The rationality of Stark-Heegner points over genus fields of real quadratic fields^{∗}

Massimo Bertolini Henri Darmon

August 27, 2012

Contents

1 A review of Stark-Heegner points 7 1.1 Modular symbols . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Double integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Indefinite integrals . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 Stark-Heegner points . . . . . . . . . . . . . . . . . . . . . . . 12

2 Hida theory 16 2.1 Hida families . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Periods attached to Hida families . . . . . . . . . . . . . . . . 17 2.3 Indefinite integrals revisited . . . . . . . . . . . . . . . . . . . 22

3 p-adic L-functions 23 3.1 The Mazur-Kitagawa p-adic L-function . . . . . . . . . . . . . 23 3.2 p-adic L-functions attached to real quadratic fields . . . . . . 24 3.3 A factorisation formula . . . . . . . . . . . . . . . . . . . . . . 27

4 Proof of the main result 27

∗appeared in Annals of Mathematics, 170 (2009), 343–369

Introduction

Let E be an elliptic curve over ℚ of conductor N, and let f be the normalised cusp form of weight 2 on Γ_{0}(N) attached to E. Suppose that there is an odd prime p||N of multiplicative reduction for E, and let K be a real quadratic field satisfying the following modified Heegner hypothesis:

1. the prime p is inert in K;

2. all the primes dividing M := N/p are split in K.

Fix an embedding of K and its p-adic completion K_{p} into ℂ_{p} := ˆ \overline{ℚ}_{p}, and let H_{p} = ℂ_{p} − ℚ_{p} denote the p-adic upper half plane. Note that since p is inert in K, the set H_{p} ∩ K is non-empty.

The first section of this paper briefly recalls the main construction of [Dar01], which associates to the cusp form f and to any τ ∈ H_{p} ∩ K a socalled Stark-Heegner point P_{τ} ∈ E(K_{p}). Conjecture 5.9 of [Dar01] predicts that some integer multiple of this point is defined over a ring class field of K depending on τ, and gives an explicit description, analogous to the Shimura reciprocity law, for the action of Gal(K^{ab}/K) on the collection of Stark-Heegner points attached to K.

The main result of the present article (Theorem 1) gives some evidence for Conjecture 5.9 of [Dar01] by showing that certain integral linear combinations of Stark-Heegner points are global points on E defined over the expected abelian extension of K. The non-vanishing of these points is also related to the first derivative at s = 1 of the Hasse-Weil L-series of E/K in the spirit of the Gross-Zagier formula, lending support for Conjecture 5.15 of [Dar01].

Before stating Theorem 1 precisely, some further notations are required. Let M_{2}(ℤ[1/p]) denote the ring of 2×2 matrices with entries in ℤ[1/p], and let R ⊂ M_{2}(ℤ[1/p]) denote the subring of matrices which are upper-triangular modulo M. The order associated to τ ∈ H_{p} ∩ K is defined to be

O_{τ} = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ R such that aτ + b = cτ^{2} + dτ\} ⊂ K,

where the inclusion on the right sends the matrix ( \begin{matrix}a & b \\ c & d \end{matrix} ) to the element

cτ + d. Via this inclusion, O_{τ} is identified with a ℤ[1/p]-order of K. Let D be the discriminant of a (not necessarily maximal) order of K, and let O_{D} denote the ℤ[1/p]-order of K of that discriminant. Set

H_{p}^{D} = {τ ∈ H_{p} ∩ K such that O_{τ} = O_{D}} .

The group

Γ := {γ ∈ R^{×} such that det(γ) = 1}  (1)

acts on H_{p} by Möbius transformations, and preserves H_{p}^{D} for any D.

Let G_{D} := Pic^{+}(O_{D}) denote the Picard group of oriented O_{D}-modules, or equivalently, the group of SL_{2}(ℤ)-equivalence classes of primitive integral binary quadratic forms of discriminant D equipped with the group law given by Gaussian composition. Class field theory defines an isomorphism

rec : G_{D}⟶Gal(H_{D}/K),

where H_{D} is the so-called narrow ring class field attached to O_{D}. The fact that p is inert in K implies that the prime ideal pO_{K} splits completely in H_{D}/K. Choose a prime of H_{D} above p. This choice determines an extension to H_{D} of the chosen embedding K⟶ℂ_{p}.

The quotient Γ\H_{p}^{D} is equipped with a natural action of G_{D} whose definition is recalled in Section 1.4, and which is written (g, τ) ↦ τ^{g}, for g ∈ G_{D} and τ ∈ Γ\H_{p}^{D}.

Conjecture 5.9 of [Dar01] predicts that some fixed multiple of P_{τ} is a global point in E(H_{D}), so P_{τ} belongs to E(H_{D}) ⊗ ℚ, and that

P_{τ^{g}} = rec(g)^{−1}(P_{τ}) for all g ∈ G_{D}.  (2)

(Note that this compatibility does not depend on the choice of embedding of H_{D} into ℂ_{p} that was made.)

Suppose now that D is the discriminant of K. A genus character of K is a quadratic unramified character of Gal(K̄/K). Such a genus character χ cuts out a biquadratic (or quadratic, in the special case where χ is the trivial character) extension of ℚ, denoted H_{χ}:

H_{χ} = ℚ(\sqrt{D_{1}}, \sqrt{D_{2}}), where D = D_{1}D_{2}.

Let χ_{1}, χ_{2}, and ϵ_{K} be the Dirichlet characters associated to the quadratic fields ℚ(\sqrt{D_{1}}), ℚ(\sqrt{D_{2}}), and K respectively. Note that χ_{1}χ_{2} = ϵ_{K}. The genus characters are in bijection with the factorisations of D into a product of two relatively prime fundamental discriminants, or, equivalently, with the unordered pairs of primitive quadratic Dirichlet characters (χ_{1}, χ_{2}) of coprime conductors satisfying χ_{1}χ_{2} = ϵ_{K}. (The trivial character χ corresponds to the factorisation D = 1 · D.) Let E(H_{χ})^{χ} denote the submodule of the Mordell-Weil group E(H_{χ}) on which Gal(H_{D}/K) acts via the character χ.

Define the point

P_{χ} = ∑_{g∈G_{D}} χ(g)P_{τ^{g}} ∈ E(K_{p}).

Equation (2) implies that an integer multiple of P_{χ} belongs to E(H_{χ})^{χ}, and Conjecture 5.15 of [Dar01] predicts that this point is of infinite order if and only if L′(E/K, χ, 1) ≠ 0.

Let q ∈ pℤ_{p} be the Tate period attached to E, and write

Φ_{Tate} : K_{p}^{×}/q^{ℤ}⟶E(K_{p})

for the Tate uniformisation (which is defined since p is inert in K and E therefore acquires split multiplicative reduction over K_{p}). Let log_{q} : K_{p}^{×}⟶K_{p} denote the branch of the p-adic logarithm satisfying log_{q}(q) = 0, and define a homomorphism

log_{E} : E(K_{p})⟶K_{p}

by the rule

log_{E}(P) := log_{q}(Φ_{Tate}^{−1}(P)).

For each m|N with gcd(m, N/m) = 1, let w_{m} denote the sign of the Fricke involution at m acting on f. Note that the modified Heegner hypothesis implies that ϵ_{K}(−M) = 1, and therefore χ_{1}(−M) = χ_{2}(−M). The main result of this article is

Theorem 1. Let χ be the genus character attached to the pair of Dirichlet characters χ_{1} and χ_{2}. Suppose that E has at least two primes of multiplicative reduction, and that χ_{1}(−M) = −w_{M}.

1. There is a global point P_{χ} ∈ E(H_{χ})^{χ} and t ∈ ℚ^{×} such that

log_{E}(P_{χ}) = t log_{E}(P_{χ}).  (3)

2. The point P_{χ} is of infinite order if and only if L′(E/K, χ, 1) ≠ 0.

Remark 2. Theorem 1 shows that P_{χ} coincides with the image of a global point in E(K_{p}) ⊗ ℚ. This implies that a suitable integer multiple of P_{χ} belongs to the natural image of E(H_{χ})^{χ} in E(K_{p}).

By way of providing a context for the proof of Theorem 1, we note the analogy between the approach that it follows and Kronecker’s “solution to Pell’s equation” in terms of special values of the Dedekind eta-function. (See Chapter IX of [We] for a historical account, and Chapter II.1 of [Sie] for a more detailed treatment.) In the classical setting considered by Kronecker, the fundamental discriminant D is taken to be negative, and corresponds to an imaginary quadratic subfield K of ℂ. The p-adic upper half plane is replaced by its archimedean counterpart H, and H^{D} is given the obvious meaning (with M = 1). This set is preserved under the action of SL_{2}(ℤ) by Möbius transformations, and the quotient SL_{2}(ℤ)\H^{D} is equipped with a natural action of the class group G_{D} of K. A quadratic character χ of G_{D}— assumed to be non-trivial, although the trivial character requires no special consideration in the setting of Stark-Heegner points—corresponds to a pair of Dirichlet characters χ_{1} and χ_{2} which are even and odd respectively, cutting out quadratic extensions K_{1} and K_{2} of ℚ. Let ϵ_{1} > 1 be the fundamental unit of the real quadratic field K_{1}, denote by h_{j} (for j = 1, 2) the class number of K_{j}, and write w_{2} for the number of roots of unity in K_{2}. After setting

η^{∗}(τ) := |D|^{−1/4}\sqrt{2y}|η(τ)|^{2},

Kronecker shows (cf. Theorem 6 of Chapter II.1 of [Sie]) that for any τ ∈ H^{D},

∑_{σ∈G_{D}} χ(σ) log η^{∗}(τ^{σ}) = −\frac{2h_{1}h_{2}}{w_{2}} log(ϵ_{1}).  (4)

This expresses a solution to the Pell equation x^{2} − D_{1}y^{2} = 1 in terms of the function η^{∗} evaluated at suitable quadratic imaginary arguments. Kronecker’s proof is obtained by combining the following three ingredients:

1. The Kronecker limit formula, which expresses the left hand side of (4) in terms of the L-series ζ(K, χ, s) = ∑_{𝔞} χ(𝔞)N(𝔞)^{−s}, where the sum is taken over all the ideals 𝔞 of K:

− ∑_{σ∈G_{D}} χ(σ) log η^{∗}(τ^{σ}) = \frac{d}{ds}ζ(K, χ, s)|_{s=0}.  (5)

2. A factorisation of ζ(K, χ, s) as a product of the Dirichlet L-series attached to χ_{1} and χ_{2}:

ζ(K, χ, s) = L(χ_{1}, s)L(χ_{2}, s).  (6)

3. Dirichlet’s class number formula which asserts that

L′(χ_{1}, 0) = h_{1} log(ϵ_{1}),  L(χ_{2}, 0) = \frac{2h_{2}}{w_{2}} .  (7)

Kronecker’s identity (4) is a direct consequence of (5), (6) and (7). It can also be understood in the framework of the theory of complex multiplication, which relates the individual quantities η^{∗}(τ^{σ}) to elliptic units defined over H_{D}. Kronecker’s approach is noteworthy in that it makes no use of the theory of complex multiplication. This represents an advantage in the setting of Stark-Heegner points attached to real quadratic fields, where no analogue of the theory of complex multiplication is known, and where the algebraicity of the individual quantities P_{τ} remains conjectural.

The proof of Theorem 1 is explained in Section 4. At this stage we limit ourselves to some general remarks on the counterparts to steps 1, 2, and 3 above in our approach.

1. The role of Kronecker’s limit formula is played by Theorem 4.1, which relates the Stark-Heegner point P_{χ} to the leading term of a Hida padic L-function L_{p}(f_{∞}/K, χ, k) attached to the datum of a Hida family {f_{∞}} interpolating f in weight two. The relation that emerges between periods of Hida families and Stark-Heegner points represents a new insight that was suggested by combining the calculations in [Das1] and [DD] with the main result of [BD98]. It is hoped that Theorem 4.1 may be of some independent interest beyond its role in the proof of Theorem 1.

2. The Hida L-function L_{p}(f_{∞}/K, χ, k) interpolates the central critical values L(f_{k}/K, χ, k/2) of the weight k specialisations of f_{∞}. This interpolation property is a direct consequence of a formula of Popa [Po] expressing these values in terms of certain “geodesic cycle integrals” attached to f and K, à la Shintani. This interpolation property is the key to expressing L_{p}(f/K, χ, k) as a product of two Mazur-Kitagawa p-adic L-functions L_{p}(f_{∞}, χ_{j}, k, s) attached to {f_{∞}} and the Dirichlet characters χ_{1} and χ_{2}, restricted to the central critical line s = k/2.

3. One is finally reduced to expressing the leading term in a neighbourhood of k = 2 of L_{p}(f_{∞}, χ_{j}, k, k/2) in terms of rational quantities and logarithms of global points. This last ingredient is supplied by Theorem 5.4 of [BD05], whose precise formulation is recalled in Section 4, and whose proof relies on a p-adic analytic construction of (classical) Heegner points coming from Shimura curve parametrisations, via the Cerednik-Drinfeld theory of p-adic uniformisation of these curves. It is this reliance on parametrisations by Shimura curves over ℚ which forces the assumption in Theorem 1 that E has at least two primes of multiplicative reduction.

Acknowledgements. The authors would like to thank the anonymous referee for detailed comments and suggestions which led to significant improvements in the exposition. The second author also acknowledges Samit Dasgupta for a stimulating collaboration [DD] which led to some key insights which are used in the present work.

1 A review of Stark-Heegner points

This section reviews the definition of Stark-Heegner points that is given in [Dar01], presenting it in a way that is adapted to the subsequent proof of Theorem 1.

1.1 Modular symbols

Let g ∈ S_{k}(Γ_{0}(N)) be a normalised cusp form of even weight k ≥ 2 on Γ_{0}(N), and let K_{g} denote the finite extension of ℚ generated by its Fourier coefficients a_{n}(g) (n ≥ 1). We view K_{g} as a subfield both of ℂ and ℂ_{p}, by fixing complex and p-adic embeddings of K_{g}.

Let F be a field, and let P_{k}(F) denote the space of homogenous polynomials in two variables of degree k − 2 with coefficients in F. It is equipped with a right action of GL_{2}(ℚ) given by the rule

(P |γ)(x, y) := P (ax + by, cx + dy), for γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) .  (8)

Let V_{k}(F) denote the F -linear dual of P_{k}(F).

A modular symbol with values in an abelian group G is a function

I : ℙ_{1}(ℚ) × ℙ_{1}(ℚ)⟶G, denoted (r, s) ↦ I{r → s},

satisfying

I{r → s} + I{s → t} = I{r → t}, for all r, s, t ∈ ℙ_{1}(ℚ).

The group GL_{2}(ℚ) acts on the space of V_{k}(ℂ)-valued modular symbols by the rule

(I|γ){r → s}(P) := I{γr → γs}(P |γ^{−1}).

The periods of the form g are encoded in such a modular symbol, denoted Ĩ_{g} and defined by

Ĩ_{g}{r → s}(P) := 2πi ∫_{r}^{s} g(z)P (z, 1)dz.

This symbol is invariant under Γ_{0}(N).

The matrix c = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) normalises Γ_{0}(N) and hence induces an invo-

lution on the space MS(V_{k}(ℂ))^{Γ_{0}(N)} of Γ_{0}(N)-invariant V_{k}(ℂ)-valued modular symbols. Let Ĩ_{g}^{+} and Ĩ_{g}^{−} denote the plus and minus eigencomponents of Ĩ_{g} for this involution.

Proposition 1.1. There exist complex periods Ω_{g}^{+} and Ω_{g}^{−} with the property that the modular symbols

I_{g}^{+} := (Ω_{g}^{+})^{−1}Ĩ_{g}^{+}, I_{g}^{−} := (Ω_{g}^{−})^{−1}Ĩ_{g}^{−}

belong to MS(V_{k}(K_{g}))). These periods can be chosen to satisfy

Ω_{g}^{+}Ω_{g}^{−} = ⟨g, g⟩,  (9)

where ⟨g, g⟩ is the Petersson scalar product of g with itself.

Proof. The proof is explained, for example, in Section 1.1 of [KZ]. (See in particular the first corollary and the third theorem in that section).

Remark 1.2. Note that the periods Ω_{g}^{±} are only well-defined up to multiplication by a non-zero scalar in K_{g}^{×}. This ambiguity seems unavoidable, because no obvious choice of Ω_{g}^{±} imposes itself naturally, even though the product Ω_{g}^{+}Ω_{g}^{−} is uniquely defined by (9).

Choose a “sign at infinity” w_{∞} ∈ {+1, −1}, and set

Ω_{g} := \{ \begin{matrix}Ω_{g}^{+} & if w_{∞} = +1; \\ Ω_{g}^{−} & if w_{∞} = −1; \end{matrix} I_{g} := \{ \begin{matrix}I_{g}^{+} & if w_{∞} = +1; \\ I_{g}^{−} & if w_{∞} = −1. \end{matrix}

Note that the modular symbol I_{g} can be viewed as an element of MS(V_{k}(ℂ_{p})) thanks to the chosen embedding of K_{g} into ℂ_{p}.

1.2 Double integrals

In the case where f is a modular form of weight 2 on Γ_{0}(N) with rational Fourier coefficients, the modular symbol I_{f} := I_{f}(1) is ℚ-valued, and can even be rescaled so that it takes values in ℤ. Assume that this has been done from now on.

A measure on ℙ_{1}(ℚ_{p}) is an element of the continuous ℚ_{p}-linear dual of the space of continuous ℚ_{p}-valued functions on ℙ_{1}(ℚ_{p}), equipped with the topology of uniform convergence. Given a measure μ, and a compact open subset U of ℙ_{1}(ℚ_{p}), set μ(U) := μ(χ_{U}), where χ_{U} denotes the characteristic function of U. The function U ↦ μ(U) is a bounded, finitely additive ℚ_{p}- valued function on the set of compact open subsets of ℙ_{1}(ℚ_{p}). Conversely, any such function gives rise to a ℚ_{p}-valued measure on ℙ_{1}(ℚ_{p}). The measure μ is said to be integral, or ℤ-valued, if μ(U) belongs to ℤ for all compact open U ⊂ ℙ_{1}(ℚ_{p}).

Recall the group Γ of equation (1) in the introduction. Fix a subset ℙ of ℙ_{1}(ℚ) on which Γ acts transitively by Möbius transformations.

The following elementary proposition is key to the definition of Stark-Heegner points attached to real quadratic fields.

Proposition 1.3. There exists a unique system of ℤ-valued measures on ℙ_{1}(ℚ_{p}), indexed by r, s ∈ ℙ and denoted μ_{f}{r → s}, satisfying the following properties.

1. For all r, s ∈ ℙ,

μ_{f}{r → s}(ℙ_{1}(ℚ_{p})) = 0, μ_{f}{r → s}(ℤ_{p}) = I_{f}{r → s}.

2. For all γ ∈ Γ, and all compact open U ⊂ ℙ_{1}(ℚ_{p}),

μ_{f}{γr → γs}(γU) = μ_{f}{r → s}(U).

Proof. The proof of this proposition is identical to that of Proposition 2.6 of [DD], which considered the case where the newform f is replaced by the logarithmic derivative of a modular unit, a weight two Eisenstein series. The main property of this Eisenstein series that is used is the fact that it is fixed by the Hecke operator U_{p}^{2}. Since this is also true of f, the proof of Proposition 2.6 can be adapted to the setting at hand with essentially no modifications.

Remark 1.4. We have chosen to consider modular symbols defined on ℙ×ℙ, largely for convenience: for example, this will guarantee the uniqueness of the “indefinite integral” of Proposition 1.5.

The measures μ_{f} can be used to define a “double multiplicative integral” attached to τ_{1}, τ_{2} ∈ H_{p} and r, s ∈ ℙ as in equation (71) of [Dar01], by setting

× ∫_{τ_{1}}^{τ_{2}}∫_{r}^{s} ω_{f} := × ∫_{ℙ_{1}(ℚ_{p})} (\frac{t − τ_{2}}{t − τ_{1}} ) dμ_{f}\{r → s\}(t).  (10)

The “multiplicative integral” notation appearing on the right indicates that a limit of Riemann products is being taken, rather than a limit of Riemann sums, i.e., that the integration is relative to the multiplicative structure on ℂ_{p}^{×}. Such a definition is made possible by the fact that for each r, s ∈ ℙ, the measure μ_{f}{r → s} is ℤ-valued, and that the integrand is a continuous K_{p}^{×}-valued function on ℙ_{1}(ℚ_{p}) relative to the natural topology on K_{p}^{×}. (For a more detailed discussion of this multiplicative integral, and its basic properties, see the discussion following Lemma 1.10 in [Dar01].)

Recall the Tate period q ∈ pℤ_{p} attached to E/ℚ_{p}, and the branch log_{q} of the p-adic logarithm sending q to 0. Define

∫_{τ_{1}}^{τ_{2}}∫_{r}^{s} ω_{f} := log_{q} (× ∫_{τ_{1}}^{τ_{2}}∫_{r}^{s} ω_{f}) .  (11)

Note that the definition of this “additive integral” differs somewhat from the definition that is given in [Dar01], where the Iwasawa branch of the p-adic logarithm satisfying log(p) = 0 is used. (Cf. equation (73) of [Dar01].)

1.3 Indefinite integrals

The following result justifies the choice of branch of p-adic logarithm that was made in (11).

Proposition 1.5. There is a unique function from H_{p}×ℙ×ℙ to ℂ_{p}, denoted

(τ, r, s) ↦ ∫ ^{τ}∫_{r}^{s} ω_{f},

satisfying

1. For all γ ∈ Γ,

∫ ^{γτ}∫_{γr}^{γs} ω_{f} = ∫ ^{τ}∫_{r}^{s} ω_{f}.

2. For all τ_{1}, τ_{2} ∈ H,

∫ ^{τ_{2}}∫_{r}^{s} ω_{f} − ∫ ^{τ_{1}}∫_{r}^{s} ω_{f} = ∫_{τ_{1}}^{τ_{2}}∫_{r}^{s} ω_{f}.

3. For all r, s, t ∈ ℙ,

∫ ^{τ}∫_{r}^{t} ω_{f} + ∫ ^{τ}∫_{t}^{s} ω_{f} = ∫ ^{τ}∫_{r}^{s} ω_{f}.

Proof. The proof of this proposition is explained in Section 3.1 of [Dar01], where it is reduced to the exceptional zero conjecture of Mazur, Tate and Teitelbaum proved by Greenberg and Stevens. (In the notations of Section 3.1 of [Dar01], and in particular of equation (162), we have

∫ ^{τ}∫_{r}^{s} ω_{f} = log_{q}(η_{f,τ}{r → s}).)

A more direct proof, albeit one whose main idea can still be traced to the calculations of Greenberg and Stevens, can be obtained by specialising the approach described in [BDI] to weight 2 modular forms.

The function which is characterised indirectly in Proposition 1.5 is called the indefinite integral attached to f. The articles [DG] and [DP] explain how Proposition 1.5 can be used to produce efficient algorithms for the numerical evaluation of the indefinite integral. Section 2.3 gives a direct formula for it in terms of the periods of the Hida family interpolating f, which is better adapted to the general calculations of this paper.

Remark 1.6. It is the existence of the indefinite integral that relies crucially on the branch of p-adic logarithm chosen in (11). Its uniqueness then follows by noting that the difference

δ{r → s} := ∫ ^{τ}∫_{r}^{s} ω_{f}^{(1)} − ∫ ^{τ}∫_{r}^{s} ω_{f}^{(2)}

of any two functions satisfying properties 1, 2 and 3 of Proposition 1.5 is independent of the choice of τ, and hence defines a Γ-invariant ℂ_{p}-valued modular symbol on ℙ × ℙ. Since Γ acts transitively on ℙ, such a symbol is determined by the homomorphism φ_{δ} : Γ⟶ℂ_{p} defined by choosing a base point r ∈ ℙ and setting φ_{δ}(γ) = δ{r → γr}. But this homomorphism is necessarily trivial, since Γ has finite abelianisation and ℂ_{p} is torsion-free.

Remark 1.7. Proposition 1.5 can be refined (cf. Theorem 5.2 of [BDG], or Theorem 3.3 of [Das2]) to yield the existence of a lattice Q ⊂ ℂ_{p}^{×} which is commensurable with q^{ℤ}, and an “indefinite multiplicative integral”

× ∫ ^{τ}∫_{r}^{s} ω_{f} ∈ ℂ_{p}^{×}/Q

satisfying the obvious multiplicative analogues of the properties listed in Proposition 1.5. (Cf. equations (163)-(165) of [Dar01].) Roughly speaking, the lattice Q appears as the obstruction to splitting a two-cocycle in H^{2}(Γ, ℂ_{p}^{×}) constructed in terms of the double multiplicative integral. Note that log_{q}(Q) = 0, and that we can write

∫ ^{τ}∫_{r}^{s} ω_{f} = log_{q} (× ∫ ^{τ}∫_{r}^{s} ω_{f}) .

1.4 Stark-Heegner points

Given τ ∈ H_{p} ∩ K, let γ_{τ} = ( \begin{matrix}a & b \\ c & d \end{matrix} ) denote the unique generator for the

stabiliser of τ in Γ satisfying

cτ + d > 1.  (12)

(Where in this inequality, we have made use of the fixed real embedding of K.) Associate to τ a multiplicative and additive period by choosing any base point r ∈ ℙ and setting

J_{τ}^{×} := × ∫ ^{τ}∫_{r}^{γ_{τ} r} ω_{f} ∈ K_{p}^{×}/Q,  J_{τ} := log_{q}(J_{τ}^{×}) = ∫ ^{τ}∫_{r}^{γ_{τ} r} ω_{f}.  (13)

Since Φ_{Tate}(Q) is contained in E(K_{p})_{tors}, the image of J_{τ}^{×} under Φ_{Tate} is welldefined in E(K_{p}) ⊗ ℚ, and is called the Stark-Heegner point attached to τ and f:

P_{τ} := Φ_{Tate}(J_{τ}^{×}),  hence log_{E}(P_{τ}) = J_{τ}.  (14)

Let D be the discriminant of K. The modified Heegner hypothesis imposed on K in the first paragraph of the introduction implies the existence of an element δ ∈ ℤ/Mℤ satisfying

δ^{2} ≡ D (mod M).

Fix such a δ once and for all. Let F^{D} denote the set of primitive binary quadratic forms Ax^{2} + Bxy + Cy^{2} of discriminant D satisfying

M|A, B ≡ δ (mod M).

The set F^{D} is preserved under the natural action of Γ_{0}(M) defined by (8). Recall the class group G_{D} of SL_{2}(ℤ)-equivalence classes of primitive binary quadratic forms of discriminant D. The natural map

Γ_{0}(M)\F^{D}⟶G_{D}

obtained by sending the class of the quadratic form Q to its corresponding SL_{2}(ℤ)-equivalence class is readily seen to be a bijection, and hence Γ_{0}(M)\F^{D} is endowed with the stucture of a principal homogenous space under G_{D}. If Q(x, y) belongs to Γ_{0}(M)\F^{D}, and σ to G_{D}, write Q^{σ} for the image of Q by σ. Let H_{D} denote the narrow Hilbert class field of K, whose Galois group is identified with G_{D}, and let

rec : G_{D}⟶Gal(H_{D}/K)

denote the isomorphism arising from the reciprocity law of global class field theory.

Given Q = Ax^{2} + Bxy + Cy^{2} ∈ Γ_{0}(M)\F^{D}, let

τ_{Q} := \frac{−B + \sqrt{D}}{2A}  (15)

be a fixed root of the quadratic polynomial Q(x, 1). Note that τ_{Q} belongs (via the fixed p-adic embedding of K) to H_{p}^{D}, and that its image in Γ\H_{p}^{D} is well-defined. Given σ ∈ G_{D}, write τ^{σ} ∈ H_{p}^{D} for the root of any quadratic form in the Γ_{0}(M)-equivalence class of Q^{σ}. This definition gives a precise meaning to the conjectural equation (2) of the Introduction.

Since the definition of Stark-Heegner points is purely p-adic analytic, little can be said about the action of Gal(H_{D}/K) on these points independently of assertion (2) in the introduction. However, something unconditional can be asserted about the action of the Frobenius element at p, denoted τ_{p} ∈ Gal(H_{D}/ℚ). Since the prime p is inert in K, the element τ_{p}, which is only defined up to conjugation, corresponds to a reflection in the generalised dihedral group Gal(H_{D}/ℚ). This reflection corresponds to the involution in Gal(K_{p}/ℚ_{p}) after fixing an embedding H_{D}⟶K_{p}, i.e., a prime of H_{D} above p. Proposition 5.10 of [Dar01] asserts the existence of an element σ_{τ} in G_{D} satisfying

τ_{p}(J_{τ}) = −w_{M}J_{τ^{στ}} ,  τ_{p}(P_{τ}) = w_{N}P_{τ^{στ}} .  (16)

Note the sign difference in the two equations, which arises from the fact that τ_{p} does not commute with Φ_{Tate} in general, but rather satisfies

τ_{p}Φ_{Tate}τ_{p} = a_{p}Φ_{Tate} = −w_{p}Φ_{Tate}.

The element σ_{τ} is denoted by σ in Proposition 5.10 of [Dar01], but we have denoted it here by σ_{τ} to emphasize its dependence on τ. Indeed, replacing τ by τ^{α}, for some α ∈ G_{D}, one can see that

σ_{τ^{α}} = σ_{τ}α^{−2}.  (17)

This identity is a consequence of the fact that τ_{p} does not commute with the elements of G_{D}, but rather satisfies

τ_{p}σ = σ^{−1}τ_{p}, for all σ ∈ G_{D}.

Equation (17) shows that the image of σ_{τ} in G_{D}/G_{D}^{2} is independent of τ ∈ Γ\H_{p}^{D}. Denote this element by σ, in keeping with the notations of Proposition 5.10 of [Dar01].

It will be necessary to have a precise description of σ. To do this, we give a formula for χ(σ), as χ runs over the characters of G_{D}/G_{D}^{2}. These characters are precisely the genus characters attached to the discriminant D, and correspond to pairs (χ_{1}, χ_{2}) of primitive quadratic Dirichlet characters of coprime conductor satisfying χ_{1}χ_{2} = ϵ_{K}. (Cf. the discussion in the Introduction preceeding the statement of Theorem 1.) The unordered pair (χ_{1}, χ_{2}) is characterised by the properties

χ(frob_{λ}) = χ(frobλ̄) = χ_{1}(ℓ) = χ_{2}(ℓ),  (18)
χ(frob_{∞}) = χ_{1}(−1) = χ_{2}(−1),

for any rational prime ℓ which splits completely as a product ℓ = λλ̄ of prime ideals of K of norm ℓ. (Here, frob_{∞} denotes the conjugacy class of complex conjugation in G_{D}.)

Proposition 1.8. For any genus character χ attached to the discriminant D, corresponding to the pair (χ_{1}, χ_{2}) of quadratic Dirichlet characters,

χ(σ) = χ_{1}(−M).

Proof. We will show that the element σ corresponds to the class in G_{D}/G_{D}^{2} of the element ∞λ_{1}^{e_{1}} ⋯ λ_{t}^{e_{t}} ∈ Pic^{+}(O), where

1. ∞ stands for the class of a principal ideal generated by an element x ∈ K of negative norm for which ord_{p}(x) is even, and corresponds to complex conjugation in G_{D}/G_{D}^{2};

2. M = ℓ_{1}^{e_{1}} ⋯ ℓ_{t}^{e_{t}} is the factorisation of M into a product of distinct prime powers, and λ_{j} is some ideal of K above ℓ_{j}.

To see this, we first recall the description of the action of G_{D} on Γ\H_{p}^{D} given in Section 2.4 of [DD]. If τ belongs to H_{p}^{D}, one may choose a representative element (denoted τ again by abuse of notation) in its Γ-orbit in such a way that both Λ_{1} = ⟨τ, 1⟩ and Λ_{2} = ⟨Mτ, 1⟩ are fractional ideals of O with τ > τ̄ and ord_{p}(τ − τ̄) even. The ratio λ = Λ_{1}/Λ_{2} is a cyclic integral O-ideal of norm M. Using the bijection \underline{τ} described in Section 2.4 of [DD], the element W_{M}(τ̄) = −1/(Mτ̄) corresponds to the pair of fractional ideals

Λ′_{1} = ⟨−x, Mxτ̄⟩, Λ′_{2} = ⟨−Mx, Mxτ̄⟩.

Now a direct calculation shows that

(Λ′_{1}, Λ′_{2}) = 𝔠 ∗ (Λ_{1}, Λ_{2}),

where

𝔠 = (x)λ̄(Λ̄_{1}/Λ_{1}).

The fractional ideal (Λ̄_{1}/Λ_{1}) = \frac{Λ_{1}Λ̄_{1}}{Λ_{1}^{2}} represents the trivial element in G_{D}/G_{D}^{2}. It follows that σ is represented by the class of (x)λ̄ in G_{D}/G_{D}^{2}, as was to be shown.

Proposition 1.8 now follows from (18).

2 Hida theory

The definition of Stark-Heegner points given in Section 1.4 is ill-suited for theoretical calculations, because of the indirect nature of the definition of the indefinite integral of Proposition 1.5. The present chapter gives a useful definition of the the indefinite integral and of Stark-Heegner points in terms of periods attached to Hida families.

2.1 Hida families

Let

Λ̃ := ℤ_{p}[[ℤ_{p}^{×}]], Λ = ℤ_{p}[[(1 + pℤ_{p})^{×}]]

denote the usual Iwasawa algebras, and let

X = hom(ℤ_{p}^{×}, ℤ_{p}^{×}) ≃ ℤ/(p − 1)ℤ × ℤ_{p}

be the space of continuous p-adic characters of ℤ_{p}^{×}, equipped with its natural topology. Elements of X can also be viewed in a natural way as continuous algebra homomorphisms from Λ̃ to ℤ_{p}. The space X contains ℤ as a dense subset by sending k ∈ ℤ to the character θ_{k}(x) := x^{k−2}. Note the shift by two in our convention, which means that k = 2 corresponds to the augmentation map on Λ̃.

Following the discussion that is made in Section 1.2 of [BD05], we associate to f a so-called Hida family

f_{∞} := ∑_{n=1}^{∞} a_{n}(k)q^{n}.

This is a formal q-expansion with coefficients in the ring A(U) of p-adic analytic functions on U, where U is an appropriate neighbourhood of 2 ∈ X. Assume for simplicity that U is contained in the residue disc of 2 modulo p − 1, and let ℤ^{≥2} denote the set of integers which are ≥ 2. The formal q-expansion f_{∞} is characterised by the following properties:

1. If k belongs to U ∩ ℤ^{≥2}, the q-expansion

f_{k} := ∑_{n=1}^{∞} a_{n}(k)q^{n}

is a normalised eigenform of weight k on Γ_{0}(N). For this reason it is referred to as the weight k specialisation of f_{∞}.

2. f_{2} = f.

Note in particular that the field K_{f_{k}} generated by the Fourier coefficients of the normalised eigenform f_{k} is a finite extension of ℚ. For each k ∈ U ∩ ℤ^{≥2}, we fix the Shimura periods Ω_{k}^{+} := Ω_{f_{k}}^{+} and Ω_{k}^{−} := Ω_{f_{k}}^{−} as in Proposition 1.1. This choice of periods allows us to talk about the V_{k}(ℂ_{p})-valued modular symbols I_{f_{k}}^{+} and I_{f_{k}}^{−} associated to each f_{k}. The modular symbol I_{f_{k}} will be taken to be I_{f_{k}}^{+} or I_{f_{k}}^{−} depending on the choice of w_{∞} that was made.

2.2 Periods attached to Hida families

Let L_{∗} := ℤ_{p}^{2} denote the standard ℤ_{p}-lattice in ℚ_{p}^{2}, and let L′_{∗} denote its set of primitive vectors, i.e., the vectors in L_{∗} which are not divisible by p. Let 𝔻 denote the space of compactly supported ℚ_{p}-valued measures on W := ℚ_{p}^{2} − {0}, and let 𝔻_{∗} denote the subspace of measures that are supported on L′_{∗}. The action of the group ℤ_{p}^{×} on W and L′_{∗} given by λ(x, y) = (λx, λy) gives rise to Λ̃ and Λ module structures on 𝔻 and 𝔻_{∗}. The module 𝔻 is also equipped with a right Λ̃-linear action of GL_{2}(ℚ_{p}) defined by the rule

∫_{W} F d(μ|γ) = ∫_{W}(F |γ^{−1})dμ,  (19)

where GL_{2}(ℚ_{p}) operates on the continuous functions on W by the rule extending (8)

(F |γ)(x, y) := F (ax + by, cx + dy), for γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) .  (20)

Denote by Γ_{0}(pℤ_{p}) the group of matrices in GL_{2}(ℤ_{p}) which are upper triangular modulo p. Our interest in the space 𝔻_{∗} lies in the fact that it is equipped, for all k ∈ ℤ^{≥2}, with a Γ_{0}(pℤ_{p})-equivariant homomorphism

ρ_{k} : 𝔻_{∗}⟶V_{k}

defined by

ρ_{k}(μ)(P) := ∫_{ℤ_{p}×ℤ_{p}^{×}} P (x, y)dμ(x, y).

(Note that ρ_{k} does not respect the full action of GL_{2}(ℤ_{p}), because the domain of integration that appears in its definition is only preserved by Γ_{0}(pℤ_{p}).) The homomorphism ρ_{k} gives rise to a homomorphism, denoted by the same letter by abuse of notation:

ρ_{k} : MS_{Γ_{0}(M)}(𝔻_{∗})⟶MS_{Γ_{0}(N)}(V_{k}(ℂ_{p})).

The space MS_{Γ_{0}(M)}(𝔻_{∗}) is equipped with a natural action of the Hecke operators, and of the operator U_{p} in particular. Let MS_{Γ_{0}(M)}^{ord}(𝔻_{∗}) denote the ordinary subspace of MS_{Γ_{0}(M)}(𝔻_{∗}).

Proposition (6.1) of [GS] asserts that this module is free and of finite rank over Λ.

The Iwasawa algebra Λ is identified in the usual way with a subring of the ring of analytic functions on X. Let Λ^{†} denote the larger ring of ℂ_{p}- valued functions on X which can be represented by a convergent power series expansion in some neighbourhood of 2 ∈ X, and set

𝔻_{∗}^{†} := 𝔻_{∗} ⊗_{Λ} Λ^{†}, MS_{Γ_{0}(M)}^{ord}(𝔻_{∗})^{†} := MS_{Γ_{0}(M)}^{ord}(𝔻_{∗}) ⊗_{Λ} Λ^{†}.

Similar notations are adopted, with the obvious meanings, when 𝔻_{∗} is replaced by 𝔻. If μ = λ_{1}μ_{1} + ⋯ + λ_{t}μ_{t}, with λ_{i} ∈ Λ^{†} and μ_{i} ∈ 𝔻, is any element of 𝔻^{†}, then there exists a neigbourhood U_{μ} of 2 ∈ X on which all the λ_{j} are defined. Call such a region a neighbourhood of regularity for μ. Given k ∈ ℤ, a function F on W is said to be homogeneous of degree k if F (λx, λy) = λ^{k}F (x, y) for all λ ∈ ℤ_{p}. Observe that for any k ∈ U_{μ} ∩ ℤ^{≥2}, and any homogenous function F (x, y) of degree k − 2, one can integrate F against μ on any compact open region X ⊂ W by the rule

∫_{X} F dμ := λ_{1}(k) ∫_{X} F dμ_{1} + ⋯ + λ_{t}(k) ∫_{X} F dμ_{t}.

The following result of Greenberg and Stevens plays a key role in the constructions of this section.

Theorem 2.1. There exists a 𝔻_{∗}^{†}-valued modular symbol μ_{∗} ∈ MS_{Γ_{0}(M)}^{ord}(𝔻_{∗})^{†} such that

1. ρ_{2}(μ_{∗}) = I_{f};

2. For all k ∈ U_{μ_{∗}} ∩ ℤ^{≥2}, there exists a scalar λ(k) ∈ ℂ_{p} such that

ρ_{k}(μ_{∗}) = λ(k)I_{f_{k}}.

Proof. This is Theorem 1.5 of [BD05], which follows from Theorem (5.13) of [GS] and whose proof is explained in Section 6 of that paper. See also [BDI] where extensions of this result to modular forms that are not necessarily ordinary are discussed.

Remark 2.2. Note that the scalar λ(k) depends on the choice of complex period used to define I_{f_{k}}. Since no attempt was made to choose these complex periods “coherently” as k varies, (and in fact, the authors ignore whether such a choice can meaningfully be made a priori) the function k ↦ λ(k) cannot be expected to extend to a continuous function on U_{μ_{∗}}.

Since R^{×} acts transitively on the set of ℤ_{p}-lattices in ℚ_{p}^{2}, and the stabiliser of L_{∗} for this action is precisely Γ_{0}(M), we may define a collection of 𝔻^{†}- valued modular symbols μ_{L} indexed by the ℤ_{p}-lattices in ℚ_{p}^{2} as in Proposition 1.8 of [BD05], by imposing the rules:

μ_{L_{∗}} = μ_{∗}, ∫_{γX}(F |γ^{−1})dμ_{γL}{γr → γs} = ∫_{X} F dμ_{L}{r → s},

for all γ ∈ Γ, for all lattices L ⊂ ℚ_{p}^{2}, and all compact open regions X ⊂ W.

Let

U := U_{μ_{∗}} = U_{μ_{L}}

denote a region of regularity for the measures μ_{L}. The measures μ_{L}{r → s} are supported on the compact subsets L′ of W, and they satisfy the following property:

Lemma 2.3. Suppose that L_{2} ⊂ L_{1} is a sublattice of index p in L_{1}. For all k ∈ U ∩ ℤ^{≥2}, for all homogenous polynomials P (x, y) of degree k − 2, and for all r, s ∈ ℙ, we have

∫_{L′_{1}∩L′_{2}} P dμ_{L_{2}}{r → s} = a_{p}(k) ∫_{L′_{1}∩L′_{2}} P dμ_{L_{1}}{r → s}.

Proof. See Lemma 1.10 of [BD05].

The basic property of the measures μ_{∗}{r → s} given in Theorem 2.1 can be re-written as

∫_{ℤ_{p}×ℤ_{p}^{×}} P (x, y)dμ_{∗}{r → s}(x, y) = λ(k)I_{f_{k}}{r → s}(P),  (21)

for all P ∈ P_{k}(ℚ), and for all r, s ∈ ℙ. It will be useful to understand the value of the the integral appearing on the left in (21), when the region of integration is taken to be the full L′_{∗} = (ℤ_{p}^{2})′ instead of the subset ℤ_{p} × ℤ_{p}^{×}.

To write down such a formula, we first remark that for each integer k > 2, the form f_{k} is old at p, and there is a unique normalised eigenform f_{k}^{♯} of weight k on Γ_{0}(M) satisfying

f_{k}(z) = f_{k}^{♯}(z) − p^{k−1}a_{p}(k)^{−1}f_{k}^{♯}(pz).

(By convention, we set f_{2}^{♯} := 0.) Let I_{f_{k}^{♯}} be the modular symbol attached to

f_{k}^{♯} via the choice of complex period Ω_{k}. This modular symbol satisfies the relation

I_{f_{k}}{r → s}(P) = I_{f_{k}^{♯}}{r → s}(P) − p^{k−2}a_{p}(k)^{−1}I_{f_{k}^{♯}}{r/p → s/p}(P (x, y/p)).

Proposition 2.4. For all k ∈ U ∩ ℤ^{≥2}, for all P ∈ P_{k}(ℂ_{p}) and for all r, s ∈ ℙ,

∫_{(ℤ_{p}^{2})′} P (x, y)dμ_{∗}{r → s}(x, y) = λ(k)(1 − a_{p}(k)^{−2}p^{k−2})I_{f_{k}^{♯}}{r → s}(P).

Proof. This proposition is inspired from Lemma 4.17 of [DD] which treats the case of the Hida family of Eisenstein series. The proof given here follows the argument of [DD] closely. (The essential, comparatively minor difference is that one has λ(k) = a_{p}(k)^{2} = 1 in the context of Lemma 4.17 of [DD].) Let L_{∗} and L_{∞} denote the lattices

L_{∗} = ℤ_{p}^{2}, L_{∞} = ℤ_{p} ⊕ pℤ_{p}.

Note that

(L′_{∗} ∩ L′_{∞}) = ℤ_{p}^{×} × pℤ_{p}, (L′_{∗} ∩ \frac{1}{p}L′_{∞}) = ℤ_{p} × ℤ_{p}^{×}.  (22)

Let θ ∈ R^{×} be any matrix of determinant p satisfying

θ(L_{∗}) = L_{∞}, θ(L_{∞}) = pL_{∗}, hence θ(ℤ_{p}^{×} × pℤ_{p}) = p(ℤ_{p} × ℤ_{p}^{×}). (23)

Observe that L′_{∗} can be written as a disjoint union of the two regions appearing in (22). Hence we may write

∫_{(ℤ_{p}^{2})′} P (x, y)dμ_{∗}{r → s}(x, y) = J_{1} + J_{2},

where

J_{1} = ∫_{ℤ_{p}×ℤ_{p}^{×}} P dμ_{f}{r → s} = λ(k)I_{f_{k}}{r → s}(P),
J_{2} = ∫_{ℤ_{p}^{×}×pℤ_{p}} P dμ_{f}{r → s}
= ∫_{p(ℤ_{p}×ℤ_{p}^{×})}(P |θ^{−1})dμ_{L_{∞}}{θr → θs}
= p^{k−2} ∫_{ℤ_{p}×ℤ_{p}^{×}}(P |θ^{−1})dμ_{\frac{1}{p} L_{∞}}\{θr → θs\}
= a_{p}(k)^{−1}p^{k−2} ∫_{ℤ_{p}×ℤ_{p}^{×}}(P |θ^{−1})dμ_{L_{∗}}{θr → θs}
= a_{p}(k)^{−1}p^{k−2}λ(k)I_{f_{k}}{θr → θs}(P |θ^{−1}),

and the penultimate equality follows from Lemma 2.3. To evaluate the contributions J_{1} and J_{2} in terms of the form f_{k}^{♯}, note that, for any choice of θ satisfying (23),

f_{k}(z) = f_{k}^{♯}(z) − p^{k−1}a_{p}(k)^{−1}(f_{k}^{♯}|θ)(z).

A direct calculation, using a change of variables, then shows that

I_{(f_{k}^{♯}|θ)}{r → s}(P) = p^{−1}I_{f_{k}^{♯}}{θr → θs}(P |θ^{−1}).

Hence

J_{1} = λ(k) (I_{f_{k}^{♯}}{r → s}(P) − p^{k−1}a_{p}(k)^{−1}I_{(f_{k}^{♯}|θ)}{r → s}(P))
= λ(k) (I_{f_{k}^{♯}}{r → s}(P) − p^{k−2}a_{p}(k)^{−1}I_{f_{k}^{♯}}{θr → θs}(P |θ^{−1})) ,

while

J_{2} = λ(k)a_{p}(k)^{−1}p^{k−2} (I_{f_{k}^{♯}}{θr → θs}(P |θ^{−1})
−p^{k−2}a_{p}(k)^{−1}I_{f_{k}^{♯}}{θ^{2}r → θ^{2}s}(P |θ^{−2})) .

Note that θ^{2} = pγ, for some γ ∈ Γ_{0}(N). Since I_{f_{k}^{†}} is Γ_{0}(N)-invariant, it follows that

I_{f_{k}^{♯}}{θ^{2}r → θ^{2}s}(P |θ^{−2}) = p^{−k+2}I_{f_{k}^{♯}}{r → s}(P),

and therefore

J_{1} + J_{2} = λ(k)(1 − a_{p}(k)^{−2}p^{k−2})I_{f_{k}^{♯}}{r → s}(P).

The proposition follows.

2.3 Indefinite integrals revisited

Recall the indefinite integral that was defined in Section 1.3. The relevance of Hida families to Stark-Heegner points can be explained by the fact that the system of distribution-valued modular symbols μ_{L}{r → s} that was introduced in Section 2.2 can be used to give a direct formula for this indefinite integral.

We content ourselves with doing this when τ belongs to H_{p} ∩ K_{p}, and hence is defined over a quadratic unramified extension of ℚ_{p}. In that case, the function

(x, y) ↦ x − τy

identifies ℚ_{p}^{2} with K_{p}. Let L_{τ} be the ℤ_{p}-lattice in ℚ_{p}^{2} defined by

L_{τ} = {(x, y) such that x − τy belongs to O_{K} ⊗ ℤ_{p}}.

The following theorem is a direct generalisation of Proposition 4.7 of [DD].

Theorem 2.5. For all τ ∈ H_{p} ∩ K_{p}, and for all r, s ∈ ℙ,

∫ ^{τ}∫_{r}^{s} ω_{f} = ∫_{L′_{τ}} log(x − τy)dμ_{L_{τ}} {r → s}(x, y),

where log : K_{p}^{×}⟶K_{p} is any branch of the p-adic logarithm.

Proof. Note that the expression (x − τy) belongs to (O_{K} ⊗ ℤ_{p})^{×} for any (x, y) ∈ L′_{τ}, and hence the integrand on the right is independent of the branch of the p-adic logarithm that was chosen to define it. To prove Theorem 2.5, it suffices to check that the three defining properties of the indefinite integral listed in Proposition 1.5 are satisfied by the expression appearing on the right in Theorem 2.5. The invariance under Γ stated as property 1 is a consequence of Proposition 4.6 of [BDI]. As for property 2, it follows from Proposition 4.7 of [BDI], which holds for any branch of the p-adic logarithm, including log_{q} for which the extra term appearing in Proposition 4.7 of [BDI] vanishes. Finally, property 3 is a direct consequence of the definitions.

Corollary 2.6. The Stark-Heegner point P_{τ} associated to Q ∈ Γ_{0}(M)\F^{D} satisfies

log_{E} P_{τ} = J_{τ} = ∫_{(ℤ_{p}^{2})′} log(x − τy)dμ_{∗}{r → γ_{τ}r}(x, y).

Proof. This follows from (14) and Theorem 2.5, after noting that L_{τ} = ℤ_{p}^{2} when τ = τ_{Q} and Q is an element of F^{D}.

3 p-adic L-functions

3.1 The Mazur-Kitagawa p-adic L-function

Let χ : (ℤ/mℤ)^{×}⟶{±1} be a primitive quadratic Dirichlet character of conductor m with χ(−1) = w_{∞}, and let

τ(χ) := ∑_{a=1}^{m} χ(a)e^{2πia/m}

denote the Gauss sum attached to χ. For each k ∈ U ∩ℤ^{≥2}, and 1 ≤ j ≤ k−1 with j odd, the expression

L^{∗}(f_{k}, χ, j) := \frac{(j − 1)!τ(χ)}{(−2πi)^{j−1}Ω_{k}} L(f_{k}, χ, j)  (24)

belongs to K_{f_{k}}; it is called the algebraic part of the special value L(f_{k}, χ, j). Recall that the period Ω_{k} that appears in the definition of L^{∗}(f_{k}, χ, j) was chosen at the end of Section 2.1 and depends on the sign w_{∞} that was fixed in that section, and therefore on the parity of χ.

One defines L^{∗}(f_{k}^{♯}, χ, j) similarly, by replacing f_{k} by f_{k}^{♯} in the definition above. Note that

L^{∗}(f_{k}, χ, j) = (1 − χ(p)a_{p}(k)^{−1}p^{k−1−j})L^{∗}(f_{k}^{♯}, χ, j).  (25)

The measures μ_{∗}{r → s} can be used to define the Mazur-Kitagawa twovariable p-adic L-function of (k, s) ∈ U × X by the rule:

L_{p}(f_{∞}, χ, k, s) = ∑_{a=1}^{m} χ(pa) ∫_{ℤ_{p}^{×}×ℤ_{p}^{×}}(x − \frac{pa}{m}y)^{s−1} y^{k−s−1}dμ_{∗} \{∞ → \frac{pa}{m}\} .

This function satisfies the following interpolation property with respect to special values of the classical L-functions L^{∗}(f_{k}, χ, j).

Theorem 3.1. Suppose that k ∈ U ∩ ℤ^{≥2}, and that 1 ≤ j ≤ k − 1 satisfies χ(−1) = (−1)^{j−1}w_{∞}. Then

L_{p}(f_{∞}, χ, k, j) = λ(k)(1 − χ(p)a_{p}(k)^{−1}p^{j−1})L^{∗}(f_{k}, χ, j).

Proof. See Theorem 1.12 of [BD05].

It will be useful to have a formula expressing L_{p}(f_{∞}, χ, k, j) in terms of the special value L^{∗}(f_{k}^{♯}, χ, j). Theorem 3.1 and equation (25) imply that

L_{p}(f_{∞}, χ, k, j) =

λ(k)(1 − χ(p)a_{p}(k)^{−1}p^{j−1})(1 − χ(p)a_{p}(k)^{−1}p^{k−1−j})L^{∗}(f_{k}^{♯}, χ, j).

In particular, specialising at j = k/2 one obtains

Corollary 3.2. Suppose that χ satisfies

χ(−1) = (−1)^{k/2−1}w_{∞}.

Then for all k ∈ U ∩ ℤ^{≥2},

L_{p}(f_{∞}, χ, k, k/2) = λ(k)(1 − χ(p)a_{p}(k)^{−1}p^{\frac{k}{2} −1})^{2}L^{∗}(f_{k}^{♯}, χ, k/2).

Note that the Euler factor appearing in this last expression is a perfect square.

3.2 p-adic L-functions attached to real quadratic fields

Given Q = Ax^{2} + Bxy + Cy^{2} in F^{D}, let τ = τ_{Q} and τ̄ be the roots of the quadratic polynomial Q(x, 1), ordered as in (15). These belong to K, and can be viewed as elements of ℂ_{p} via the chosen embedding of K into ℂ_{p}.

The quadratic form Q has a stabiliser in Γ_{0}(M) of rank one, generated by the element γ_{τ} ∈ Γ_{0}(M) normalised as in (12). Let ϵ = cτ + d denote the corresponding fundamental unit of K. By analogy with the definition of the Mazur-Kitagawa p-adic L-function, it is tempting to associate to f_{∞} and Q a “two-variable p-adic L-function” by the rule:

L_{p}(f_{∞}, Q, k, s) := A^{\frac{k}{2} −1} ∫_{(ℤ_{p}^{2})′}(x − τy)^{s−1}(x − τ̄y)^{k−s−1}dμ_{∗}\{r → γ_{τ}r\}(x, y).

Note that this expression depends on the choice of base point r in a crucial way: replacing r by r′ ∈ ℙ has the effect of modifying L_{p}(f_{∞}, Q, k, s) by the term

A^{\frac{k}{2} −1}(1 − ϵ^{k−2s}) ∫_{(ℤ_{p}^{2})′}(x − τy)^{s−1}(x − τ̄y)^{k−s−1}dμ_{∗}\{r → r′\}(x, y).

It follows that the restriction of L_{p}(f_{∞}, Q, k, s) to the central critical line s = k/2 is independent of the choice of r. This motivates the following definitions.

Definition 3.3. Let r ∈ ℙ be any base point. The partial square root p-adic L-function attached to f_{∞} and Q is the function of k ∈ U defined by

L_{p}(f_{∞}, Q, k) := ∫_{(ℤ_{p}^{2})′} Q(x, y)^{\frac{k−2}{2}} dμ_{∗}\{r → γ_{τ}r\}(x, y).

Let χ be a fixed genus character of G_{D}. This character is said to be even (resp. odd) if it cuts out a totally real, resp. imaginary, quadratic extension of K. If χ is even (resp. odd), then the associated Dirichlet characters χ_{1} and χ_{2} are both even (resp. odd). Recall the sign at infinity w_{∞} that was chosen in defining the modular symbols I_{f_{k}} and in choosing the Shimura period Ω_{k}.

Definition 3.4. Let χ be a genus character. Assume that w_{∞} = 1 if χ is even, and that w_{∞} = −1 if χ is odd.

1. The square root p-adic L- function attached to f_{∞} and χ is the function of k ∈ U defined by

L_{p}(f_{∞}/K, χ, k) := ∑_{σ∈G_{D}} χ(σ)L_{p}(f_{∞}, Q^{σ}, k).

2. The p-adic L-function attached to f_{∞} and χ is the function of k ∈ U

defined by

L_{p}(f_{∞}/K, χ, k) := L_{p}(f_{∞}/K, χ, k)^{2}.

We now prove the interpolation property for L_{p}(f_{∞}/K, χ, k) which justifies its designation as a p-adic L-function.

Theorem 3.5. For all k ∈ U ∩ ℤ^{≥2},

L_{p}(f_{∞}/K, χ, k) = λ(k)^{2}(1 − a_{p}(k)^{−2}p^{k−2})^{2}D^{\frac{k−2}{2}} L^{∗}(f_{k}^{♯}/K, χ, k/2), where

L^{∗}(f_{k}^{♯}/K, χ, k/2) = \frac{(\frac{k}{2} − 1)!^{2}\sqrt{D}}{(2πi)^{k−2}Ω_{k}^{2}}L(f_{k}^{♯}/K, χ, k/2).  (26)

Proof. By definition, we have:

L_{p}(f_{∞}/K, Q, k) = ∫_{(ℤ_{p}^{2})′} Q(x, y)^{\frac{k−2}{2}} dμ_{∗}\{r → γ_{Q}r\}(x, y)
= λ(k)(1 − a_{p}(k)^{−2}p^{k−2})I_{f_{k}^{♯}}\{r → γ_{Q}r\}(Q^{\frac{k−2}{2}} ),

where the last equality follows from Proposition 2.4. Hence L_{p}(f_{∞}/K, χ, k)

= λ(k)^{2}(1 − a_{p}(k)^{−2}p^{k−2})^{2} ( ∑_{σ∈G_{D}} χ(σ)I_{f_{k}^{♯}}\{r → γ_{Q^{σ}}r\} ((Q^{σ})^{\frac{k−2}{2}} ))^{2}
= λ(k)^{2}(1 − a_{p}(k)^{−2}p^{k−2})^{2}(2πi)^{2}Ω_{k}^{−2}𝕃_{χ},  (27)

where

𝕃_{χ} = ( ∑_{σ∈G_{D}} χ(σ) ∫_{z_{0}}^{γ_{Q}z_{0}} f_{k}^{♯}(z)Q^{σ}(z, 1)^{\frac{k−2}{2}} dz)^{2} .

The crucial ingredient in the proof of Theorem 3.5 is Theorem 6.3.1 of [Po], which asserts that

𝕃_{χ} = D^{\frac{k−1}{2}} (2π)^{−k} (\frac{k − 2}{2})!^{2}L(f_{k}^{♯}/K, χ, k/2).  (28)

It follows readily from (27) and (28) that

L_{p}(f_{∞}/K, χ, k) = λ(k)^{2}(1 − a_{p}(k)^{−2}p^{k−2})^{2}D^{\frac{k−2}{2}} L^{∗}(f_{k}^{♯}/K, χ, k/2), as was to be shown.

3.3 A factorisation formula

We come to the following factorisation relating the Mazur-Kitagawa p-adic L-functions (more precisely, their restrictions to the central critical line) with the p-adic L-functions attached to the real quadratic field K in the previous section.

Theorem 3.6. For all k ∈ U,

L_{p}(f_{∞}/K, χ, k) = D^{\frac{k−2}{2}} L_{p}(f_{∞}, χ_{1}, k, k/2)L_{p}(f_{∞}, χ_{2}, k, k/2).

Proof. By comparing the Euler product expansions on both sides (and noting that Ind_{G_{K}}^{G_{ℚ}}(χ) = χ_{1} ⊕ χ_{2}) we see that

L(f_{k}^{♯}/K, χ, k/2) = L(f_{k}^{♯}, χ_{1}, k/2)L(f_{k}^{♯}, χ_{2}, k/2).  (29)

Gauss’s classical calculation of quadratic Gauss sums, in the case of fundamental discriminants, shows that

τ(χ_{i}) = \sqrt{D_{i}}, i = 1, 2.  (30)

Combining (29) and (30) with definitions (24) and (26), we see that

L^{∗}(f_{k}^{♯}/K, χ, k/2) = L^{∗}(f_{k}^{♯}, χ_{1}, k/2)L^{∗}(f_{k}^{♯}, χ_{2}, k/2).

Also, because p is inert in K, the Dirichlet characters χ_{1} and χ_{2} satisfy χ_{1}(p) = −χ_{2}(p), and hence the product of the Euler factors at p appearing in Corollary 3.2 with χ replaced by χ_{1} and χ_{2} is equal to the Euler factor appearing in Theorem 3.5. Hence Theorem 3.5 and Corollary 3.2 imply that for all k ∈ U ∩ ℤ^{≥2},

L_{p}(f_{∞}/K, χ, k) = D^{\frac{k−2}{2}} L_{p}(f_{∞}, χ_{1}, k, k/2)L_{p}(f_{∞}, χ_{2}, k, k/2). (31)

Since U ∩ ℤ^{≥2} is dense in U, and the two sides of (31) are continuous on U, they necessarily agree on this region.

4 Proof of the main result

We begin by noting the following connection between the Stark-Heegner point P_{χ} and the leading term of the p-adic L-function that was introduced in Section 3.2. This relation can be viewed as a somewhat exotic p-adic variant of the Gross-Zagier formula since it relates Stark-Heegner points to derivatives of p-adic L-series. It is even more closely in the spirit of the main Theorem of [BD98] and its extension to Hida families stated in Theorem 4.9 of [BD05].

Recall first from Definition 3.3 that, for all Q ∈ F^{D},

L_{p}(f_{∞}/K, Q, 2) = ∫_{(ℤ_{p}^{2})′} dμ_{∗}{r → γ_{τ}r}(x, y) = μ_{f}{r → γ_{τ}r}(ℙ_{1}(ℚ_{p})) = 0.

It follows that L_{p}(f_{∞}/K, χ, 2) = 0 for all characters χ of G_{D}.

Theorem 4.1. For all genus characters χ of G_{D},

\frac{d}{dk} L_{p}(f_{∞}/K, χ, k)_{k=2} = \frac{1}{2} (1 − χ_{1}(−M)w_{M}) log_{E}(P_{χ}).

Proof. By definition for each Q ∈ Γ_{0}(M)\F^{D} with associated roots τ and τ̄, we have

\frac{d}{dk} L_{p}(f_{∞}, Q, k)_{k=2} = \frac{1}{2} ∫_{(ℤ_{p}^{2})′} (log(x − τy) + log(x − τ̄y)) dμ_{∗}\{r → γ_{τ}r\}
= \frac{1}{2}(J_{τ} + τ_{p}J_{τ})
= \frac{1}{2}(J_{τ} − w_{M}J_{τ^{στ}}),

where the last equality is a consequence of equation (16). It follows from Proposition 1.8 that

\frac{d}{dk} L_{p}(f_{∞}/K, χ, k)_{k=2} = \frac{1}{2} (1 − χ_{1}(−M)w_{M}) ( ∑_{σ∈G_{D}} χ(σ)J_{τ^{σ}}) ,

as was to be proved.

Corollary 4.2. For all genus characters χ of G_{D},

\frac{d^{2}}{dk^{2}} L_{p}(f_{∞}/K, χ, k)_{k=2} = \{ \begin{matrix}2 log_{E}^{2}(P_{χ}) & if χ_{1}(−M) = −w_{M} \\ 0 & if χ_{1}(−M) = w_{M}. \end{matrix}

We are now ready to prove Theorem 1 of the introduction.

Theorem 4.3. Let χ be the genus character attached to the pair of Dirichlet characters χ_{1} and χ_{2}. Suppose that E has at least two primes of multiplicative reduction, and that χ_{1}(−M) = −w_{M}.

1. There is a point P_{χ} ∈ E(H_{χ})^{χ} and t ∈ ℚ^{×} such that

log_{E}(P_{χ}) = t log_{E}(P_{χ}).

2. The point P_{χ} is of infinite order if and only if L′(E/K, χ, 1) ≠ 0.

Proof. The proof proceeds in three stages, which are parallel to Kronecker’s “solution to Pell’s equation” described in the last paragraph of the introduction.

1. Corollary 4.2 expresses the logarithm of the Stark-Heegner point P_{χ} in terms of special values of L-series:

log_{E}^{2}(P_{χ}) = \frac{1}{2} \frac{d^{2}}{dk^{2}} L_{p}(f_{∞}/K, χ, k)_{k=2}.  (32)

2. Theorem 3.6 asserts that

L_{p}(f_{∞}/K, χ, k) = D^{\frac{k−2}{2}} L_{p}(f_{∞}, χ_{1}, k, k/2)L_{p}(f_{∞}, χ_{2}, k, k/2). (33)

Hence we are now reduced to understanding the leading terms of the Mazur-Kitagawa p-adic L-functions attached to χ_{1} and χ_{2} in a neighbourhood of k = 2.

3. Let

sign(E, χ_{j}) := −w_{N}χ_{j}(−N)

denote the sign in the functional equation for the complex L-series L(E, χ_{j}, s). Since χ_{1}(−N)χ_{2}(−N) = ϵ_{K}(−N) = −1, it follows that sign(E, χ_{1}) and sign(E, χ_{2}) are opposite. Order χ_{1} and χ_{2} in such a way that

sign(E, χ_{1}) = −1,  sign(E, χ_{2}) = 1.  (34)

Then χ_{1}(−N) = w_{N}, and the running hypothesis that χ_{1}(−M) = −w_{M} implies that

χ_{1}(p) = −w_{p} = a_{p}.  (35)

Therefore the Mazur-Kitagawa p-adic L-function L_{p}(f, χ_{1}, k, s) has an exceptional zero at (k, s) = (2, 1). Conditions (34) and (35) imply that we are in the situation where Theorem 5.4 of [BD05] can be applied to L_{p}(f, χ_{1}, k, k/2). Hence, this p-adic analytic function vanishes to order ≥ 2 at k = 2, and there is a global point P_{χ_{1}} ∈ E(ℚ(\sqrt{D_{1}})) and a rational number ℓ_{1} ∈ ℚ^{×} satisfying the following three properties:

A.

\frac{d^{2}}{dk^{2}} L_{p}(f_{∞}, χ_{1}, k, k/2)_{k=2} = ℓ_{1} log^{2}(P_{χ_{1}}).  (36)

B. The point P_{χ_{1}} is of infinite order if and only if L′(E, χ_{1}, 1) ≠ 0.

C. The rational number ℓ_{1} satisfies

ℓ_{1} ≡ L^{∗}(f, ψ, 1) (mod (ℚ^{×})^{2}),  (37)

for any primitive quadratic Dirichlet character ψ for which L(f, ψ, 1) ≠ 0 and such that

ψ(ℓ) = χ_{1}(ℓ) for ℓ|M, ψ(p) = −χ_{1}(p).

On the other hand, the quantity

L_{p}(f_{∞}, χ_{2}, 2, 1) = 2L^{∗}(E, χ_{2}, 1) =: 2ℓ_{2}  (38)

is a rational number, which is non-zero if and only of L(E, χ_{2}, 1) does not vanish. Note that in this case ℓ_{1}ℓ_{2} is a rational square by (37). Choose t ∈ ℚ^{×} in such a way that

t^{2} = \{ \begin{matrix}ℓ_{1}ℓ_{2} & if ℓ_{2} ≠ 0; \\ 1 & otherwise, \end{matrix}

and let

P_{χ} := \{ \begin{matrix}P_{χ_{1}} & if L′(E/K, χ, 1) = L′(E, χ_{1}, 1)L(E, χ_{2}, 1) ≠ 0, \\ 0 & otherwise, \end{matrix}

Equations (32), (33), (36) and (38) imply Theorem 4.3, after possibly adjusting the sign of t.

Remark 4.4. The condition χ_{1}(−M) = −w_{M} that is imposed in Theorem 4.3 is needed both in the first and third steps of the argument. When χ_{1}(−M) = w_{M}, the signs in the functional equations for both L_{p}(f_{k}, χ_{1}, s) and L_{p}(f_{k}, χ_{2}, s) are −1.

1. In the case of L_{p}(f_{k}, χ_{1}, s), this arises from the fact that the sign in the classical functional equation for L(f_{k}, χ_{1}, s) is −1, while L_{p}(f_{k}, χ_{1}, s) does not have an exceptional zero, because χ_{1}(p) = w_{p} = −a_{p}.

2. For L_{p}(f_{∞}, χ_{2}, k, s), the classical L-function L(f_{k}, χ_{2}, s) vanishes to even order, but its p-adic counterpart has an exceptional zero and therefore vanishes to odd order at the central critical point. (This latter situation is precisely the one that was studied by Greenberg and Stevens in [GS], where the vanishing of L_{p}(f_{∞}, χ_{2}, k, s) on the central critical line was used to prove the “exceptional zero conjecture” of Mazur, Tate and Teitelbaum.)

Thus, in the setting where χ_{1}(−M) = w_{M}, equation (33) implies that the p-adic L-function L_{p}(f_{∞}/K, χ, k) vanishes identically. Hence no arithmetic information is to be extracted from this function, and a proof of Theorem 1 would seem to require a different approach. (An eventual extension of equation (36) to the setting of Hilbert modular forms attached to the real quadratic field K seems a promising avenue.)

Remark 4.5. In the case where χ is not quadratic, the crucial factorisation (33) ceases to be available. This reflects the presence of a serious obstacle, and the ideas explored in his paper appear to shed no light on the algebraicity of the individual Stark-Heegner points P_{τ} when G_{D} is not of exponent 2. Indeed, the reader will have noted that no “genuinely new” Stark-Heegner points are produced in this article. Rather, it is shown that where the classical theory of Heegner points and the theory of Stark-Heegner points intersect—namely, genus fields—the Stark-Heegner points can be expressed in terms of Heegner points. To go beyond genus characters, one would probably need to develop a theory of “real multiplication” yielding an independent construction of Stark-Heegner points and an “explicit class field theory” for real quadratic fields.

References

[BD98] M. Bertolini and H. Darmon. Heegner points, p-adic L-functions and the Cerednik-Drinfeld uniformization, Inventiones Math 131 (1998) 453–491.

[BD05] M. Bertolini and H. Darmon. Hida families and rational points on elliptic curves. Submitted.

[BDG] M. Bertolini, H. Darmon, and P. Green. Periods and points attached to quadratic algebras. Proceedings of an MSRI workshop on special values of Rankin L-series, H. Darmon and S. Zhang, eds. 323–382.

[BDI] M. Bertolini, H. Darmon and A. Iovita. Modular forms on definite quaternion algebras and Teitelbaum’s conjecture. To be submitted.

[Dar01] H. Darmon. Integration on H_{p} × H and arithmetic applications. Annals of Mathematics 154 (2001) 589–639.

[Das1] S. Dasgupta. Gross-Stark units, Stark-Heegner points, and class fields of real quadratic fields. PhD thesis, Berkeley, May 2004.

[Das2] S. Dasgupta. Stark-Heegner points on modular Jacobians. Ann. Sci. Ecole Norm. Sup. (4) 38 (2005), no. 3, 427–469.

[DD] H. Darmon and S. Dasgupta. Elliptic units for real quadratic fields. Annals of Mathematics 163 (2006) 301–345.

[DG] H. Darmon and P. Green. Elliptic curves and class fields of real quadratic fields: algorithms and evidence. Experimental Mathematics, 11:1 (2002) 37-55.

[DP] H. Darmon and R. Pollack. The efficient calculation of Stark-Heegner points via overconvergent modular symbols. Israel Journal of Mathematics, to appear.

[GS] R. Greenberg, G. Stevens, p-adic L-functions and p-adic periods of modular forms. Invent. Math. 111 (1993), no. 2, 407–447.

[KZ] W. Kohnen and D. Zagier. Modular forms with rational periods. In Modular forms (Durham, 1983), 197–249, Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Horwood, Chichester, 1984.

[Po] A. Popa. Central values of Rankin L-series over real quadratic fields. Compositio Mathematica, to appear.

[Sie] C.L. Siegel. Advanced analytic number theory. Second edition. TIFR Studies in Mathematics, 9. TIFR, Bombay, 1980.

[We] A. Weil. Elliptic Functions according to Eisenstein and Kronecker. Classics in Mathematics, Springer-Verlag, 1976.