Euler systems and the Birch and Swinnerton-Dyer conjecture Henri Darmon (joint work with Massimo Bertolini, Victor Rotger)

The Birch and Swinnerton-Dyer conjecture for an elliptic curve E/ℚ asserts that

(1)  ord_{s=1}L(E, s) = rank(E(ℚ)),

where L(E, s) is the Hasse-Weil L-function attached to E. The scope of the conjecture can be broadened somewhat by introducing an Artin representation

(2)  ϱ : G_{ℚ} ⟶ Aut(V_{ϱ}) ≃ GL_{n}(ℂ),

and studying the Hasse-Weil-Artin L-function L(E, ϱ, s), namely, the L-function attached to H_{et}^{1}(Eℚ̄, ℚ_{p}) ⊗ V_{ϱ}, viewed as a (compatible system of) p-adic representations. The “equivariant Birch and Swinnerton-Dyer conjecture” states that

(3)  ord_{s=1}L(E, ϱ, s) = dim_{ℂ} hom_{G_{ℚ}}(V_{ϱ}, E(H) ⊗ ℂ),

where H is a finite extension of ℚ through which ϱ factors. Denote by BSD_{r}(E, ϱ) the assertion that the right-hand side of (3) is equal to r when the same is true of the left-hand side. Virtually nothing is known about BSD_{r}(E, ϱ) when r > 1. For r ≤ 1, there are the following somewhat fragmentary results, listed in roughly chronological order:

Theorem (Gross-Zagier 1984, Kolyvagin 1989) If ϱ is induced from a ring class character of an imaginary quadratic field, and r ≤ 1, then BSD_{r}(E, ϱ) holds.

Theorem A (Kato, 1990) If ϱ is abelian (i.e., corresponds to a Dirichlet character), then BSD_{0}(E, ϱ) holds.

Theorem B (Bertolini-Darmon-Rotger, 2011) If ϱ is an odd, irreducible, twodimensional representation whose conductor is relatively prime to the conductor of E, then BSD_{0}(E, ϱ) holds.

Theorem C (Darmon-Rotger, 2012) If ϱ = ϱ_{1} ⊗ ϱ_{2}, where ϱ_{1} and ϱ_{2} are odd, irreducible, two-dimensional representations of G_{ℚ} satisfying:

(1) det(ϱ_{1}) = det(ϱ_{2})^{−1}, so that ϱ is isomorphic to its contragredient representation;

(2) ϱ is regular, i.e., there is a σ ∈ G_{ℚ} for which ϱ(σ) has distinct eigenvalues;

(3) the conductor of ϱ is prime to that of E; then BSD_{0}(E, ϱ) holds.

This lecture endeavoured to explain the proofs of Theorems A, B, and C, emphasising the fundamental unity of ideas underlying all three.

The key ingredients are certain global cohomology classes

κ(f, g, h) ∈ H^{1}(ℚ, V_{f} ⊗ V_{g} ⊗ V_{h}(c))

attached to triples (f, g, h) of modular forms of respective weights (k, ℓ, m); here V_{f}, V_{h} and V_{g} denote the Serre-Deligne representations attached to f, g and h, and it is assumed that the triple tensor product of Galois representations admits a Kummer-self-dual Tate twist, denoted V_{f} ⊗ V_{g} ⊗ V_{h}(c). (This is true when the product of nebentype characters associated to f, g and h is trivial.)

When f, g and h are all of weight two and level dividing N, and f is cuspidal, associated to an elliptic curve E, say, the class κ(f, g, h) admits a geometric construction via p-adic étale regulators/Abel-Jacobi images of

(1) Beilinson-Kato elements in the higher Chow group CH^{2}(X_{1}(N), 2) of the modular curve X_{1}(N), when g and h are Eisenstein series of weight two arising as logarithmic derivatives of suitable Siegel units;

(2) Beilinson-Flach elements in the higher Chow group CH^{2}(X_{1}(N)^{2}, 1) when g is cuspidal and h is an Eisenstein series;

(3) Gross-Kudla-Schoen diagonal cycles in the Chow group CH^{2}(X_{1}(N)^{3}), when all forms are cuspidal.

When g and h are of weight one rather than two, and hence, are associated to certain (possibly reducible) odd two-dimensional Artin representations, the construction of κ(f, g, h) via K-theory and algebraic cycles ceases to be available. The class κ(f, g, h) is obtained instead by a process of p-adic analytic continuation, interpolating the geometric constructions at all classical weight two points of Hida families passing through g and h in weight one, and then specialising to this weight. The resulting κ(f, g, h) is called the generalised Kato class atttached to the triple (f, g, h) of modular forms of weights (2, 1, 1).

The generalised Kato classes arising from (p-adic limits of) Beilinson-Kato elements, Beilinson-Flach elements, and Gross-Kudla-Schoen cycles are germane to the proofs of Theorems A, B and C respectively. The key point in all three proofs is an explicit reciprocity law which asserts that the global class κ(f, g, h) is noncristalline at p pecisely when the classical central critical value L(f ⊗ g ⊗ h, 1) = L(E, ϱ, 1) is non-zero. The non-cristalline classes attached to (f, g, h) (of which there are actually four, attached to various choices of ordinary p-stabilisations of g and h) can then be used (by a standard argument involving local and global Tate duality) to conclude that the natural inclusion of E(H) into E(H ⊗ ℚ_{p}) becomes zero when restricted to ϱ_{g} ⊗ ϱ_{h}-isotypic components, and hence, that hom_{G_{ℚ}}(V_{ϱ}, E(H) ⊗ ℂ) is trivial when L(E, ϱ, 1) ≠ 0.

The lecture strived to set the stage for the two that immediately followed, which were both devoted to further developments arising from these ideas:

(1) Victor Rotger’s lecture studied the generalised Kato classes κ(f, g, h) when L(f, g, h, 1) = 0. In that case, they belong to the Selmer group of E/H, and can be viewed as p-adic avatars of L′′(E, ϱ, 1);

(2) Sarah Zerbes’ lecture reported on [LLZ1], [LLZ2], [KLZ] in which the study of Beilinson-Flach elements undertaken in [BDR] is generalised, extended and refined. By making more systematic use of the Euler system properties of Beilinson-Flach elements, notably the possibility of “tame deformations” at primes ℓ ≠ p, the article [KLZ] is also able to establish strong finiteness results for the relevant ϱ-isotypic parts of the Shafarevich-Tate group of E over H, in the setting of Theorem B.

References

[BDR] M. Bertolini, H. Darmon, and V. Rotger, Beilinson-Flach elements and Euler systems II: the Birch and Swinnerton-Dyer conjecture for Hasse-Weil-Artin L-series, submitted.

[DR] H. Darmon and V. Rotger, Diagonal cycles and Euler systems II: the Birch and Swinnerton-Dyer conjecture for Hasse-Weil-Artin L-series, submitted.

[Ka] K. Kato. p-adic Hodge theory and values of zeta functions of modular forms, Cohomologies p-adiques et applications arithmétiques III, Astérisque no. 295 (2004).

[KLZ] G. Kings, D. Loeffler and S. Zerbes, Rankin-Selberg Euler systems and p-adic interpolation, submitted.

[LLZ1] A. Lei, D. Loeffler, and S. Zerbes, Euler systems for Rankin-Selberg convolutions, Annals of Mathematics, to appear.

[LLZ2] A. Lei, D. Loeffler, and S. Zerbes, Euler systems for modular forms over imaginary quadratic fields, submitted.