n}] ⊗_{ℤ_{p}} ℤ_{p,χ}(1)(χ^{−1}))
= H^{1}(ℚ, Λ_{cyc} ⊗_{ℤ_{p}} ℤ_{p,χ}(1)(χ^{−1})),

where

• ℤ_{p}[G_{n}] is the group ring of G_{n} := Gal(F_{n}/ℚ) = (ℤ/p^{n}ℤ)^{×}, equipped with the tautological action of G_{ℚ} in which σ ∈ G_{ℚ} acts via multiplication by its image in G_{n}, and the identification (12) follows from Shapiro’s lemma;

• Λ_{cyc} = \underleftarrow{lim}_{n} ℤ_{p}[G_{n}] = ℤ_{p}[[ℤ_{p}^{×}]] is the completed group ring of ℤ_{p}^{×} equipped with the

similar “tautological” action of G_{ℚ}. The Galois module Λ_{cyc} can be viewed as a p-adic interpolation of the Tate twists ℤ_{p}(k) for all k ∈ ℤ. More precisely, given k ∈ ℤ and a Dirichlet character ξ of p-power conductor, let ν_{k,ξ} : Λ ⟶ ℤ_{p,ξ} be the ring homomorphism sending the group-like element a ∈ ℤ_{p}^{×} to a^{k−1}ξ^{−1}(a). It induces a G_{ℚ}-equivariant specialisation map

ν_{k,ξ} : Λ_{cyc} ⟶ ℚ_{p,ξ}(k − 1)(ξ^{−1}),

giving rise to a collection of global cohomology classes

κ_{k,χξ} := ν_{k,ξ}(κ_{χ,∞}) ∈ H^{1}(ℚ, ℚ_{p,χξ}(k)((χξ)^{−1})).

These classes can be viewed as the “arithmetic specialisations” of the p-adic family κ_{χ,∞} of cohomology classes.

Given any Dirichlet character η with η(p) ≠ 1, let F_{p,η} be the finite extension of ℚ_{p} cut out by the corresponding Galois character and denote by G_{η} = Gal(F_{p,η}/ℚ_{p}) its Galois group. Restriction to F_{p,η} composed with δ^{−1} leads to the identifications

H^{1}(ℚ_{p}, ℚ_{p,η}(1)(η)) = H^{1}(F_{p,η}, ℚ_{p,η}(1)(η))^{G_{η}} = (H^{1}(F_{p,η}, ℚ_{p}(1)) ⊗_{ℚ_{p}} ℚ_{p,η}(η))^{G_{η}}
= (O_{F_{p,η}}^{×} ⊗_{ℚ_{p}} ℚ_{p,η}(η))^{G_{η}} .

Applying the p-adic logarithm log_{p} : O_{F_{p,η}}^{×} → F_{p,η} to this last module leads to the map

(13)  log_{η} : H^{1}(ℚ_{p}, ℚ_{p,η}(1)(η)) ⟶ (F_{p,η} ⊗_{ℚ_{p}} ℚ_{p,η}(η))^{G_{η}} = (ℂ_{p} ⊗_{ℚ_{p,η}} ℚ_{p,η}(η))^{G_{ℚp}},

where the Tate-Sen isomorphism F_{p,η} = ℂ_{p}^{Gal(ℚ̄_{p}/F_{p,η})} has been used to make the last identification.

The module 𝔻_{ℂ_{p}}(ℚ_{p,η}(η)) := (ℂ_{p} ⊗_{ℚ_{p,η}} ℚ_{p,η}(η))^{G_{ℚp}} appearing as the target in (13) is the Dieudonné module (with ℂ_{p} as “period ring”) attached to the Artin representation ℚ_{p,η}(η). It is a one-dimensional ℚ_{p,η}-vector space generated by the “Gauss sum”

𝔤(η) = ∑_{a=1}^{m−1} ζ_{m}^{a} ⊗ η(a)

which can thus be viewed as a “p-adic period” attached to the Galois representation ℚ_{p,η}(η). Theorem 1.1 can be re-phrased as the following relationship between the classes κ_{1,χξ} and the values of the Kubota Leopoldt L-function at s = 1, twisted eventually by finite order characters:

(14)  L_{p}(1, χξ) = −\frac{(1 − χξ(p)p^{−1})}{(1 − (χξ)^{−1}(p))} × \frac{log_{χξ}(κ_{1,χξ})}{𝔤(χξ)}.

Note in particular that the global classes κ_{1,χξ} determine the Kubota-Leopoldt L-function completely, since an element of the Iwasawa algebra has finitely many zeroes.

For all k ≥ 1 and characters η of conductor prime to p, Bloch and Kato have defined a generalisation of the map log_{η} of (13) for the representation ℚ_{p,η}(k)(η), in which ℂ_{p} is replaced by the larger Fontaine period ring B_{dR}:

(15)  log_{k,η} : H^{1}(ℚ_{p}, ℚ_{p,η}(k)(η)) ⟶ 𝔻_{dR}(ℚ_{p,η}(k)(η)) := (B_{dR} ⊗_{ℚ_{p}} ℚ_{p,η}(k)(η))^{G_{ℚp}}.

(See for instance [Kato], [Cz1, §2.6]; we have implicitly used the fact that for k ≥ 1, all extensions of ℚ_{p} by ℚ_{p}(k) are cristalline, and that Fil^{0}𝔻_{dR}(ℚ_{p,η}(k)(η)) = 0.) The target of the logarithm map is a one-dimensional ℚ_{p,η}-vector space with a canonical generator t^{−k}𝔤(η), where t ∈ B_{dR} is Fontaine’s p-adic analogue of 2πi on which G_{ℚ_{p}} acts as σt = χ_{cyc}(σ)t. The Bloch-Kato logarithm log_{k,η} with k = 1 is related to the map log_{η} of (13) by the formula

log_{η} = t log_{1,η},

and (14) admits the following extension for all k ≥ 1 (cf. [PR4, 3.2.3 ]):

(16)  L_{p}(k, χ) = \frac{(1 − χ(p)p^{−k})}{(1 − χ^{−1}(p)p^{k−1})} × \frac{(−t)^{k}}{(k − 1)!𝔤(χ)} × log_{k,χ}(κ_{k,χ}).

When k ≤ 0, the source and target of the logarithm map are both zero, and (16) does not extend to the negative integers. An interpretation of L_{p}(k, χ) can be given in terms of the dual exponential map of Bloch-Kato,

(17)  exp_{k,η}^{∗} : H^{1}(ℚ_{p}, ℚ_{p,η}(k)(η)) ⟶ 𝔻_{dR}(ℚ_{p,η}(k)(η)),

obtained by dualising the map exp_{1−k,η^{−1}} := log_{1−k,η^{−1}}^{−1} and combining local Tate duality with the natural duality between 𝔻_{dR}(ℚ_{p,η}(1−k)(η^{−1})) and 𝔻_{dR}(ℚ_{p,η}(k)(η)). The kernel of the map exp_{k,η}^{∗} consists precisely of the extensions of ℚ_{p} by ℚ_{p,η}(k)(η) which are cristalline, and exp_{k,η}^{∗} is an isomorphism of one-dimensional ℚ_{p,η}-vector spaces for all k ≤ 0. The following theorem for k ≤ 0 [PR4, 3.2.2] is one of the simplest instances of so-called reciprocity laws relating L-values to distinguished global cohomology classes with values in the associated p-adic representation:

(18)  L_{p}(k, χ) = \frac{(1 − χ(p)p^{−k})}{(1 − χ^{−1}(p)p^{k−1})} × \frac{(−k)!t^{k}}{𝔤(χ)} × exp_{k,χ}^{∗}(κ_{k,χ}).

Equations (16) and (18) give a satisfying intepretation of L_{p}(k, χ) at all integers k ∈ ℤ in terms of the global classes κ_{k,χ}. In particular the global classes κ_{k,χ} are non-trivial, and in fact non-cristalline, whenever L_{p}(k, χ) ≠ 0 and k < 0. Since k is then in the region of classical interpolation defining L_{p}(s, χ), the non-vanishing of L_{p}(k, χ) is directly related to the behaviour of the corresponding classical L-function. In this region of classical interpolation, the complex L-value attached to the p-adic representation ℚ_{p,χ}(k)(η) is thus given new meaning, as the obstruction to the global class κ_{k,χ} ∈ H^{1}(ℚ, ℚ_{p,χ}(k)(χ)) being cristalline at p.

1.2. Elliptic units. In addition to the cusps, modular curves are endowed with a second distinguished class of algebraic points: the CM points attached to the moduli of elliptic curves with complex multiplication by an order in a quadratic imaginary field K. The values of modular units at such points give rise to units in abelian extensions of K, the so-called elliptic units, which play the same role for abelian extensions of K as circular units in the study of cyclotomic fields. The resulting p-adic families of global cohomology classes are the main ingredient in the seminal work of Coates and Wiles on the arithmetic of elliptic curves with complex multiplication.

The Eisenstein series E_{k,χ} of (1), viewed as a function of a variable τ in the complex upper half-plane H by setting q = e^{2πiτ}, is given by the well-known formula

(19)  E_{k,χ}(τ) := N^{k}𝔤(χ̄)^{−1} \frac{(k − 1)!}{(2πi)^{k}}\begin{matrix}∑ & ′ \\ (m,n)∈Nℤ×ℤ \end{matrix} \frac{χ̄(n)}{(mτ + n)^{k}} ,

where 𝔤(χ̄) is the Gauss sum defined in (8), and the superscript ′ indicates that the sum is to be taken over the non-zero lattice vectors in Nℤ × ℤ.

Assume for simplicity that K has class number one, trivial unit group O_{K}^{×} = ±1, and odd discriminant D < 0. Assume also that there is an integral ideal–and hence, an element 𝔫 of O_{K}–satisfying

(20)  O_{K}/𝔫 = ℤ/Nℤ.

One then says that the Eisenstein series E_{k,χ} satisfies the Heegner hypothesis relative to K. Under this hypothesis, the even character χ gives rise to a finite order character χ_{𝔫} of conductor 𝔫 on the ideals of K by the rule

(21)  χ_{𝔫}((α)) := χ̄(α mod 𝔫).

After writing

(22)  τ_{𝔫} = \frac{b + \sqrt{D}}{2N},  where 𝔫 = ℤN + ℤ\frac{b + \sqrt{D}}{2},

a direct calculation using (19) shows that

(23)  E_{k,χ}(τ_{𝔫}) = N^{k}𝔤(χ̄)^{−1} \frac{(k − 1)!}{(2πi)^{k}}L(K, χ_{𝔫}, k, 0),

where for all k_{1}, k_{2} ∈ ℤ with k_{1} + k_{2} > 2,

L(K, χ_{𝔫}, k_{1}, k_{2}) := \begin{matrix}∑ & ′ \\ α∈O_{K} \end{matrix} χ_{𝔫}(α)α^{−k_{1}}ᾱ^{−k_{2}}.

Note that L(K, χ_{𝔫}, s) := \frac{1}{2}L(K, χ_{𝔫}, s, s), viewed as a function of a complex variable s, is the usual Hecke L-function attached to the finite order character χ_{𝔫}. The relation (23) between values of E_{k,χ} at CM points and L-series of quadratic imaginary fields is the direct counterpart of (1) relating the value at the cusps with Dirichlet L-series. For the purpose of p-adic interpolation it will be convenient, at least initially, to consider the values L(K, χ_{𝔫}, k, 0) rather than the values L(K, χ_{𝔫}, k, k).

Under the moduli interpretation of Γ_{1}(N)\H = Y_{1}(N)(ℂ), the point τ ∈ H corresponds to the pair (ℂ/ℤ ⊕ ℤτ, 1/N) consisting of an elliptic curve over ℂ with a marked point of order N. The point τ_{𝔫} corresponds in this way to the pair (ℂ/𝔫̄^{−1}, 1/N). Since 𝔫̄^{−1} is a fractional ideal of O_{K}, the elliptic curve A = ℂ/𝔫̄^{−1} has complex multiplication by O_{K} and hence has a model defined over K and even over O_{K}, while 1/N represents an 𝔫-torsion point of A, denoted t_{𝔫}, and hence is defined over the ray class field K_{𝔫} of K of conductor 𝔫. In particular the point P_{𝔫} of X_{1}(N) attached to the pair (A, t_{𝔫}) is rational over K_{𝔫}.

Choose a Néron differential ω_{A} ∈ Ω^{1}(A/O_{K}) and let Ω_{K} · O_{K} ⊂ ℂ be the associated period lattice. This determines the complex number Ω_{K} uniquely up to sign, once ω_{A} has been chosen. Following Katz, an algebraic modular form of weight k on Γ_{1}(N) can be viewed as a function f on the isomorphism classes of triples (E, t, ω_{E}) consisting of an elliptic curve (E, t) with Γ_{1}(N)-level structure and a choice of regular differential ω_{E} on E, satisfying a weight −k-homogeneity condition under scaling of ω_{E}. The convention relating both points of view is that f(ℂ/ℤ + ℤτ, 1/N, dz) := f(τ) for τ ∈ H, where dz is the standard differential on ℂ/Λ whose period lattice is equal to Λ. In particular,

(24)  E_{k,χ}(A, t_{𝔫}, ω_{A}) = E_{k,χ} (ℂ/ℤ + ℤτ_{𝔫}, 1/N, Ω_{K} 𝔫̄dz) = \frac{E_{k,χ}(τ_{𝔫})}{(Ω_{K} 𝔫̄)^{k}},

where by an abuse of notation 𝔫̄ is identified with one of its generators.

Because E_{k,χ} is defined over the field ℚ_{χ} generated by the values of χ and the triple (A, t_{𝔫}, ω_{A}) is defined over K_{𝔫}, the quantities in (24) are algebraic, and in fact belong to the compositum K_{𝔫,χ} of K_{𝔫} and ℚ_{χ}. Therefore by (23), the normalised L-value

(25)  \frac{E_{k,χ}(τ_{𝔫})}{(Ω_{K} 𝔫̄)^{k}} = 𝔫^{k}𝔤(χ̄)^{−1} \frac{(k − 1)!}{(2πi · Ω_{K})^{k}} L(K, χ_{𝔫}, k, 0)

also belongs to K_{𝔫,χ}.

Fix a prime p ∤ N, and let X_{01}(p, N) be the modular curve attached to Γ_{0}(p) ∩ Γ_{1}(N). Recall the affinoid region A of Section 1.1 obtained by deleting the supersingular residue discs from X_{1}(N). The natural algebraic projection X_{01}(p, N) ⟶ X_{1}(N) admits a rigid analytic section s : A ⟶ X_{01}(p, N)(ℂ_{p}) over this ordinary locus, which sends the point attached to an ordinary pair (E, t) with Γ_{1}(N)-level structure to the point attached to the triple (E, C, t) where C is the canonical subgroup of E of order p. The ordinary pstabilisation E_{k,χ}^{(p)} of E_{k,χ} defined in (2) can thus be viewed either as a classical modular form on X_{01}(p, N) or as a p-adic modular form on X_{1}(N) by pulling back to A via s.

Assume further that p = 𝔭𝔭̄ splits in K, let K ↪ ℚ_{p} be the embedding of K into its completion at 𝔭, and extend this to an embedding ı_{p} : ℚ̄ ↪ ℂ_{p}. The elliptic curve A/ℂ_{p} deduced from A via ı_{p} is ordinary at p, and its canonical subgroup of order p is equal to the group scheme of its 𝔭-division points. In particular, this group is defined over K. It follows that the image s(P_{𝔫}) of P_{𝔫} = (A, t_{𝔫}) belongs to X_{01}(p, N)(K_{𝔫}). A direct calculation shows that it is represented by the pair (ℂ/ℤ + ℤτ_{𝔭𝔫}, \frac{𝔭̄_{𝔫}^{−1}}{pN}), where τ_{𝔭𝔫} ∈ Γ_{01}(p, N)\H is defined by the same equation as (22) but with 𝔫 replaced by 𝔭𝔫:

(26)  τ_{𝔭𝔫} = \frac{b + \sqrt{D}}{2Np},  where 𝔭𝔫 = ℤpN + ℤ\frac{b + \sqrt{D}}{2},

and 𝔭̄_{𝔫}^{−1} is any element of O_{K}/𝔫O_{K} which is congruent to 𝔭̄^{−1}. The value of the p-adic modular form E_{k,χ}^{(p)} on the ordinary triple (A, t_{𝔫}, ω_{A}) can therefore be calculated as

E_{k,χ}^{(p)}(A, t_{𝔫}, ω_{A}) = E_{k,χ}^{(p)} (ℂ/ℤ + ℤτ_{𝔭𝔫}, \frac{𝔭̄_{𝔫}^{−1}}{pN} , 𝔭̄𝔫̄Ω_{K}dz) = \frac{χ_{𝔫}(𝔭̄)E_{k,χ}^{(p)}(τ_{𝔭𝔫})}{(𝔭̄𝔫̄Ω_{K})^{k}}
(27)  = (1 − χ_{𝔫}^{−1}(𝔭)𝔭^{k}/p) × 𝔫^{k}𝔤(χ̄)^{−1} \frac{(k − 1)!}{(2πi · Ω_{K})^{k}} L(K, χ_{𝔫}, k, 0),

where the second and third occurrence of E_{k,χ}^{(p)} are treated as classical modular forms of level pN, and (27) follows from a direct calculation based on (19) and (26). The fact that all the coefficients in the Fourier expansion (3) of E_{k,χ}^{(p)} extend to p-adic analytic functions of k on weight space W = ℤ/(p−1)ℤ×ℤ_{p} and that the pair (A, t_{𝔫}) belongs to A suggests that the right-hand side of (27), normalised by a suitable p-adic period, should admit a similar prolongation. More precisely, the formal completion Â of A_{O_{ℂp}} along its identity section is isomorphic to the formal multiplicative group Ĝ_{m}, and upon choosing an isomorphism ı : Â ⟶ Ĝ_{m} we may define a p-adic period Ω_{p} ∈ ℂ_{p}^{×} by the rule

ω_{A} = Ω_{p} · ω_{can},

where ω_{can} := ı^{∗} \frac{dt}{t}, with \frac{dt}{t} the canonical differential on Ĝ_{m}, plays the role of the complex differential dz in the p-adic setting. The function L_{p}(K, χ_{𝔫}, k) of k defined by

(28)  L_{p}(K, χ_{n}, k) = E_{k,χ}^{(p)}(A, t_{𝔫}, ω_{can}) = Ω_{p}^{k} · E_{k,χ}^{(p)}(A, t_{𝔫}, ω_{A}),

extends to a p-adic analytic function of k ∈ W and is equal (up to the p-adic period Ω_{p}^{k}) to the right-hand side of (27) for all integers k ≥ 2. It is called the Katz one-variable p-adic L-function attached to K and to the character χ_{𝔫}.

Recall the Siegel units g_{a} and g_{a}^{(p)} defined in equations (6) and (7) of Section 1.1. Evaluating these functions at the pair (A, t_{𝔫}) gives rise to the elliptic units

u_{a,𝔫} := g_{a}(A, t_{𝔫}) = g_{a}(τ_{𝔫}), u_{a,𝔫}^{(p)} := g_{a}^{(p)}(A, t_{𝔫}) = g_{a}^{(p)}(τ_{𝔭𝔫}) = u_{a,𝔫}^{σ_{𝔭}−p}

in O_{K_{𝔫}}^{×}, where σ_{𝔭} ∈ Gal(K_{𝔫}/K) denotes the Frobenius element at 𝔭.

The following result of Katz (as well as its proof) is the direct counterpart of Leopoldt’s formula (Theorem 1.1) in which cusps are replaced by CM points and circular units by elliptic units.

Theorem 1.2 (Katz). Let χ be a non-trivial even primitive Dirichlet character of conductor N and let K be a quadratic imaginary field equipped with an ideal 𝔫 satisfying O_{K}/𝔫 = ℤ/Nℤ. Let χ_{𝔫} be the ideal character of K associated to the pair (χ, 𝔫) as in (21).

Then

L_{p}(K, χ_{𝔫}, 0) = −\frac{(1 − χ_{𝔫}(𝔭)p^{−1})}{𝔤(χ̄)} × ∑_{a=1}^{N−1} χ^{−1}(a) log_{p} u_{a,𝔫}.

Proof. Setting k = 0 in (28) gives

L_{p}(K, χ_{𝔫}, 0) = E_{0,χ}^{(p)}(A, t_{𝔫}),

where E_{0,χ}^{(p)}(A, t_{𝔫}) refers to E_{0,χ}^{(p)}(A, t_{𝔫}, ω) for any choice of regular differential ω on A, since E_{0,χ} is of weight zero and this value is therefore independent of ω. But by equations (11) and (9) in the proof of Theorem 1.1,

E_{0,χ}^{(p)}(A, t_{𝔫}) = h_{χ}^{(p)}(A, t_{𝔫}) = \frac{1}{p𝔤(χ̄)} × ∑_{a=1}^{N−1} χ^{−1}(a) log_{p} g_{a}^{(p)}(A, t_{𝔫})

= \frac{1}{p𝔤(χ̄)} × ∑_{a=1}^{N−1} χ^{−1}(a) log_{p} u_{a,𝔫}^{σ_{𝔭}−p} = \frac{χ_{𝔫}(𝔭) − p}{p𝔤(χ̄)} × ∑_{a=1}^{N−1} χ^{−1}(a) log_{p} u_{a,𝔫}.

The theorem follows. □

The calculations above can be extended by introducing the Katz two-variable p-adic L-function L_{p}(K, χ_{𝔫}, k_{1}, k_{2}) which interpolates the values of L(K, χ_{𝔫}, k_{1}, k_{2}) as k_{1} and k_{2} both vary over weight space. These more general L-values are related to the values at CM points of the non-holomorphic Eisenstein series

E_{k_{1},k_{2},χ}(τ) = N^{k_{1}+k_{2}}𝔤(χ̄)^{−1} \frac{(k_{1} − 1)!}{(2πi)^{k_{1}}}(τ − τ̄)^{k_{2}}\begin{matrix}∑ & ′ \\ (m,n)∈Nℤ×ℤ \end{matrix} \frac{χ̄(n)}{(mτ + n)^{k_{1}}(mτ̄ + n)^{k_{2}}}

by the formula generalising (23)

(29)  E_{k_{1},k_{2},χ}(τ_{𝔫}) = N^{k_{1}}𝔤(χ̄)^{−1} \frac{(k_{1} − 1)!}{(2πi)^{k_{1}}} \sqrt{D}^{k_{2}}L(K, χ_{𝔫}, k_{1}, k_{2}).

The function E_{k_{1},k_{2},χ} is a real analytic function on H which transforms like a modular form of weight k_{1}−k_{2} and character χ under the action of Γ_{0}(N). Although it is non-holomorphic in general, it can sometimes be expressed as the image of holomorphic modular forms under iterates of the Shimura-Maass derivative operator

(30)  δ_{k} = \frac{1}{2πi} (\frac{d}{dτ} + \frac{k}{τ − τ̄})

sending real analytic modular forms of weight k to real analytic modular forms of weight k + 2. More precisely, after setting δ_{k}^{r} = δ_{k+2r−2} ∘ ⋯ ∘ δ_{k+2} ∘ δ_{k}, a direct calculation reveals that

δ_{k}^{r}E_{k,χ} = E_{k+r,−r,χ}.

A nearly holomorphic modular form of weight k on Γ_{1}(N) is a linear combination

f = ∑_{i=1}^{t} δ_{k−2j_{i}}^{j_{i}}f_{i}, f_{i} ∈ M_{k−2j_{i}}(Γ_{1}(N)),

where the f_{i} are classical modular forms of weight k − 2j_{i} on Γ_{1}(N). If the Fourier expansions of all the f_{i} at all the cusps have Fourier coefficients in a field L, then the nearly holomorphic modular form f is said to be defined over L.

Shimura proved that nearly holomorphic modular forms of weight k which are defined over ℚ̄ take algebraic values at CM triples like (A, t_{𝔫}, ω_{A}). More precisely, if f is defined over L ⊂ ℚ̄, then

(31)  f(A, t_{𝔫}, ω_{A}) := \frac{f(τ_{𝔫})}{(𝔫̄Ω_{K})^{k}} belongs to LK_{𝔫}.

A conceptual explanation for this striking algebraicity result rests on the relationship between nearly holomorphic modular forms of weight k and global sections of an algebraic vector bundle arising from the relative de Rham cohomology of the universal elliptic curve over Y_{1}(N), and on the resulting interpretation of the Shimura-Maass derivative in terms of the Gauss-Manin connection on this vector bundle. See for instance Section 1.5 of [BDP] or Section 2.4 of [DR1] for a brief account of this circle of ideas, and Section 10.1 of [Hi2] for a more elementary treatment.

Specialising (31) to the setting where f is the Eisenstein series E_{k+r,−r,χ} of weight k + 2r with k ≥ 2 and r ≥ 0, and invoking (29) leads to the conclusion that the special values

(32)  E_{k+r,−r,χ}(A, t_{𝔫}, ω_{A}) = 𝔫^{k+r}𝔫̄^{−r}𝔤(χ̄)^{−1} \frac{(k + r − 1)!}{(2πi)^{k+r}Ω_{K}^{k+2r}} \sqrt{D}^{−r}L(K, χ_{𝔫}, k + r, −r)

belong to the compositum K_{𝔫,χ} of K_{𝔫} and ℚ_{χ}, just as in (25), for all k ≥ 2 and r ≥ 0.

In light of this algebraicity result, it is natural to attempt to interpolate these values p-adically as a function of both k and r in weight space. This p-adic interpolation rests on the fact that the Shimura-Maass derivative admits a counterpart in the realm of p-adic modular forms: the Atkin-Serre operator d which raises the weight by two and acts as the differential operator d = q \frac{d}{dq} on q-expansions. If f is a classical modular form of weight k with rational Fourier coefficients, the nearly holomorphic modular form δ_{k}^{r}f and the p-adic modular form d^{r}f are objects of a very different nature, but nonetheless their values agree on ordinary CM triples (where it makes sense to compare them) so that in particular

(33)  δ_{k}^{r}f(A, t_{𝔫}, ω_{A}) = d^{r}f(A, t_{𝔫}, ω_{A}),  for all r ≥ 0.

The reason, which is explained for instance in Section 1.5 of [BDP], is that d admits the same algebraic description as δ_{k} in terms of the Gauss-Manin connection on a relative de Rham cohomology sheaf, with the sole difference that the (non-holomorphically varying) Hodge decomposition on the complex de Rham cohomology of the fibers is replaced in the p-adic setting by the Frobenius decomposition of the de Rham cohomology of the universal (ordinary) elliptic curve over A. But the functorial action of the endomorphism algebra on algebraic de Rham cohomology causes these two decompositions to agree for ordinary CM elliptic curves.

Equation (33) means that, as far as values at ordinary CM triples are concerned, the p-adic modular form d^{r}E_{k,χ} is a perfect substitute for E_{k+r,−r,χ} = δ_{k}^{r}E_{k,χ}. The Fourier expansion of this p-adic avatar is given, for r ≥ 1, by

(34)  d^{r}E_{k,χ} = ∑_{n=1}^{∞} n^{r}σ_{k−1,χ}(n)q^{n}.

The coefficients of q^{n} when p|n do not extend to a p-adic analytic function of (k, r) ∈ W^{2}, and this difficulty persists after replacing E_{k,χ} by its ordinary p-stabilisation E_{k,χ}^{(p)}. One is therefore led to consider instead the so-called p-depletion of E_{k,χ}, defined by

(35)  E_{k,χ}^{[p]}(τ) = E_{k,χ}(τ) − (1 + χ(p)p^{k−1})E_{k,χ}(pτ) + χ(p)p^{k−1}E_{k,χ}(p^{2}τ),

so that E_{k+r,−r,χ}^{[p]} := δ_{k}^{r}E_{k,χ}^{[p]}–a nearly holomorphic modular form of level Np^{2}–is given by

(36) E_{k+r,−r,χ}(τ) − (p^{r} + χ(p)p^{k+r−1})E_{k+r,−r,χ}(pτ) + χ(p)p^{k+2r−1}E_{k+r,−r,χ}(p^{2}τ),

while its p-adic avatar d^{r}E_{k,χ}^{[p]}–a p-adic modular form of level N–has Fourier expansion

d^{r}E_{k,χ}^{[p]} = ∑_{p∤n}^{∞}n^{r}σ_{k−1,χ}(n)q^{n}.

The coefficients in this expansion do extend to p-adic analytic functions of k and r on weight space W = ℤ/(p − 1)ℤ × ℤ_{p}. Just as in the construction of L_{p}(K, χ_{𝔫}, k), this suggests that the function L_{p}(K, χ_{𝔫}, k_{1}, k_{2}) defined (for k ≥ 2 and r ≥ 0) by the rule

\frac{L_{p}(K, χ_{𝔫}, k + r, −r)}{Ω_{p}^{k+2r}} := d^{r}E_{k,χ}^{[p]}(A, t_{𝔫}, ω_{A}) = E_{k+r,−r,χ}^{[p]} (ℂ/ℤ + ℤτ_{𝔭^{2}𝔫}, \frac{𝔭̄_{𝔫}^{−2}}{p^{2}N} , 𝔭̄^{2}𝔫̄Ω_{K}dz)

= (1 − χ_{𝔫}^{−1}(𝔭)𝔭^{k+r}𝔭̄^{−r}/p) × (1 − χ_{𝔫}(𝔭̄)𝔭^{r}𝔭̄^{−k−r}) × (37)

𝔫^{k+r}𝔫̄^{−r}𝔤(χ̄)^{−1} \frac{(k + r − 1)!}{(2πi)^{k+r}Ω_{K}^{k+2r}} \sqrt{D}^{−r}L(K, χ_{𝔫}, k + r, −r)

extends to an analytic function of (k, r) ∈ W^{2}. The function L_{p}(K, χ_{𝔫}, k_{1}, k_{2}) is called the Katz two-variable p-adic L-function attached to the character χ_{𝔫}. Note that the restriction of L_{p}(K, χ_{𝔫}, k_{1}, k_{2}) to the line k_{2} = 0 is related to the Katz one-variable p-adic L-function by the rule

L_{p}(K, χ_{𝔫}, k, 0) = (1 − χ_{𝔫}(𝔭̄)𝔭̄^{−k})L_{p}(K, χ_{𝔫}, k).

The ratio of the two sides (which can be seen to be a p-adic analytic function of k, since 𝔭̄ belongs to O_{K_{𝔭}}^{×}) reflects the difference between working with the ordinary p-stabilisation E_{k,χ}^{(p)} and the p-depletion E_{k,χ}^{[p]}.

The following variant of Katz’s Theorem 1.2 expresses the special value L_{p}(K, χ_{𝔫}^{−1}, 1, 1) in terms of the elliptic units u_{a,𝔫}^{∗} := g_{a}(w(A, t_{𝔫})), where w is the Atkin-Lehner involution such that g_{a}(w(A, t_{𝔫})) = g_{a}(−1/Nτ_{𝔫}).

Theorem 1.3 (Katz). With notations as in Theorem 1.2,

L_{p}(K, χ_{𝔫}^{−1}, 1, 1) = (1 − χ_{𝔫}^{−1}(𝔭̄))(1 − χ_{𝔫}(𝔭)/p) × ∑_{a=1}^{N−1} χ^{−1}(a) log_{p} u_{a,𝔫}^{∗}.

Proof. By setting k = 2 and r = −1 in equation (37) (and replacing χ_{𝔫} by χ_{𝔫}^{−1}), we obtain

L_{p}(K, χ_{𝔫}^{−1}, 1, 1) = d^{−1}E_{2,χ^{−1}}^{[p]}(A, t_{𝔫}, ω_{A}).

The p-adic modular form of weight 0 which appears on the right hand side of this identity has the same Fourier expansion as the p-depletion of the Eisenstein series E_{0}(χ^{−1}, 1) introduced in the remark following Theorem 1.1 of Section 1. Combined with [Hi1, Lemma 5.3], we thus see that

L_{p}(K, χ_{𝔫}^{−1}, 1, 1) = 𝔤(χ̄)wE_{0,χ}^{[p]}(A, t_{𝔫}, ω_{A}),

where w is the Atkin-Lehner involution at N introduced above. The result then follows from an almost identical calculation to the proof of Theorem 1.2. □

In stark analogy with the case of circular units, the expressions

u_{χ_{𝔫}} := ∏_{a=1}^{N−1} u_{a,𝔫}^{χ^{−1}(a)} = ∏_{σ∈Gal(K_{𝔫}/K)}(σu_{1,𝔫})^{χ_{𝔫}(σ)}

are (formal ℚ_{χ}-linear combination of) special units in K_{𝔫} lying in the χ_{𝔫}^{−1}-eigenspace for the natural action of G_{K}.

These units arise as the “bottom layer” of a norm-coherent family of elliptic units over the two-variable ℤ_{p}-extension K_{∞} of K. The same construction as in equation (12) of Section 1.1 leads to a global cohomology class

κ_{χ_{𝔫},∞} ∈ H^{1}(K, Λ_{K}(χ_{𝔫})),

where Λ_{K} = ℤ_{p}[[Gal(K_{∞}/K)]] is the two-variable Iwasawa algebra attached to K, equipped with its tautological G_{K}-action. The Galois module Λ_{K}(χ_{𝔫}) gives a p-adic interpolation of the Hecke characters of the form χ_{𝔫}ϕ where ϕ is a Hecke character of p-power conductor.

In particular, if ψ is a Hecke character of infinity type (k_{1}, k_{2}) arising as a specialisation of Λ_{K}(χ_{𝔫}), the global class

κ_{ψ} ∈ H^{1}(K, ℚ_{p}(ψ))

obtained by specialising κ_{χ_{𝔫},∞} at ψ, although it arises from elliptic units, encodes arithmetic information about a Galois representation V_{ψ} of K attached to a Hecke character of possible infinite order.

The reciprocity law of Coates and Wiles expresses the Katz two-variable p-adic Lfunction L_{p}(K, χ_{𝔫}, k_{1}, k_{2}) as the image under a “big exponential map” of the global class κ_{𝔫,∞}. By an analogue of (18) of Section 1.1, the special value L(ψ^{−1}, 0) can then be interpreted as the obstruction to the class κ_{ψ} being cristalline at 𝔭, whenever ψ lies in the range of classical interpolation for the Katz p-adic L-function.

A classical result of Deuring asserts that the p-adic Tate module of a CM elliptic curve, viewed as a representation of G_{K}, is always of the form V_{ψ} for a suitable Hecke character ψ of K of infinity type (1, 0). The global class κ_{ψ} for such a ψ acquires a special interest in relation to the Birch and Swinnerton-Dyer conjecture for the elliptic curve A_{ψ} attached to it. This connection between L(A_{ψ}, 1) and the singular parts of global classes κ_{ψ} was used by Coates-Wiles to give what historically was the first broad piece of convincing supporting evidence for the conjecture of Birch and Swinnerton-Dyer, notably the implication

L(A, 1) ≠ 0 ⟹ A(K) ⊗ ℚ = {0},

for any elliptic curve A/ℚ with complex multiplication by K.

1.3. Heegner points. This section replaces the Eisenstein series E_{k,χ} of Section 1.2 by a cusp form of weight k. The argument used in the proof of Theorem 1.3 then leads naturally to the p-adic Gross-Zagier formula of [BDP] expressing the special values of certain p-adic Rankin L-series attached to f and K in terms of the p-adic Abel-Jacobi images of so-called generalised Heegner cycles.

For simplicity, let f ∈ S_{k}(N) be a normalized cuspidal eigenform of even weight k on Γ_{1}(N) with rational Fourier coefficients and trivial nebentypus character. (When k = 2, the form f is therefore associated to an elliptic curve E/ℚ.) Let K be a quadratic imaginary field satisfying all the hypotheses of Section 1.2, including the Heegner assumption (20). We will also assume that p = 𝔭𝔭̄ is, as before, a rational prime which splits in K and does not divide N.

By analogy with the construction of the Katz two-variable p-adic L-function, it is natural to consider the quantities

(38)  δ_{k}^{r}f(A, t_{𝔫}, ω_{A}) = d^{r}f(A, t_{𝔫}, ω_{A}),

which belong to K_{𝔫} for all r ≥ 0. The role of the relatively elementary formula (32) of Section 1.2 relating such quantities to L-values when f is an Eisenstein series is played in this context by a seminal formula of Waldspurger, whose importance for the arithmetic study of generalised Heegner cycles and points would be hard to overstate, even though its proof lies beyond the scope of this survey.

Waldspurger’s formula relates (38) to the L-function of f twisted (over K) by certain unramified Hecke characters of K. If ϕ is such a character (viewed as a mutiplicative function on fractional ideals of K in the usual way) then the L-series L(f, K, ϕ, s) of f/K twisted by ϕ is defined (for s ∈ ℂ in some right-half plane) by the Euler product

L(f, K, ϕ, s) = ∏_{𝔩}[(1 − α_{N𝔩}(f) · ϕ(𝔩)N𝔩^{−s})(1 − β_{N𝔩}(f) · ϕ(𝔩)N𝔩^{−s})]^{−1}

taken over the prime ideals 𝔩 in O_{K}, where α_{ℓ}(f) and β_{ℓ}(f) are the roots of the Hecke polynomial x^{2} − a_{ℓ}(f)x + ℓ^{k−1} for f at ℓ, and we set α_{N𝔩} := α_{ℓ}(f)^{t} and β_{N𝔩} := β_{ℓ}(f)^{t} if N𝔩 = ℓ^{t}. Rankin’s method can be used to show that L(f, K, ϕ, s) admits an analytic continuation to the entire complex plane.

If k_{1} and k_{2} are integers with the same parity, let ϕ_{k_{1},k_{2}} be the unramified Hecke character of K of infinity type (k_{1}, k_{2}) defined on fractional ideals by the rule

(39)  ϕ_{k_{1},k_{2}}((α)) = α^{k_{1}}ᾱ^{k_{2}},

and set

L(f, K, k_{1}, k_{2}) := L(f, K, ϕ_{k_{1},k_{2}}^{−1}, 0).

Because

L(f, K, k_{1} + s, k_{2} + s) = L(f, K, ϕ_{k_{1},k_{2}}^{−1}, s),

we may view L(f, K, k_{1}, k_{2}) as a function on pairs (k_{1}, k_{2}) ∈ ℂ^{2} with k_{1} − k_{2} ∈ 2ℤ. The functional equation relates the values L(f, K, k_{1}, k_{2}) and L(f, K, k − k_{2}, k − k_{1}) and hence corresponds to the reflection about the line k_{1} +k_{2} = k in the (k_{1}, k_{2})-plane, preserving the perpendicular lines k_{1} −k_{2} = k +2r for any r ∈ ℤ. The restriction of L(f, K, k_{1}, k_{2}) to such a line therefore admits a functional equation whose sign depends on r in an interesting way. More precisely, it turns out that assumption (20) forces this sign to be −1 when 1 − k ≤ r ≤ −1, and to be +1 for other integer values of r. In particular, the central critical value L(f, K, k + r, −r) vanishes for reasons of sign when 1 − k ≤ r ≤ −1, but is expected to be non-zero for infinitely many r ≥ 0. The quantities L(f, K, k + r, −r) are precisely the special values that arise in the formula of Waldspurger, which asserts (cf. [BDP, Thm. 5.4]) that

(40)  (δ_{k}^{r}f(A, t_{𝔫}, ω_{A}))^{2} = 1/2 · (2π/\sqrt{D})^{k+2r−1}r!(k+r−1)! · \frac{L(f, K, k + r, −r)}{(2πi · 𝔫̄Ω_{K})^{2(k+2r)}}.

Note the square that appears on the left-hand side. It is an enlightening exercise to recover equation (32) of Section 1.2 (up to an elementary constant) by setting f = E_{k,χ} and replacing ϕ_{k+r,−r} by χ_{𝔫}ϕ_{k+r,−r} in equation (40). This suggests that Waldspurger’s formula should be viewed as the natural extension of (32) to the setting where Eisenstein series are replaced by cusp forms. An important difference with the setting of Section 1.2 is that the quantities δ_{k}^{r}f(A, t_{𝔫}, ω_{A}) only encode the central critical values L(f, K, k + r, −r). In the setting of Eisenstein series the analogous value L(E_{k,χ}, K, (χ_{𝔫}ϕ_{k+r,−r})^{−1}, 0) is also central critical, but breaks up as a product

L(E_{k,χ}, K, (χ_{𝔫}ϕ_{k+r,−r})^{−1}, 0) = L(K, (χ_{𝔫}ϕ_{k+r,−r})^{−1}, 0) × L(K, (χ_{𝔫}^{−1}ϕ_{1−k−r,1+r})^{−1}, 0)

whose two factors, which are values of L-functions of Hecke characters, are interchanged by the functional equation and are non-self dual in general.

In order to interpolate the values d^{r}f(A, t_{𝔫}, ω_{A}) p-adically, we replace, exactly as in (35), the modular form f by its p-depletion

(41)  f^{[p]}(τ) := f(τ) − a_{p}(f)f(pτ) + p^{k−1}f(p^{2}τ).

A direct calculation using (41) shows that for all r ≥ 0,

(42)  d^{r}f^{[p]}(A, t_{𝔫}, ω_{A}) = (1 − a_{p}𝔭^{r}𝔭̄^{−k−r} + 𝔭^{k+2r−1}𝔭̄^{−k−2r−1})d^{r}f(A, t_{𝔫}, ω_{A}).

Since the collection of modular forms indexed by the parameter r

(43)  d^{r}f^{[p]}(q) = ∑_{(n,p)=1} n^{r}a_{n}(f)q^{n}

is a p-adic family of modular forms in the sense defined in Section 1.1, it follows, just as in Section 1.2, that the product of (42) by the p-adic period Ω_{p}^{k+2r} extends to a p-adic analytic function of r ∈ W = (ℤ/(p − 1)ℤ) × ℤ_{p}. In light of (38) and (40), the quantity defined by

(44)  L_{p}(f, K, k + r, −r) := Ω_{p}^{2(k+2r)} × d^{r}f^{[p]}(A, t_{𝔫}, ω_{A})^{2}

is referred to as the anticyclotomic p-adic L-function attached to f and K.

The value L_{p}(f, K, k + r, −r) of this p-adic L-function for r < 0 is defined by p-adic continuity rather than by the direct interpolation of classical L-values. For r ∈ [1 − k, −1], the quantity L_{p}(f, K, k + r, −r) should be regarded as a genuinely p-adic avatar of the special value L(f/K, ϕ_{k+r,−r}^{−1}, 0)—or rather, of the first derivative L′(f/K, ϕ_{k+r,−r}^{−1}, 0) since the value vanishes for parity reasons. The main result of [BDP] relates it to the image under the p-adic Abel-Jacobi map of the generalised Heegner cycles introduced in loc.cit.

We will now state the main result of [BDP] in the illustrative special case where k = 2 and f is attached to an elliptic curve E of conductor N. Let P_{K} ∈ J_{0}(N)(K) be the class of the degree 0 divisor (A, t_{𝔫}) − (∞) in the Jacobian variety J_{0}(N) of X_{0}(N), let P_{f,K} denote its image in E(K) under the modular parametrisation

φ_{E} : J_{0}(N) ⟶ E

arising from the modular form f, and let ω_{E} be the regular differential on E satisfying φ_{E}^{∗}(ω_{E}) = ω_{f} := (2πi)f(τ)dτ.

Theorem 1.4. Let f ∈ S_{2}(N) be a normalized cuspidal eigenform of level Γ_{0}(N) with N prime to p and let K be a quadratic imaginary field equipped with an integral ideal 𝔫 satisfying O_{K}/𝔫 = ℤ/Nℤ. Then

L_{p}(f, K, 1, 1) = (\frac{1 − a_{p}(f) + p}{p})^{2} log_{p}(P_{K,f})^{2},

where log_{p} is the formal group logarithm on E associated with the regular differential ω_{E}.

Sketch of proof. By (44) with k = 2 and r = −1,

(45)  L_{p}(f, K, 1, 1) = d^{−1}f^{[p]}(A, t_{𝔫}, ω_{A})^{2}.

As in the proof of Theorems 1.1, 1.2 and 1.3, the result will be obtained by interpreting the p-adic modular function F^{[p]} := d^{−1}f^{[p]} as the rigid analytic primitive of the differential ω_{f^{[p]}} on A which vanishes at the cusp ∞, and relating this rigid analytic function to the Coleman primitive of ω_{f}.

Recall the lift of Frobenius Φ on the ordinary locus A ⊂ X_{0}(N)(ℂ_{p}). A direct computation shows that

(46)  Φω_{f} = pω_{V f}

as sections of Ω_{X_{0}(N)}^{1} over A, where V is the operator on p-adic modular forms acting as f(q) ↦ f(q^{p}) on q-expansions.

Let F denote the Coleman primitive of ω_{f} on A. It is a locally analytic function on A, which is well-defined up to a constant and satisfies dF = ω_{f} on W_{ϵ}. By definition of the Coleman primitive,

F (A, t_{𝔫}) = log_{ω_{f}} (P_{K}) = log_{p}(P_{K,f}),

where log_{ω_{f}} is the p-adic logarithm on J_{0}(N) attached to the differential ω_{f}, and we have used the fact that φ_{E}(P_{K}) = P_{K,f} and φ_{E}^{∗}(ω_{E}) = ω_{f} to derive the second equality. The result now follows from noting that

F^{[p]}(A, t_{𝔫}) = (1 − a_{p}(f)/p + 1/p)F (A, t_{𝔫}).

□

Like circular units and elliptic units, the Heegner point P_{K,f} ∈ E(K) arises naturally as a universal norm of a compatible system of points defined over the so-called anticyclotomic ℤ_{p}-extension of K. This extension, denoted K_{∞}^{−}, is contained in the twovariable ℤ_{p}-extension K_{∞} of Section 1.2 and is the largest subextension which is Galois over ℚ and for which Gal(K/ℚ) acts as −1 on Gal(K_{∞}^{−}/K) via conjugation. After letting Λ_{K}^{−} = ℤ_{p}[[Gal(K_{∞}^{−}/K)]] be the Iwasawa algebra attached to this extension, equipped with its tautological action of G_{K}, this norm-compatible collection of Heegner points can be parlayed into the construction of a global cohomology class

κ_{f,K,∞} ∈ H^{1}(K, V_{p}(E) ⊗ Λ_{K}^{−}),

where V_{p}(E) is the Galois representation attached to the p-adic Tate module of E.

The module Λ_{K}^{−}⊗_{ℤ_{p}}V_{p}(E) is a deformation of V_{p}(E) which p-adically interpolates the twists of V_{p}(E) by the anticyclotomic Hecke characters ϕ_{r,−r}, and hence the class κ_{f,K,∞} admits specialisations

κ_{f,K,r} ∈ H^{1}(K, V_{p}(E) ⊗ ℚ_{p}(ϕ_{r,−r})).

When r = 0 this class arises from the image of Heegner points under the Kummer map, and hence is cristalline at p. Theorem 1.4 asserts that its p-adic logarithm is related to the values of the anti-cyclotomic p-adic L-function L_{p}(f, K, 1 + r, 1 − r) at the point (1, 1) that lies outside the range of classical interpolation.

In contrast, the classes κ_{f,K,r} need not be cristalline at p when r > 0, and the formalism described in the previous sections suggests that the image of κ_{f,K,r} under the dual exponential map at p should be related to the value L_{p}(f, K, 1 + r, 1 − r), a simple non-zero multiple of L(f, K, ϕ_{r,−r}^{−1}, 1) since (1 + r, 1 − r) lies in the range of classical interpolation defining the anticyclotomic p-adic L-function. One thus expects, just as in Sections 1.1 and 1.2, a direct relation between the images of the “higher weight” specializations κ_{f,K,r} under the Bloch-Kato dual exponential map and the central critical values L(f, ϕ_{−r,r}, 1). Current ongoing work of the second author aims to exploit the classes κ_{f,K,r} and Kolyvagin’s “method of Euler systems”, as summarised in [Kato, §13], for example, to derive new cases of the Bloch-Kato conjecture of the form

(47)  L(f, ϕ_{r,−r}, 1) ≠ 0 ⟹ Sel_{K}(V_{p}(E)(ϕ_{r,−r})) = {0},  (r > 0).

Using Hida families and a p-adic deformation along them of the Euler system of Heegner points due to Howard [How], one can more generally hope to establish the analogue of (47) for p-ordinary cuspidal eigenforms of even weight k ≥ 2. This program is being carried out in [Cas] based in part on a suitable extension of the circle of ideas described in this section.

2. Euler systems of Garrett-Rankin-Selberg type

The three formulae described in Chapter 1 relate the p-adic logarithms of circular units, elliptic units and Heegner points to values of associated p-adic L-functions at points that lie outside their range of classical interpolation. The construction of the p-adic L-function in all three cases rests on formulae for critical L-values in terms of the values of modular forms at distinguished points of modular curves, namely cusps or CM points.

There is a different but equally useful class of special value formulae arising from the Rankin-Selberg method. The prototypical such formula is concerned with a triple (f, g, h) of eigenforms of weights k, ℓ, and m respectively with k = ℓ+m+2r and r ≥ 0. It involves the Petersson scalar product

I(f, g, h) := ⟨f, g × δ_{m}^{r}h⟩,

and relates the square of this quantity to the central critical value L(f ⊗ g ⊗ h, \frac{k+ℓ+m−2}{2}) of the convolution L-function attached to f, g and h. An overarching theme of Chapter 2 is that the quantity I(f, g, h) can be p-adically interpolated as f, g and h are made to vary over a suitable set of classical specialisations of Hida families. In particular, when f, g and h are of weight two, forcing r to tend p-adically to −1 in weight space, the resulting p-adic limit of I(f, g, h), denoted I_{p}(f, g, h)—a p-adic L-value—acquires an interpretation as the Bloch-Kato p-adic logarithm of a global cohomology class arising from a suitable geometric construction, thereby motivating the study of the following Euler systems:

(1) When g and h are Eisenstein series, the invariant I_{p}(f, g, h) is related in Section 2.1 to the p-adic regulator

reg_{p}{u_{g}, u_{h}}(η_{f}),

were u_{g} and u_{h} are the modular units whose logarithmic derivatives are equal to g and h respectively, {u_{g}, u_{h}} ∈ K_{2}(X_{1}(N)) is the Beilinson element in the second K-group of X_{1}(N) formed essentially by taking the cup-product of these two units, and η_{f} is a suitable class in H_{dR}^{1}(X_{1}(N)) attached to f. The p-adic regulator has a counterpart in p-adic étale cohomology and the images of the Beilinson elements under this map lead to a system of global cohomology classes which underlie Kato’s study of the Mazur-Swinnerton-Dyer p-adic L-function attached to classical modular forms via the theory of Euler systems.

(2) When only h is an Eisenstein series and f and g are cuspidal, the invariant I_{p}(f, g, h) is again related, in Section 2.2, to a p-adic regulator of the form

reg_{p}(Δ_{u_{h}})(η_{f} ∧ ω_{g}),

where Δ_{u_{h}} is a Beilinson-Flach element in K_{1}(X_{1}(N) × X_{1}(N)) attached to the modular unit u_{h} viewed as a function on a diagonally embedded copy of X_{1}(N) ⊂ X_{1}(N)^{2}. The Euler system of Beilinson-Flach elements is obtained by replacing the p-adic regulator by its p-adic étale counterpart; some of its possible arithmetic applications are discussed in Section 2.2.

(3) When f, g and h are all cusp forms, the invariant I_{p}(f, g, h) is related in Section

2.3 to

AJ_{p}(Δ)(η_{f} ∧ ω_{g} ∧ ω_{h}),

where

AJ_{p} : CH^{2}(X_{1}(N)^{3})_{0} ⟶ Fil^{2}(H_{dR}^{3}(X_{1}(N)^{3}))^{∨}

is the p-adic Abel-Jacobi map, and Δ is the Gross-Kudla-Schoen cycle obtained by a simple modification of the diagonal cycle in X_{1}(N)^{3}. The resulting Euler system of Gross-Kudla-Schoen diagonal cycles and some of its eventual arithmetic applications are described in Section 2.3.

2.1. Beilinson-Kato elements. Let f ∈ S_{2}(N) be a cuspidal eigenform on Γ_{0}(N) (not necessarily new of level N), and let p be an odd prime not dividing N. Assume that p is ordinary for f, relative to a fixed embedding of ℚ̄ into ℚ̄_{p}. Denote by L_{p}(f, s) the Mazur-Swinnerton-Dyer p-adic L-function attached to f. This p-adic L-function is defined as in [MTT] via the p-adic interpolation of the complex central critical values L(f, ξ, 1), with ξ varying among the Dirichlet characters of p-power conductor.

One goal of this section is to explain the connection between the value of L_{p}(f, s) at the point s = 2, lying outside the range of classical interpolation for L_{p}(f, s), and the image of so-called Beilinson-Kato elements by the p-adic syntomic regulator on K_{2} of the modular curve of level N. The resulting formula is a p-adic analogue of Beilinson’s theorem [Bei2] relating L(f, 2) to the complex regulator of the Beilinson-Kato elements considered above.

Before stating the main theorem of this section, we recall the definition of the p-adic regulator, following [Bes3] and [BD1] (to which we refer for more detailed explanations). The reader should notice that this definition builds upon the techniques of p-adic integration that played a crucial role in the preceding sections.

Write Y for the open modular curve Y_{1}(N) over ℚ, and X for its canonical compactification X_{1}(N); furthermore, denote by Ȳ and X̄ the extension to ℚ̄ of Y and X, respectively. Let F be a field of characteristic 0, and let Eis_{2}(Γ_{1}(N), F) denote the F -vector space of weight 2 Eisenstein series on Γ_{1}(N) with Fourier coefficients in F. There is a surjective homomorphism

(48)  \xymatrix{O× ; Ȳ ⊗ F \ar[r]^{dlog} & Eis_{2}(Γ_{1}(N), F),}

sending a modular unit u (or rather, an F -linear combination of such) to the Eisenstein

series \frac{1}{2πi} \frac{u′(z)}{u(z)}. Given u_{1}, u_{2} ∈ O× ; Ȳ, write

{u_{1}, u_{2}} ∈ K_{2}(Ȳ) ⊗ ℚ

for their Steinberg symbol in the second K-group of Ȳ. Similarly to Section 1.1, let A be the ordinary locus of Y (viewed here as a rigid analytic curve over ℂ_{p}), obtained by removing both the supersingular and the cuspidal residue discs. It is equipped with a system of wide open neighborhoods W_{ϵ}. Let Φ be the canonical lift of Frobenius on A, and let Φ × Φ be the corresponding lift of Frobenius on A × A. Choose a polynomial P (t) ∈ ℚ[t] satisfying

• P (Φ × Φ) annihilates the class of \frac{du_{1}}{u_{1}} ⊗ \frac{du_{2}}{u_{2}} in H_{rig}^{2}(W_{ϵ} × W_{ϵ}),

• P (Φ) annihilates the target of the residue map on H_{rig}^{1}(W_{ϵ}) at the supersingular annuli,

• P (Φ) is invertible on H_{dR}^{1}(X).

Such a P exists because the eigenvalues of Φ on H_{dR}^{1}(X) have complex absolute value \sqrt{p}, while Φ acts on the target of the residue map, and on H_{rig}^{2}(W_{ϵ} × W_{ϵ}), with eigenvalues of strictly larger modulus. The first condition on P implies the existence of a rigid analytic one-form ρ_{P} in Ω^{1}(W_{ϵ}^{2}) such that

(49)  dρ_{P} = P (Φ × Φ) (\frac{du_{1}}{u_{1}} ⊗ \frac{du_{2}}{u_{2}} ) .

The form ρ_{P} is well-defined up to closed rigid one-forms on W_{ϵ} × W_{ϵ}. Fix a base point x ∈ W_{ϵ}, and set

ξ̃_{P,x} := (δ^{∗} − h_{x}^{∗} − v_{x}^{∗})ρ_{P} ∈ Ω^{1}(W_{ϵ}),

where δ(w) := (w, w), h_{x}(w) := (w, x) and v_{x}(w) := (x, w) are the diagonal, horizontal and vertical inclusions of W_{ϵ} in W_{ϵ}^{2}, respectively. One checks that the image ξ_{P,x} of ξ̃_{P,x} in H_{rig}^{1}(W_{ϵ}) does not depend on the choice of a form ρ_{P} satisfying equation (49), and vanishes at the supersingular annuli by the second condition imposed on P. It follows that ξ_{P,x} can be identified via restriction with an element of the algebraic de Rham cohomology group H_{dR}^{1}(X). Moreover, setting ξ_{x} := P (Φ)^{−1}ξ_{P,x} in view of the third condition on P, we find that the class ξ_{x} is independent of the choice of P. The p-adic regulator of {u_{1}, u_{2}} is defined as

(50)  reg_{p}{u_{1}, u_{2}} := ξ_{x} ∈ H_{dR}^{1}(X).

By Poincaré duality, the p-adic regulator reg_{p}{u_{1}, u_{2}} can and will be identified with a linear functional on H_{dR}^{1}(X).

We are now ready to state the main theorem of this section. Let χ be a primitive, even Dirichlet character of conductor N. Recall the Eisenstein series E_{2,χ} appearing in equation (1) of Section 1.1, and let u_{χ} be a modular unit satisfying

(51)  dlog(u_{χ}) = E_{2,χ}.

Normalize the Petersson scalar product on real analytic modular forms of weight k and character ψ on Γ_{0}(N) by setting

(52)  ⟨f_{1}, f_{2}⟩_{k,N} := ∫_{Γ_{0}(N)\H} y^{k}\overline{f_{1}(z)}f_{2}(z)\frac{dxdy}{y^{2}} .

Write α_{p}(f), resp. β_{p}(f) for the unit, resp. non-unit root of the Frobenius polynomial x^{2} − a_{p}(f)x + p associated to f. Consider the unit root subspace

H_{dR}^{1}(X)^{f,ur} ⊂ H_{dR}^{1}(X)^{f}

of the f-isotypic part of H_{dR}^{1}(X), on which Frobenius acts as multiplication by α_{p}(f). We attach to f a canonical element η_{f}^{ur} of H_{dR}^{1}(X)^{f,ur} in the following way (cf. [BD1], Sections 2.5 and 3.1 for more details). First, we define an anti-holomorphic differential

(53)  η_{f}^{ah} := ⟨f, f⟩_{2,N}^{−1} · f̄(z)dz̄.

The differential η_{f}^{ah} gives rise to a class in H_{dR}^{1}(X_{ℂ}), whose natural image in H^{1}(X_{ℂ}, O_{X}) is in fact defined over ℚ̄ via our fixed embedding of ℚ̄ into ℂ. Using now the embedding of ℚ̄ into ℂ_{p} we obtain a class η_{f} in H^{1}(X, O_{X}), and a lift η_{f}^{ur} of η_{f} to H_{dR}^{1}(X)^{f,ur}.

Theorem 2.1. The equality

L_{p}(f, 2) · \frac{L(f, χ, 1)}{Ω_{f}^{+}} = (2iN^{−2}𝔤(χ))(1 − β_{p}(f)p^{−2})(1 − β_{p}(f)) · reg_{p}\{u_{χ^{−1}}, u_{χ}\}(η_{f}^{ur})

holds, where Ω_{f}^{+} is a real period attached to f as in Section 2.3 of [BD1], and 𝔤(χ) is the Gauss sum defined in equation (8). Remark 2.2. A version of Theorem 2.1 has been obtained by Brunault [Br], as a consequence of Kato’s reciprocity law [Kato]. A different proof, proposed by Bannai-Kings [BK] and Niklas [Nik], relies on the Eisenstein measures introduced by Panchiskine. The approach sketched here is based on the methods of [BD1], depending crucially on Hida’s p-adic deformation of f.

The key steps in this approach consist in:

• The p-adic approximation of the value L_{p}(f, 2) (lying outside the range of classical interpolation of L_{p}(f, s)) by means of values in the range of classical interpolation of the so-called Mazur-Kitagawa p-adic L-function.

• The description of the Mazur-Kitagawa p-adic L-function as a factor of a p-adic Rankin L-series L_{p}(f, E_{χ}) associated to the convolution of the Hida families f and E_{χ} interpolating in weight 2 the ordinary forms f and E_{2,χ}, respectively. (This factorisation follows from a corresponding factorisation of complex special values.)

• The explicit evaluation of L_{p}(f, E_{χ}) at the weights (2, 2) (lying outside the range of classical interpolation for this p-adic L-function), which yields an expression directly related to the p-adic regulator reg_{p}{u_{χ^{−1}}, u_{χ}}(η_{f}^{ur}) described above.

Write U_{f} ⊂ (ℤ/(p − 1)ℤ) × ℤ_{p} for the weight space attached to f, and denote by f_{k} ∈ S_{k}(N) the classical eigenform whose ordinary p-stabilisation is equal to the weight k specialisation of f, for all k in the space of classical weights U_{f,cl} := U_{f} ∩ℤ^{≥2}. (In particular, f_{2} = f.) Let L_{p}(f_{k}, ρ, s) be the Mazur-Swinnerton-Dyer p-adic L-function [MTT] associated to f_{k} and to a Dirichlet character ρ (equal to 1 or to χ in our study). Thus L_{p}(f_{k}, ρ, s) interpolates the special values L_{p}(f_{k} ⊗ ξρ, j), for 1 ≤ j ≤ k − 1 and ξ in the set of Dirichlet characters of p-power conductor. As k varies, the p-adic L-functions L_{p}(f_{k}, ρ, s) can be patched together to yield the Mazur-Kitagawa two-variable p-adic L-function L_{p}(f, ρ)(k, s), defined on the domain U_{f} × ℤ_{p}. For k ∈ U_{f,cl}, one has the identity

L_{p}(f, ρ)(k, s) = λ(k) · L_{p}(f_{k}, ρ, s),

where λ(k) is a p-adic period equal to 1 at k = 2 and non-vanishing in a neighborhood of k = 2. Note that L_{p}(f, 2) can be described as the p-adic limit, as (k, ℓ) ∈ U_{f,cl} × [1, k/2] tends to (2, 2), of the values L_{p}(f, 1)(k, k/2 + ℓ − 1) occurring in the range of classical interpolation for L_{p}(f, 1).

Let E_{χ} be the Hida family of Eisenstein series whose weight ℓ ∈ ℤ^{≥2} specialisation is equal to the ordinary p-stabilisation of the Eisenstein series E_{ℓ,χ}. We recall the definition [BD1] of the two-variable Rankin p-adic L-function L_{p}(f, E_{χ})(k, ℓ), where (k, ℓ) belongs to U_{f} × ℤ_{p}. It is defined via the p-adic interpolation of the critical values

L(f_{k} ⊗ E_{ℓ,χ}, k/2 + ℓ − 1), ℓ ∈ [1, k/2].

Set t := k/2 − ℓ (so that t ≥ 0). Let δ_{ℓ} be the Shimura-Maass differential operator of equation (30), mapping the space M_{ℓ}^{nh}(N, ϕ) of weight ℓ nearly holomorphic modular forms on Γ_{0}(N) and character ϕ to M_{ℓ+2}^{nh}(N, ϕ), and let δ_{ℓ}^{t} denote its t-fold iterate δ_{ℓ+2t−2} ∘…∘δ_{ℓ}. Theorem 2 of [Sh1] yields the special value formula

(54)  L(f_{k} ⊗ E_{ℓ,χ}, k/2 + ℓ − 1) = B_{N,k,ℓ,χ} · ⟨f_{k}(z), (δ_{ℓ}^{t}E_{ℓ,χ^{−1}}) · E_{ℓ,χ}⟩_{k,N},

where B_{k,ℓ,χ} is an explicit non-zero algebraic constant depending on N, k, ℓ and χ. Set Ξ_{k,ℓ} := (δ_{ℓ}^{t}E_{ℓ,χ^{−1}}) · E_{ℓ,χ}, and denote by Ξ_{k,ℓ}^{hol} its holomorphic projection. Hence

⟨f_{k}(z), Ξ_{k,ℓ}⟩_{k,N} = ⟨f_{k}(z), Ξ_{k,ℓ}^{hol}⟩_{k,N},

and (54) implies that the ratio

\frac{L(f_{k} ⊗ E_{ℓ,χ}, k/2 + ℓ − 1)}{⟨f_{k}, f_{k}⟩_{k,N}}

is algebraic. (More precisely, it belongs to the extension of ℚ generated by the Fourier coefficients of f_{k} and the values of χ.) Viewing Ξ_{k,ℓ}^{hol} as a p-adic modular form, the calculations of Section 3.1 of [BD1] – see in particular equation (46)– show that the ratios

\frac{⟨f_{k}(z), Ξ_{k,ℓ}^{hol}⟩_{k,N}}{⟨f_{k}, f_{k}⟩_{k,N}} ,

normalised by multiplying them by suitable Euler factors, are interpolated p-adically by a p-adic L-function L_{p}(f, E_{χ})(k, ℓ). As a by-product of this construction, one obtains the formula (cf. equation (50) of loc. cit.)

(55)  L_{p}(f, E_{χ})(2, 2) = \frac{1}{1 − β_{p}(f)^{2}p^{−1}} ⟨η_{f}^{ur}, e_{ord}(d^{−1}E_{2,χ^{−1}}^{[p]} · E_{2,χ})⟩_{Y} ,

where e_{ord} denotes Hida’s ordinary projector, E_{2,χ^{−1}}^{[p]} is the p-depletion of E_{2,χ^{−1}} defined in equation (35), d = q \frac{d}{dq} is the Atkin-Serre derivative operator, and ⟨ , ⟩_{Y} is the natural Poincaré pairing on Y. The right-hand side of (55) is equal to reg_{p}{u_{χ^{−1}}, u_{χ}}(η_{f}^{ur}) up to an Euler factor (whose precise form is given in equation (60) of loc. cit.).

Remark 2.3. Note that the rigid-analytic function d^{−1}E_{2,χ^{−1}}^{[p]} is the Coleman primitive of E_{2,χ^{−1}}^{[p]}, so that equation (55) expresses values of p-adic L-functions in terms of the theory of p-adic integration. This feature is in common with the formulae presented in the previous sections, with the notable difference that in this case (and in the cases described in the following sections) it is a so-called “iterated Coleman integral” that makes its appearance.

As a final step, we remark that the factorisation of complex L-functions

L(f_{k} ⊗ E_{ℓ,χ}, k/2 + ℓ − 1) = L(f_{k}, k/2 + ℓ − 1) · L(f_{k}, χ, k/2)

implies directly the factorisation of p-adic L-functions

(56)  L_{p}(f, E_{χ})(k, ℓ) = η(k) · L_{p}(f, 1)(k, k/2 + ℓ − 1) · L_{p}(f, χ)(k, k/2),

where η(k) is a p-adic analytic function whose exact value at 2 is determined in Theorem 3.4 of loc. cit. Theorem 2.1 follows by combining equations (55) and (56), given the interpretation of the right-hand side of (55) as a p-adic regulator. This concludes our outline of the proof of Theorem 2.1.

We now turn to a brief discussion of the theory of Euler systems, which in the current context aims to relate the values of L_{p}(f, s) at integer points to a collection of classes in various continuous Galois cohomology groups associated to f. Let F be a p-adic field containing the values of χ, and let δ_{χ^{±}} be the image of u_{χ^{±}} in H_{et}^{1}(Y, F (1)) arising from Kummer theory. Define the (p-adic) étale regulator

reg_{et}{u_{χ^{−1}}, u_{χ}} := δ_{χ^{−1}} ∪ δ_{χ} ∈ H_{et}^{2}(Y, F (2)) = H^{1}(ℚ, H_{et}^{1}(Ȳ , F (2))),

where the last identification is a consequence of the Hochschild-Serre spectral sequence. Consider the isomorphism

(57)  log_{Y,2} : H^{1}(ℚ_{p}, H_{et}^{1}(Ȳ , F (2))) ⟶ D_{dR}(H_{et}^{1}(Ȳ , F (2))) = H_{dR}^{1}(Y/F),

where the first map is the Bloch-Kato logarithm (which is an isomorphism in our setting), and the second equality follows from the comparison theorem between étale and de Rham cohomology. The map (57) sends the restriction at p of the étale regulator, denoted reg_{et,p}, to the p-adic regulator:

(58)  log_{Y,2}(reg_{et,p}{u_{χ^{−1}}, u_{χ}}) = reg_{p}{u_{χ^{−1}}, u_{χ}}

(cf. Proposition 9.11 and Corollary 9.10 of [Bes2]). Thus Theorem 2.1 can be rephrased as a relation between the value L_{p}(f, 2) and the Bloch-Kato logarithm of the étale regulator:

(59) L_{p}(f, 2) · \frac{L(f, χ, 1)}{Ω_{f}^{+}} = (2iN^{−2}𝔤(χ))(1 − β_{p}(f)p^{−2})(1 − β_{p}(f))
× log_{Y,2}(reg_{et,p}{u_{χ^{−1}}, u_{χ}})(η_{f}^{ur}).

The identity (59) should be viewed as the analogue of equation (14) of Section 1.1. More general versions of (59) can be obtained by replacing u_{χ} by the modular units corresponding to the Eisenstein series E_{2}(χ_{1}, χ_{2}) defined in Section 1.1.

Extend F so that it contains the Fourier coefficients of f, and fix a Galois and Hecke equivariant (for the prime-to-N Hecke operators) projection

π_{f} : H_{et}^{1}(Ȳ , F) ⟶ V_{f},

where V_{f} is the Galois representation attached to f. Let κ_{f} be the image in H^{1}(ℚ, V_{f}(2)) of reg_{et}{u_{χ^{−1}}, u_{χ}} by the natural map induced by π_{f}. Although κ_{f} depends on the choice of an auxiliary character χ, this dependency may be eliminated by “stripping off” the scalar \frac{L(f,χ,1)}{Ω_{f}^{+}} ∈ F, which can assumed to be non-zero by a judicious choice of χ.

Kato shows that κ_{f} = κ_{f,0} is the bottom element of a norm-compatible system of classes κ_{f,n} ∈ H^{1}(ℚ(μ_{p^{n}}), V_{f}(2)), constructed from Eisenstein series of level divisible by p^{n} (cf.

[Kato] and [Br]). The formalism outlined in Section 1.1 identifies the system of classes (κ_{f,n})_{n≥0} with an element κ_{f,∞} of H^{1}(ℚ, Λ_{cyc} ⊗ V_{f}(2)). Setting

κ_{f,ξ}(k) := ν_{k,ξ}(κ_{f,∞}) ∈ H^{1}(ℚ, ℚ_{p,ξ} ⊗ V_{f}(1 + k))

for the “weight k specialisation” of κ_{f,∞}, it turns out that the image of κ_{f,ξ}(1) by the Bloch-Kato logarithm encodes the values L_{p}(f, ξ, 2), whereby generalising (59) (cf. [Kato], (16.6)). In this way, L_{p}(f, s) can be “read-off” from the class κ_{f,∞}. The work [BD2] undertakes the task of obtaining such a description of L_{p}(f, s) by extending the techniques of [BD1] outlined above.

More generally, Perrin-Riou’s description of L_{p}(f, s) as the image of κ_{f,∞} by a “big” logmap (interpolating the Bloch-Kato logarithms) allows one to recover the values L_{p}(f, ξ, 1+ k) for all k ≥ 1 from the logarithmic images of the classes κ_{f,ξ}(k) (cf. for example [Br], Theorem 23, [PR2], 3.3.10 and [Cz1]).

Consider now the dual exponential map (in the case k = 0)

exp_{0,ξ}^{∗} : H^{1}(ℚ_{p}, ℚ_{p,ξ} ⊗ V_{f}(1)) ⟶ D_{dR}(ℚ_{p,ξ} ⊗ V_{f}(1)).

Kato’s explicit reciprocity law describes L_{p}(f, ξ, 1) in terms of exp_{0,ξ}^{∗}(κ_{f,ξ}) (cf. for example [PR2], §2.1). In light of the interpolation formula for L_{p}(f, s), this gives the chain of equivalences

(60)  exp_{0,ξ}^{∗}(κ_{f,ξ}) ≠ 0  ⇔  L_{p}(f, ξ, 1) ≠ 0  ⇔  L(f, ξ, 1) ≠ 0.

When combined with Kolyvagin’s theory of Euler systems, (60) implies the following case of the Birch and Swinnerton-Dyer conjecture (where ξ can be assumed to be an arbitrary Dirichlet character):

(61)  L(f, ξ, 1) ≠ 0  ⇒  Hom(ℂ(ξ), E(ℚ̄) ⊗ ℂ) = 0.

2.2. Beilinson-Flach elements. Let f ∈ S_{2}(Γ_{1}(N), χ_{f}) and g ∈ S_{2}(Γ_{1}(N), χ_{g}) be two cusp forms. In this section we discuss the p-adic Beilinson formula of [BDR1] for the value of the p-adic Rankin L-series attached to f ⊗ g at the non-critical value s = 2. Roughly speaking, this formula is achieved by applying the formalism of the previous section with the p-adic family of Eisenstein series E_{χ} replaced by the Hida family g interpolating the cusp form g.

The characters χ_{f} and χ_{g} are taken to have modulus N, so χ_{f}(p) = 0 for p | N. We assume that the forms f and g are normalized eigenforms of level N (not necessarily new), including for the operators U_{p} for p | N. Recall the imprimitive Rankin L-series associated to f and g:

\begin{aligned}L^{imp}(f ⊗ g, s) := ∏_{p} & [(1 − α_{p}(f)α_{p}(g)p^{−s})(1 − α_{p}(f)β_{p}(g)p^{−s}) \\ & ×(1 − β_{p}(f)α_{p}(g)p^{−s})(1 − β_{p}(f)β_{p}(g)p^{−s})]^{−1} , \end{aligned} ; (62)

where α_{p}(f), β_{p}(f) are the roots of the Hecke polynomial x^{2} −a_{p}(f)x+χ_{f}(p)p of f if p ∤ N, and (α_{p}(f), β_{p}(f)) = (a_{p}(f), 0) if p | N. We adopt similar notations for g. The Rankin series L^{imp}(f ⊗ g, s) is the imprimitive L-function associated to the tensor product V_{f} ⊗ V_{g} of the motives attached to f and g, and differs from the full L(f ⊗ g, s) only possibly in Euler factors at primes p | N.

Using the Rankin-Selberg method, Shimura gave an explicit formula for L^{imp}(f ⊗ g, s) in terms of the Petersson inner product (52). Let χ = (χ_{f}χ_{g})^{−1}, and let E_{χ}(z, s − 1) be the non-holomorphic Eisenstein series of weight zero and character χ:

E_{χ}(z, s − 1) = \begin{matrix}∑ & ′ \\ (m,n)∈Nℤ×Z \end{matrix} χ^{−1}(n) Im(z)^{s}|mz + n|^{−2s}.

Shimura’s formula reads (cf. for example equation (4) on [BDR1], with k = ℓ = 2)

(63)  L^{imp}(f ⊗ g, s) = \frac{1}{2} \frac{(4π)^{s}}{Γ(s)} ⟨f^{∗}(z), E_{χ}(z, s − 1)g(z)⟩_{2,N},

where f^{∗}(z) = \overline{f(−\overline{z})} ∈ S_{2}(N_{f}, χ_{f}^{−1}) denotes the cusp form obtained by conjugating the Fourier coefficients of f. Since non-holomorphic Eisenstein series satisfy a functional equation relating s and 1−s, Shimura’s formula leads to a functional equation for L^{imp}(f ⊗g, s) relating the values at s and 3 − s.

Let X denote the modular curve X_{1}(N). Beilinson gave a geometric interpretation for L^{imp}(f ⊗ g, s) at the near central point s = 2 in terms of higher Chow groups on the surface S = X × X. The Rankin L-function L^{imp}(f ⊗ g, s) has no critical points, and in particular Beilinson’s formula concerns the non-critical point s = 2. The higher Chow group CH^{2}(S, 1) ≅ K_{1}(S) is defined to be the homology (in the middle) of the Gersten complex

K_{2}(K(S)) ⟶^{∂} ⊕_{Z⊂S}K(Z)^{∗} ⟶^{div} ⊕_{P ∈S}ℤ.

Here Z ranges over irreducible curves in S, and P ranges over closed points in S. The map denoted div sends a rational function to its divisor. The map ∂ sends a symbol {f, g} ∈ K_{2}(K(S)) associated to pair of functions x, y ∈ K(S)^{∗} to the tame symbol

(64)  ∂(\{x, y\}) = (u_{Z})_{Z⊂S},  u_{Z} = (−1)^{ν_{Z}(x)ν_{Z}(y)} \frac{x^{ν_{Z}(y)}}{y^{ν_{Z}(x)}} .

Let u ∈ K(X)^{∗} be a modular unit as in (48), i.e. a rational function on X whose divisor is supported on the cusps. By viewing u as a rational function on the diagonal Δ ⊂ S, one can define certain distiguished elements Δ_{u} ∈ CH^{2}(S, 1) as follows.

Lemma 2.4. Given a modular unit u on X, there exists an element of the form

(65)  Δ_{u} = (Δ, u) + ∑ a_{i}(Z_{i}, u_{i}) ∈ CH^{2}(S, 1) ⊗ ℚ,

where a_{i} ∈ ℚ and each Z_{i} ⊂ S is a horizonal or vertical divisor, i.e. a curve of the form X × P or P × X for a point P ∈ X.

Definition 2.5. An element of the form Δ_{u} as in (65) is called a Beilinson–Flach element in CH^{2}(S, 1) ⊗ ℚ.

We will be interested in the modular unit u_{χ} such that dlog u_{χ} = E_{2,χ} as in (51) and the associated Beilinson–Flach element Δ_{u_{χ}}.

There is a complex regulator map

reg_{ℂ} : CH^{2}(S, 1) → (Fil^{1}H_{dR}^{2}(S/ℂ))^{∨}

defined by

reg_{ℂ} (∑(Z_{i}, u_{i})) (ω) = \frac{1}{2πi} ∫_{Z′_{i}} ω log |u_{i}|,

where ω is a smooth (1,1)-form on S and Z′_{i} ⊂ Z_{i} is the locus on which u_{i} is regular. We may now state Beilinson’s formula

Theorem 2.6 (Beilinson). We have

\frac{L^{imp}(f ⊗ g, 2)}{⟨f^{∗}, f^{∗}⟩_{2,N}} = C_{χ}reg_{ℂ}(Δ_{u_{χ}})(ω_{g} ⊗ η_{f}^{ah})

where C_{χ} = (8i)π^{3}[Γ_{0}(N) : Γ_{1}(N)(±)]^{−1}N^{−2}𝔤(χ^{−1}), ω_{g} = 2πig(z)dz and

η_{f}^{ah} = ⟨f^{∗}, f^{∗}⟩_{2,N}^{−1} f̄^{∗}(z)dz̄

as in (53).

The main theorem of [BDR1] is a p-adic analogue of Beilinson’s formula. Before stating this result, it is worth noting that the very definition of the p-adic Rankin L-series associated to f ⊗ g is subtle for a reason mentioned before: the classical Rankin L-series L(f ⊗ g, s) has no critical values. When f and g are ordinary at a prime p ∤ N, Hida showed how to define a p-adic Rankin L-series as follows. Let f and g denote the Hida families whose weight two specializations are f and g, respectively. Then for weights k and ℓ such that 2 ≤ ℓ ≤ s ≤ k − 1, the values L^{imp}(f_{k} ⊗ g_{ℓ}, s) are critical. Hida proved that there exists a p-adic L-function L_{p}(f, g)(k, ℓ, s) interpolating the values

(66)  \frac{L^{imp}(f_{k} ⊗ g_{ℓ}, s)}{(2πi)^{ℓ+2s−1}⟨f_{k}, f_{k}⟩_{k,N}} ∈ \overline{ℚ} for 2 ≤ ℓ ≤ s ≤ k − 1.

Note that the roles of f and g are not symmetric in this definition, and we therefore obtain two p-adic Rankin L-series associated to f and g, namely:

L_{p}^{f}(f ⊗ g, s) := L_{p}(f, g, 2, 2, s), L_{p}^{g}(f ⊗ g, s) := L_{p}(g, f, 2, 2, s).

Just as we saw in Section 2.1 regarding the p-adic regulator attached to K_{2}(Y_{1}(N)), there is a p-adic (or “syntomic”) regulator attached to K_{1}(S) ≅ CH^{2}(S, 1). This is a map

(67)  reg_{p} : CH^{2}(S, 1) → Fil^{1}H_{dR}^{2}(S/ℚ_{p})^{∨}

defined by Besser [Bes4] in terms of Coleman’s theory of p-adic integration. As for the p-adic regulator discussed in Section 2.1, the map reg_{p} of (67) satisfies the property that it factors through the étale regulator via the Bloch–Kato logarithm. The main theorem of [BDR1] is as follows.

Theorem 2.7. We have

L_{p}^{f}(f ⊗ g, 2) = \frac{E(f, g, 2)}{E(f)E^{∗}(f)}reg_{p}(Δ_{u_{χ}})(ω_{g} ⊗ η_{f}^{ur}),

where

E(f, g, 2) = (1 − β_{p}(f)α_{p}(g)p^{−2})(1 − β_{p}(f)β_{p}(g)p^{−2})
× (1 − β_{p}(f)α_{p}(g)χ(p)p^{−1})(1 − β_{p}(f)β_{p}(g)χ(p)p^{−1})
E(f) = 1 − β_{p}(f)^{2}χ_{f}^{−1}(p)p^{−2}
E^{∗}(f) = 1 − β_{p}(f)^{2}χ_{f}^{−1}(p)p^{−1}

and η_{f}^{ur} is as in Section 2.1 (see the discussion following (53)).

As indicated earlier, the proof of theorem 2.7 follows along that of Theorem 2.1, with the Hida family E_{χ} replaced by g; we therefore indicate now only a few salient aspects. First, one uses Shimura’s generalization of (63) for the critical values of L(f_{k} ⊗ g_{ℓ}, s)—essentially (54) with E_{ℓ,χ} replaced by g_{ℓ}—and interprets the right side of this formula in terms of the Poincaré pairing on algebraic de Rham cohomology. This pairing can be realized in p-adic de Rham cohomology, and one obtains that the algebraic numbers (66) vary p-adic analytically, up to the multiplication of appropriate Euler factors. This defines a p-adic L-function L_{p}(f, g)(k, ℓ, s), whose value at k = ℓ = s = 2 is given by the formula

(68)  L_{p}(f, g)(2, 2, 2) = \frac{1}{E^{∗}(f)} ⟨η_{f}^{ur}, e_{ord}(d^{−1}E_{2,χ}^{[p]} · g)⟩_{X} .

Theorem 2.7 is then deduced by using Besser’s work [Bes4], which relates the right-hand side of equation (68) to the p-adic regulator.

We conclude this section with a brief discussion of various works in progress regarding the construction of an Euler system of Beilinson–Flach elements along the p-power level tower of self-products of modular curves X_{1}(Np^{r}) × X_{1}(Np^{r}). Such an Euler system is being studied by Lei, Loeffler and Zerbes [LLZ] and by Bertolini, Darmon, and Rotger [BDR2], and holds the promise of several arithmetic applications.

Firstly, such an Euler system would yield a generalization of Theorem 2.7 that varies in p-adic families. This generalization would in particular apply when f and g have level divisible by p, and s is a general arithmetic weight (e.g. s(x) = x^{j}ψ(x) for a p-power conductor Dirichlet character ψ and integer j). Using this generalization, Dasgupta has indicated a proof of a factorization formula for the Rankin square L_{p}(f ⊗ f, s) in terms of the p-adic L-function of the symmetric square of f and a Kubota–Leopoldt p-adic Lfunction [Das]; as proven by Citro, a suitable twist of this factorization formula implies Greenberg’s trivial zero conjecture for the p-adic L-function of the adjoint of f [Cit].

The construction an Euler system of Beilinson–Flach elements is also the subject of independent work in progress by Lei, Loeffler, and Zerbes [LLZ] with the goal of studying the Iwasawa theory of modular forms over quadratic imaginary fields.

Finally, we mention the application of an Euler system of Beilinson–Flach elements toward the rank zero BSD conjecture, which serves as the motivation of [BDR1]; details will appear in [BDR2]. Let E/ℚ be an elliptic curve, and let f ∈ S_{2}(Γ_{0}(N)) denote its associated modular form with rational Fourier coefficients. The image of the Beilinson– Flach element Δ_{u_{χ}} ∈ CH^{2}(S, 1) under the étale regulator yields a class in H^{1}(ℚ, H_{et}^{2}(S × \overline{ℚ}, ℤ_{p}(2))). By projecting onto the (f, g)-isotypic component of H_{et}^{2}(S × \overline{ℚ}, ℤ_{p}), we obtain a class

κ_{E}(g) ∈ H^{1}(ℚ, V_{E} ⊗ V_{g}(1))

where V_{g} = H_{et}^{1}(X × \overline{ℚ}, ℤ_{p})^{g} denotes the Galois representation coming from the g-isotypic component of the first étale cohomology of the modular curve X and V_{E} ≅ V_{f}(1) denotes the (rational) p-adic Tate-module of E.

The interpolation of the Beilinson–Flach elements Δ_{u_{χ}} into an Euler system allows for the construction of a p-adic family of classes

κ_{E}(g) ∈ H^{1}(ℚ, V_{E,g})

where V_{g} is Hida’s two-dimensional Λ-adic Galois representation associated to the family g, and

V_{E,g} := V_{E} ⊗_{ℤ_{p}} V_{g} ⊗_{Λ} Λ_{cyc},

where Λ_{cyc} is as in §1.1 and the tensor product V_{g} ⊗_{Λ} Λ_{cyc} is taken with respect to a suitable algebra homomorphism Λ ↪ Λ ⊗_{ℤ_{p}} Λ.

An appropriate generalization of Theorem 2.7 relates the image of κ_{E}(g) under the Bloch–Kato logarithm to the p-adic L-function L_{p}(f, g)(k, ℓ, s) when k = 2.

Let us now suppose that the specialization of g in weight 1 is a classical cusp form, and hence its associated Galois representations V_{g_{1}} is an odd 2-dimensional Artin representation (i.e. has finite image), which we denote by ρ. The specialization of the class κ_{E}(g) in weight 1 need not be crystalline at p, since it is defined as the specialization of a p-adic family of classes at a non-classical weight, and is not directly defined via a geometric construction. In fact, one shows that κ_{E}(g)_{1} is crystalline at p if and only if L(E, ρ, 1) = 0. Similarly to (61), Kolyvagin’s theory of Euler systems can then be used to deduce the following case of the BSD conjecture for E (cf. [BDR2]):

L(E, ρ, 1) ≠ 0 ⟹ Hom(ρ, E(\overline{ℚ}) ⊗ ℂ) = 0.

2.3. Gross-Kudla-Schoen cycles. The setting of this section is obtained from that of Section 2.1 and 2.2 by replacing, in the triple of modular forms one starts with, Eisenstein series with cusp forms. Thus, let f, g, h ∈ S_{2}(N) be a triple of normalized cuspidal eigenforms of weight 2, level N and nebentypus characters χ_{f}, χ_{g}, and χ_{h}, respectively. Assume χ_{f} · χ_{g} · χ_{h} is the trivial character, so that the tensor product V_{f,g,h} := V_{f} ⊗ V_{g} ⊗ V_{h} of the compatible system of Galois representations associated by Shimura to f, g and h is self-dual and the Garrett-Rankin L-function L(f, g, h, s) of V_{f,g,h} satisfies a functional equation relating the values s and 4 − s.

Let ℚ_{f,g,h} = ℚ({a_{n}(f), a_{n}(g), a_{n}(h)}_{n≥1}) denote the field generated by the Fourier coefficients of f, g and h. For the sake of simplicity in the exposition, let us also assume that N is square-free, the three eigenforms are new in that level, and a_{ℓ}(f)a_{ℓ}(g)a_{ℓ}(h) = −1 for all primes ℓ | N. The results described in this section hold (suitably adapted) in much greater generality (cf. [DR1]) – a fact that is important to bear in mind for the arithmetic applications we shall discuss at the end of this section.

Fix a prime p ∤ N and an embedding ℚ_{f,g,h} ↪ ℂ_{p} for which the three newforms are ordinary, and let f : Ω_{f} ⟶ ℂ_{p}[[q]] denote the Hida family of overconvergent p-adic modular forms passing though f. The space Ω_{f} is a finite rigid analytic covering of the weight space W = ℤ/(p − 1)ℤ × ℤ_{p}.

Write Ω_{f,cl} ⊂ Ω_{f} for the subset of points x whose image in ℤ_{p}^{×} is an integer κ(x) ≥ 2 and let f_{x} ∈ S_{κ(x)}(N) denote the eigenform whose p-stabilisation equals the specialization of f at x. Adopt similar notations for g and h.

Single out one of the three eigenforms, say f. Building on Hida’s definition of the p-adic Rankin L-function discussed in Section 2.2, Harris and Tilouine [HaTi] defined a p-adic L-function of three variables, denoted

(69)  ℒ_{p} ^{f}(f, g, h) : Ω_{f} × Ω_{g} × Ω_{h} ⟶ ℂ_{p}.

See also [DR1, §4] for a description of (69) which corrects the Euler factor in the p-adic interpolation formula arising in [HaTi].

This p-adic L-function interpolates the square-root of the central critical value of the complex L-function L(f_{x}, g_{y}, h_{z}, s), as (x, y, z) ranges over those triples (x, y, z) ∈ Ω_{f,cl} × Ω_{g,cl} × Ω_{h,cl} such that κ(x) ≥ κ(y) + κ(z). To construct ℒ_{p} ^{f}(f, g, h), one invokes the work of Garrett [Gar], Harris and Kudla [HaKu], which shows that these central critical values are equal, up to certain explicit periods, to the algebraic number

J(f_{x}, g_{y}, h_{z}) := (\frac{⟨f_{x}^{∗}, δ^{t}(g_{y}) × h_{z}⟩_{N}}{⟨f_{x}^{∗}, f_{x}^{∗}⟩_{N}} )^{2} ∈ ℚ̄,

where recall f_{x}^{∗} is the eigenform obtained from f_{x} by complex conjugating its Fourier coefficients, δ = δ_{κ(y)} denotes the Shimura-Maass operator of (30) and t := \frac{κ(x)−κ(y)−κ(z)}{2} ≥ 0 (cf. also [DR1, Thm. 4.4]). After multiplying J(f_{x}, g_{y}, h_{z}) by a suitable Euler factor at p, these quantities vary continuously and interpolate to a function on Ω_{f} × Ω_{g} × Ω_{h} denoted ℒ_{p} ^{f}(f, g, h).

Note that the original triple (f, g, h) corresponds to a point (x_{0}, y_{0}, z_{0}) above the triple of weights (2, 2, 2) which lies outside the region of interpolation used to define ℒ_{p} ^{f}(f, g, h). By an abuse of notation, let us simply write ℒ_{p} ^{f}(f, g, h)(2, 2, 2) for the value of this function at the point (x_{0}, y_{0}, z_{0}).

Similarly to Sections 2.1 and 2.2, our goal here is to report on a formula which describes ℒ_{p} ^{f}(f, g, h)(2, 2, 2) as the image of a certain cycle on the cube of the modular curve X = X_{1}(N)/ℚ under the p-adic syntomic Abel-Jacobi map. As before, the resulting formula is a p-adic analogue of a complex one: in this case Yuan-Zhang-Zhang’s theorem [YZZ] relating the first derivative L′(f, g, h, 2) to the Beilinson-Bloch’s height of Δ.

The cycle in question is essentially the diagonal X_{123} = {(x, x, x), x ∈ X} in X^{3}, which has to be modified to make it null-homologous. More precisely, fix the cusp ∞ ∈ X at infinity as base point and, following Gross, Kudla and Schoen [GrKu], [GrSc], define Δ to be the class in the Chow group CH^{2}(X^{3}) of codimension 2 cycles in X^{3} up to rational equivalence of the formal sum

X_{123} − X_{12} − X_{13} − X_{23} + X_{1} + X_{2} + X_{3},

where X_{1} = {(x, ∞, ∞), x ∈ X}, X_{12} = {(x, x, ∞), x ∈ X} and likewise for the remaining summands. One checks that, for any of the standard cohomology theories (e.g. algebraic de Rham, Betti, p-adic syntomic, p-adic étale), the class of Δ in the cohomology group H^{4}(X^{3}) vanishes. Thus Δ belongs to the subgroup CH^{2}(X^{3})_{0} of null-homologous cycles in CH^{2}(X^{3}), which is the source of the various Abel-Jacobi maps available, e.g.:

(70)

AJ_{ℂ} : CH^{2}(X^{3})_{0} ⟶ Fil^{2}H_{dR}^{3}(X^{3} × ℂ)^{∨}/H_{3}(X^{3}(ℂ), ℤ)
AJ_{syn,p} : CH^{2}(X^{3})_{0} ⟶  Fil^{2}H_{dR}^{3}(X^{3} × ℚ_{p})^{∨}
AJ_{et,p} : CH^{2}(X^{3})_{0} ⟶ H^{1}(G_{ℚ}, H_{et}^{3}(X^{3} × ℚ̄, ℤ_{p}(2))).

Since it is an essential ingredient of the formula that we aim to describe here, we recall Besser’s description [Bes1] of the image of Δ under the p-adic syntomic Abel-Jacobi map. Specialized to our setting, [Bes1, Theorem 1.2] shows that AJ_{syn,p}(Δ) is a ℚ_{p}-valued functional

(71)  AJ_{syn,p}(Δ) : Fil^{2}H_{dR}^{3}(X^{3}) ⟶ ℚ_{p}

which can be described purely in terms of Coleman integration. In loc.cit., Besser recasts Coleman’s integration theory of 1-forms on curves in a cohomological guise, and exploits this interpretation to provide a generalization of the notion of Coleman’s primitive to forms on higher-dimensional varieties V/ℚ_{p} admitting a smooth model V over ℤ_{p}. The spaces in which Besser’s primitives live are called finite polynomial cohomology groups, denoted H_{fp}^{i}(V, n) for indices i, n ≥ 0, and his formalism gives rise to a canonical projection p : H_{fp}^{i}(V, n) → Fil^{n}H_{dR}^{i}(V).

In the case of a curve, like our X equipped with its standard integral model X /ℤ_{p}, taking i = n = 1 yields an exact sequence 0 → ℚ_{p} → H_{fp}^{1}(X , 1) →^{p} Ω^{1}(X) → 0 where any pre-image ω̃ ∈ H_{fp}^{1}(X , 1) of a regular 1-form ω on X may be identified with a choice of a Coleman primitive of ω. That the kernel of p is ℚ_{p} agrees with the well-known fact that such primitives are well-defined only up to constants.

As it will suffice for our purposes, we content here to describe the restriction of (71) to the Künneth component Fil^{2}(H_{dR}^{1}(X)^{⊗3}). Up to permutations of the three variables, the typical element in this space is of the type η_{1} ⊗ ω_{2} ⊗ ω_{3}, where η_{1} is a class in H_{dR}^{1}(X) and ω_{2}, ω_{3} ∈ Ω^{1}(X) are regular differential 1-forms. [Bes1, Theorem 1,2] asserts that

(72) AJ_{p}(Δ)(η_{1} ⊗ ω_{2} ⊗ ω_{3}) = ∫_{Δ} η_{1} ⊗ ω_{2} ⊗ ω_{3} := ∑_{∅≠I⊆{1,2,3}} sign(I)tr_{I}(ι_{I}^{∗}(η̃_{1} ∪ ω̃_{2} ∪ ω̃_{3})),

where η̃_{1} ∈ H_{fp}^{1}(X , 0), ω̃_{2}, ω̃_{3} ∈ H_{fp}^{1}(X , 1) are choices of primitives of η_{1}, ω_{2}, ω_{3}, respectively, and η̃_{1} ∪ ω̃_{2} ∪ ω̃_{3} ∈ H_{fp}^{3}(X^{3}, 2) is their cup-product. Moreover, in the above formula we set sign(I) = (−1)^{|I|+1}, ι_{I} : X ↪ X^{3} denotes the natural inclusion which maps X onto the curve X_{I} ⊂ X^{3} and tr_{I} : H_{fp}^{3}(X , 2) ⟶^{∼} ℚ_{p} is the canonical trace isomorphism of [Bes1, Prop. 2.5 (4)]. While each of the terms tr_{I}(ι_{I}^{∗}(η̃_{1} ∪ ω̃_{2} ∪ ω̃_{3})) does depend on the choice of primitives, one checks that their sum does not.

We are finally in position to state the main formula alluded to at the beginning.

Theorem 2.8. For each ϕ ∈ {f, g, h}, let ω_{ϕ} ∈ Ω^{1}(X) denote the regular 1-form associated to ϕ, and η_{f}^{ur} ∈ H_{dR}^{1}(X)^{f,ur} be the unique class in the unit root subspace of the f-isotypical component of H_{dR}^{1}(X) such that ⟨ω_{f}, η_{f}^{ur}⟩ = 1. Let also α_{p}(ϕ), β_{p}(ϕ) be the two roots of the Hecke polynomial x^{2} − a_{p}(ϕ)x + p, labelled in such a way that α_{p}(ϕ) is a p-adic unit. Then the equality

(73)  ℒ_{p} ^{f}(f, g, h)(2, 2, 2) = \frac{E(f, g, h)}{E_{0}(f)E_{1}(f)}AJ_{p}(Δ)(η_{f}^{ur} ⊗ ω_{g} ⊗ ω_{h})

holds, where

E(f, g, h) := (1 − β_{p}(f)α_{p}(g)α_{p}(h)p^{−2}) × (1 − β_{p}(f)α_{p}(g)β_{p}(h)p^{−2})
× (1 − β_{p}(f)β_{p}(g)α_{p}(h)p^{−2}) × (1 − β_{p}(f)β_{p}(g)β_{p}(h)p^{−2}) ,
E_{0}(f) := (1 − β_{p}^{2}(f)χ_{f}^{−1}(p)p^{−1}), E_{1}(f) := (1 − β_{p}^{2}(f)χ_{f}^{−1}(p)p^{−2}).

This statement holds in greater generality for eigenforms f, g, h of possibly different primitive levels N_{f}, N_{g}, N_{h}, and different weights k, ℓ, m, provided none of the weights is larger than or equal to the sum of the other two. We refer the reader to [DR1] for the precise formulation; here we limit ourselves to provide an overall description of the proof of Theorem 2.8.

To show the identity (73), one first shows that

(74)  AJ_{p}(Δ)(η_{f}^{ur} ⊗ ω_{g} ⊗ ω_{h}) = ⟨η_{f}^{ur}, P (Φ)^{−1}ξ⟩

where, similarly as in the previous sections, Φ is a lift of Frobenius to the system {W_{ϵ}}_{ϵ>0} of wide open neighborhoods of the ordinary locus of X(ℂ_{p}), P (t) ∈ ℂ_{p}[t] is a polynomial satisfying

• P (Φ × Φ) annihilates the class of ω_{g} ⊗ ω_{h} in H_{rig}^{2}(W_{ϵ}^{2}),

• P (Φ) acts invertibly on H_{dR}^{1}(X),
• P (Φ) annihilates H_{dR}^{2}(X/ℚ_{p}) ≃ ℚ_{p}(−1),

and, given a rigid analytic primitive ρ ∈ Ω_{rig}^{1}(W_{ϵ}^{2}) of P (Φ × Φ)(ω_{g} ⊗ ω_{h}) and the choice of the cusp ∞ at infinity as base point, we set

ξ = (δ^{∗} − h_{∞}^{∗} − v_{∞}^{∗})ρ ∈ H_{dR}^{1}(X) ⊂ H_{rig}^{1}(W_{ϵ}).

This is proved in [DR1, Ch. 3] in two steps: a formal calculation permits first to relate the image under the Abel-Jacobi map of the diagonal cycle Δ on X^{3} to the diagonal D on the square X^{2} of the curve; this leads to a simplification, which allows to apply Besser’s machinery [Bes1] to compute AJ_{p}(Δ)(η_{f}^{ur} ⊗ω_{g} ⊗ω_{h}) purely in terms of classes of differential 1-forms on X, yielding (74).

In view of this formula, it is then not difficult to show that, up to an explicit Euler factor at p, (74) is equal to

(75)  ⟨η_{f}^{ur}, e_{ord}(d^{−1}(g^{[p]}) · h)⟩

where, as in §2.1, e_{ord} denotes Hida’s ordinary projector, g^{[p]} := ∑_{p∤n} a_{n}(g)q^{n} is the pdepletion of g and d := qd/dq is Serre’s derivative operator. That the left-hand side of (73) equals (75), again up to an explicit p-multiplier, follows from the explicit calculations involved in the very construction of the p-adic L-function ℒ_{p} ^{f}(f, g, h).

We bring (75) to the reader’s attention not only because it stands as the basic bridge between the two sides of the equality in (73), but also because this is a quantity which is amenable to effective numerical approximations, thanks to the work of A. Lauder (see [Lau] for more details).

This can be used in turn to design algorithms for computing p-adic numerical approximations to Chow-Heegner and Stark-Heegner points on elliptic curves over various number fields. We refer the reader to [DR2] and the forthcoming works of the third and sixth author, in collaboration with A. Lauder, for some of these constructions. To illustrate the method, we just mention here that the prototypical construction arises when f is taken to be the eigenform associated to an elliptic curve E/ℚ and g = h. In this setting Zhang [Zh] introduced a rational point P_{g,f} ∈ E(ℚ), whose formal group logarithm can be computed as

(76)  log_{ω_{f}} (P_{g,f}) = −2\frac{E_{1}(g)}{E(g, g, f)}⟨η_{g}^{ur}, e_{ord}(d^{−1}(g^{[p]}) · f)⟩,

by invoking Theorem 2.8 and the results of [DRS1].

Motivated by the analogies between the description of P_{g,f} given in [DRS1] by means of Chen’s complex iterated integrals and the above formula, we refer to (75) as a p-adic iterated integral.

We close this section by discussing briefly the Euler system underlying the diagonal cycle Δ, and the arithmetic applications that Theorem 2.8 has in this context.

Assume f has rational Fourier coefficients and trivial nebentypus, and let E/ℚ be the (isogeny class of the) elliptic curve associated to it by the Eichler-Shimura construction. If instead of applying the p-adic syntomic Abel-Jacobi map, one considers the image of Δ under the p-adic étale Abel-Jacobi map AJ_{et,p} recalled in (70), one obtains a global cohomology class with values in H_{et}^{3}(X^{3} × ℚ̄, ℤ_{p}(2)). After projecting to the (f, g, h)- isotypical component of this Galois module, we obtain an element

(77)  κ_{E}(g, h) ∈ H^{1}(ℚ, V_{p}(E) ⊗ V_{g} ⊗ V_{h}(1)),

where, for any of the forms ϕ = f, g, h, V_{ϕ} := H_{et}^{1}(X × ℚ̄, ℤ_{p})^{ϕ} and V_{p}(E) ≃ V_{f}(1) is the Galois representation associated to the p-adic Tate module of E.

Let now Λ_{ϕ} be the finite extension of the Iwasawa algebra Λ = ℤ_{p}[[ℤ_{p}^{×}]] corresponding to the space Ω_{ϕ} as considered at the beginning of the section for ϕ = f. Hida constructed a rank two Galois representation V_{ϕ} over Λ_{ϕ}, interpolating the Galois representations associated by Deligne to each of the classical specializations of \underline{ϕ}.

As in §1.1, let Λ_{cyc} denote the Λ-adic Galois representation whose underlying module is Λ itself, equipped with the Galois action induced by the character

Gal (ℚ̄/ℚ) ⟶ Gal (ℚ(μ_{p^{∞}})/ℚ) ⟶^{χ_{cyc}} ℤ_{p}^{×} ↪ Λ^{×},

where χ_{cyc} is the cyclotomic character and the latter inclusion maps an element z ∈ ℤ_{p}^{×} to the corresponding group-like element [z] ∈ Λ^{×}. Define

V_{f,g,h} := V_{p}(E) ⊗_{ℤ_{p}} (V_{g} ⊗_{Λ} V_{h}) ⊗_{Λ} Λ_{cyc}

where the tensor products over Λ are taken with respect to algebra homomorphisms

Λ ↪ Λ ⊗ Λ ⊆ Λ_{g} ⊗ Λ_{h}, Λ ↪ Λ ⊗ Λ ⊆ (Λ_{g} ⊗_{Λ} Λ_{h}) ⊗ Λ_{cyc}

such that for all classical points (x, y, z) of weights k, ℓ, m ≥ 2 with m = k + ℓ − 2, the specialization of V_{f,g,h} at (x, y, z) is isomorphic as a G_{ℚ}-module to V_{f_{x}} ⊗V_{g_{y}} ⊗V_{h_{z}}(k+ℓ−2).

One of the main results of [DR2] is the construction of a cohomology class

(78)  κ_{E}(g, h) ∈ H^{1}(ℚ, V_{f,g,h})

which satisfies the following interpolation property. Let (y, z) ∈ Ω_{g} ×_{Ω} Ω_{h} be a pair of classical points over a weight in Ω of the form z ↦ ξ(z)z^{2} for some Dirichlet character ξ of p-power conductor. Then the specialization of κ_{E}(g, h) at (y, z) satisfies

ν_{y,z}(κ_{E}(g, h) = E_{y,z} · κ_{E}(g_{y}, h_{z}).

for some explicit Euler factor E_{y,z}.

The most interesting arithmetical applications of the Λ-adic cohomology class (78) arise when we deform it to points (y, z) of weight 1 which are classical. Indeed, assume that for such a pair, the specializations g_{y} and h_{z} are the q-expansions of classical eigenforms of weight 1; in this case their associated Galois representations are Artin representations, denoted ρ_{y} and ρ_{z} respectively. By specializing κ_{E}(g, h) to this pair, we obtain a cohomology class κ_{E}(g_{y}, h_{z}) ∈ H^{1}(ℚ, V_{p}(E) ⊗ ρ_{y} ⊗ ρ_{z}). Note that this class needs not be cristalline at p, since it did not arise from a geometric construction, but rather by deforming p-adically a collection of geometric classes. The goal of [DR2] is to show that

(79)  κ_{E}(g_{y}, h_{z}) is cristalline at p if and only if L(E, ρ_{y} ⊗ ρ_{z}, 1) = 0.

The main arithmetical application of (79) is the following instance of the Birch and Swinnerton-Dyer conjecture:

Theorem 2.9. Let E be an elliptic curve over ℚ and let

ρ_{1}, ρ_{2} : G_{ℚ} ⟶ GL_{2}(ℂ)

be two continuous odd Galois representations, attached to weight one modular forms g and h respectively. Assume det(ρ_{1}) · det(ρ_{2}) = 1 and there exists σ ∈ G_{ℚ} for which ρ_{1} ⊗ ρ_{2}(σ) has distinct eigenvalues. If L(E ⊗ ρ_{1} ⊗ ρ_{2}, 1) ≠ 0, then

dim_{ℂ}(hom(ρ_{1} ⊗ ρ_{2}, E(ℚ̄) ⊗ ℂ)) = 0.

Conclusion

The contents of this survey are summarized in the table below, whose rows correspond to the six Euler systems covered in each section. The first and second column list the Euler system and its associated p-adic L-function, while the third gives the p-adic special value formula relating the two. The last column indicates the (eventual) application of each Euler system to the Birch and Swinnerton-Dyer conjecture. In this column, BSD_{r}(E, ρ) refers to the implication

ord_{s=1}L(E, ρ, s) = r ⇒ dim_{ℂ} hom_{G_{ℚ}}(ρ, E(K_{ρ}) ⊗ ℂ) = r,

where E is an elliptic curve over ℚ,

ρ : G_{ℚ} ⟶ GL_{n}(ℂ)

is an Artin representation, and r ≥ 0 is an integer. The letter A alludes to an elliptic curve with complex multiplication, and the letter E to a general (modular) elliptic curve over ℚ. Likewise, the symbol ρ_{ψ} refers to the induction from an imaginary quadratic field K to ℚ of a dihedral character of K, while χ refers to a one-dimensional representation of G_{ℚ} (i.e., a Dirichlet character) and ρ, ρ_{1} and ρ_{2} refer to general odd irreducible two-dimensional Artin representations of ℚ, with the sole constraint that ρ_{1} ⊗ ρ_{2} has trivial determinant.

Euler system p-adic L-function  p-adic formula  BSD application
Circular units Kubota-Leopoldt  Leopoldt’s theorem None

p-adic L-function

Elliptic units Katz two-variable  Katz’s p-adic  Coates-Wiles:
p-adic L-function  Kronecker limit  BSD_{0}(A, 1).

formula

Heegner points Anticyclotomic  p-adic Gross-Zagier Gross-Zagier,
p-adic L-function  theorem of [BDP] Kolyvagin:
of [BDP]  BSD_{0}(E, ρ_{ψ}) and
BSD_{1}(E, ρ_{ψ}).
Beilinson-Kato Mazur-  p-adic Beilinson  Kato:
elements  Swinnerton-Dyer-  formula: [Kato],  BSD_{0}(E, χ).
Panciskin  [Br], [Nik], [BD1]
Beilinson-Flach Hida’s p-adic p-adic Beilinson BSD_{0}(E, ρ).
elements  Rankin L-function formula: [BDR1]
Gross-Kudla- Hida-Harris-Tilouine p-adic Gross-Kudla BSD_{0}(E, ρ_{1} ⊗ ρ_{2})
Schoen cycles triple product  formula [DR1]

p-adic L-function

We emphasise that the p-adic Gross-Zagier formulae of [BDP] and of [DR1] alluded to in lines 3 and 6 of the table are not the only, or indeed even the most natural, generalisations of the Gross-Zagier formula to these settings. Perrin-Riou’s p-adic Gross-Zagier formula of [PR1], which relates the first derivative of a suitable p-adic L-function at a point which lies in its range of classical interpolation to the p-adic height of a Heegner point, bears a more visible analogy with its original complex counterpart, while the formulae of [BDP] and [DR1] are the direct generalisation of the p-adic formulae of Leopoldt and Katz and are better adapted to certain Euler system arguments.

This survey takes the view that an Euler system is a collection of global cohomology classes which can be related to L-functions in a precise way and can be made to vary in (p-adic) families. The possibility of p-adic variation is an essential feature because it allows the construction of global classes which do not directly arise, in general, from a geometric construction involving étale Abel-Jacobi images of algebraic cycles or étale regulators of elements in K-theory, but rather from p-adic limits of such classes. Frequently, the obstruction to such “p-adic limits of geometric classes” being cristalline at p is encoded in a classical critical L-value, thereby tying this L-value to a global object which can be used to bound the associated Bloch-Kato Selmer group.

One aspect of the picture which has been deliberately downplayed is the idea that Euler systems should arise from norm-compatible collections of global elements defined over a varying collection of abelian extensions of a fixed ground field. This feature is clearly present in the first five examples considered in this survey, but not in the sixth, where the only variables of “p-adic deformation” are the weight variables arising in Hida theory. Over the years, p-adic families of automorphic forms have been studied for a wide variety of reductive groups. This raises the hope that Gross-Kudla-Schoen diagonal cycles will point the way to further fruitful examples of Euler systems, involving for instance (p-adically varying families of) algebraic cycles on Shimura varieties of unitary or orthogonal type. Examples of this kind would comfort the authors in their belief that Euler systems are far more ubiquitous than would appear from the limited panoply of known instances described in this survey.

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M.B.: Milano, Italy

  E-mail address: massimo.bertolini@unimi.it

F.C.: Montreal, Canada

  E-mail address: castella@math.mcgill.ca

H. D.: Montreal, Canada

  E-mail address: darmon@math.mcgill.ca

S.D.: Santa Cruz, Unites States

  E-mail address: samit dasgupta@yahoo.com

K.P.: Ann Arbor, United States

  E-mail address: kartik.prasanna@gmail.com

V.R.: Barcelona, Spain

  E-mail address: victor.rotger@upc.edu