Arithmetic of elliptic curves over function fields

Transcription

1 Arithmetic of elliptic curves over function fields Massimo Bertolini and Rodolfo Venerucci The goal of this seminar is to understand some of the main results on elliptic curves over function fields of positive characteristic. It can roughly be divided into two parts. (1) The Birch and Swinnerton-Dyer conjecture. Unlike the case of number fields, a lot is known about the Birch and Swinnerton-Dyer conjecture over function fields. Let k be the function field of a curve over the finite field F q, where q = p d, and let E be an elliptic curve over k. A result by Tate and Milne shows that the the rank of E(k) is always bounded above by the order of vanishing at s = 1 of the L-function L(E/k, s). Moreover, the equality holds if and only if the Shafarevich Tate group of E/k is finite. Note that L(E/k, s) is a simpler object than its counterpart over number fields, since it can be described as a rational function in q s by results of Grothendieck, Deligne, et al. The Shafarevich Tate group is isomorphic to the Brauer group of the surface E over F q associated to E, and the Birch and Swinnerton-Dyer conjecture for E is equivalent to the Tate conjecture for E. (2) Modularity. Let E/k be as above. Assume that E is not isotrivial (i.e. it is not isomorphic to a constant curve over a finite extension of k). Let be a prime of k at which the j-invariant of E has a pole, so that E/k is a Tate curve. The properties of L(E/k, s) mentioned before imply, by results of Deligne and Jacquet Langlands, that L(E/k, s) is associated to a cuspidal automorphic representation of GL 2 over k. This implies the existence of a modular parametrisation from the jacobian of a Drinfeld modular curve to E. Drinfeld modular curves parametrise rank 2 Drinfeld modules, and admit a -adic uniformisation by the Drinfeld upper half plane associated to k. This uniformisation is the analogue for function fields of the Cerednik-Drinfeld uniformisation studied in our previous seminar on Scholze s paper on the Lubin Tate tower. Moreover, Drinfeld modules are closely related to special instances of the Drinfeld shtoukas we studied during this semester. Lectures Let k = F q (C) be the function field of a (smooth, projective and geometrically irreducible) curve over F q. Fix a closed point of C (that is a place of k), and let O = Γ(C, O C ), k and C denote respectively the ring of elements of k regular away from, the completion of k at and the completion of an algebraic 1

2 2 MASSIMO BERTOLINI AND RODOLFO VENERUCCI closure of k. For simplicity by elliptic curve over k we mean an elliptic curve E/k which is not isotrivial (that is whose j-invariant does not belong to F q ). April 20: Elliptic curves over function fields and the BSD conjecture. This talk recalls the basic properties of elliptic curves E defined over the function field k. It discusses the Mordell Weil theorem for E(k), introduces the L-function L(E/k, s) and Shafarevich Tate group X(E/k), and states the (weak and refined) Birch and Swinnerton-Dyer conjecture for E/k. References: Lecture 1 of [Ulm11]. The original Séminaire Bourbaki of Tate [Tat95] is also an excellent and quite concise reference, for this talk and more generally for the first half of the seminar. April 27: Zeta functions and the Weil conjectures. The Weil conjectures for projective surfaces over finite fields play a dominant role in the proof of the known results on the Birch and Swinnerton-Dyer conjecture over function fields. This talk recalls the main results on the cohomology of surfaces over finite fields (Lefschetz formula, Poincaré duality, purity, etc.) and how the Weil conjectures result from them. References: Lecture 0, Sections 3 and 4 of [Ulm11] contain a brief summary. For detailed expositions see [FK88, Chapter IV], [Mil80, Chapter VI, Sections 12 and 13] and the references therein. May 4. The Shioda Tate formula. The talk describes a formula due to Shioda and Tate, which places the Birch and Swinnerton-Dyer conjecture for elliptic curves over function fields in the framework of a conjecture of Tate for surfaces over finite fields. To an elliptic curve E/k one associates its minimal model π : E C, a smooth projective surface over F q with generic fibre E. The Shioda Tate formula relates the rank r(e) of E(k) to the rank r(e) of the Neron Severi group NS(E) of E. More precisely the difference r(e) r(e) is expressed as a sum of local terms depending on the geometry of the singular fibres of π. The same sum of local terms gives the discrepancy between the order of vanishing of L(E/k, s) at s = 1 and the order of the pole at s = 1 of the zeta function ζ(e, s) of E/F q. As a consequence the weak BSD conjecture for E/k is equivalent to the equality (1) ord s=1 ζ(e, s) = rank Z NS(E). Tate conjectured that the previous equality holds for every nice surface over F q. References: Section 4 of [Ulm14] explains the proof of the Shioda Tate formula. For the comparison between ord s=1 L(E/k, s) and ord s=1 ζ(e, s) see the proof of Proposition 6.7 of loc. cit., Proposition 3.3 of [Gor79] or [Ulm11, Lecture 3, Proposition 6.1]. The preprint [Tam16] by M. Tamiozzo (ask Matteo for a copy) is also an excellent reading. May 11. Brauer groups and the Tate conjecture. Let E be a nice surface over F q. By exploiting the cycle class map from NS(E) to the l-adic cohomology of E, one shows that the inequality holds in (1), and Tate s conjecture is equivalent to the finiteness of the Brauer group Br(E) of E, which in turn is equivalent to the finiteness of its l-primary part Br(E) l for any rational prime l. The talk gives an outline of the proof of these results. References: Sections 5 and 6 of [Ulm14] and Lecture 2 (especially Sections 9 and 10) of [Ulm11]. It would be nice to discuss here also Theorem 6.4 of [Ulm14] on the refined Artin Tate conjecture.

3 ARITHMETIC OF ELLIPTIC CURVES OVER FUNCTION FIELDS 3 May 18. Brauer groups and Shafarevich Tate groups. Let E/k be an elliptic curve and let E C be its minimal model. A result of Artin Tate and Grothendieck proves that Br(E) is isomorphic to X(E/k). The talk sketches a proof of this result and discusses the applications to the (weak and refined) Birch and Swinnerton-Dyer conjecture. References: a (long and likely difficult) detailed proof of the isomorphism between Br(E) and X(E/k) is in Section 4 of [Gro68], while Section 5.3 of [Ulm14] gives a brief sketch of the proof. For the applications to the weak (resp., refined) BSD conjecture see [Ulm14, Section 2.2.2] and [Ulm11, Lecture 3, Section 8] (resp., Sections and of [Ulm14]). May 25. Analytic modularity of elliptic curves. In this talk one uses results of Grothendieck, Weil and Jacquet Langlands to prove the analytic modularity of an elliptic curve over E/k. This means that L(E/k, s) equals the L-function L(ϕ E, s) of an automorphic cusp form on GL 2 (k). On the one hand this uses the circle of ideas introduced in the first part of the seminar (especially in «Zeta functions and the Weil conjectures»), on the other hand it serves as a motivation for the the second part of the seminar. References: The overall strategy is explained in Section 3.2 of [Ulm04], to which we refer for precise references on the work of Weil and Jacquet Langlands. Lecture 4, Sections 1 and 2 of [Ulm11] contain a clear exposition of the needed results of Grothendieck. June 8. Drinfeld upper half-plane. This talk starts the second part of the seminar, dedicated to Drinfeld modular varieties and geometric modularity of elliptic curves. It introduces the analytic (i.e. local) theory of Drinfeld modular curves, which are obtained as arithmetic quotients of the Drinfeld upper half-plane. The Drinfeld upper half plane over C is defined by H = C k. It admits a natural reduction map r : H T into the Bruhat Tits tree T of PGL 2 (k ), which in turn gives rise to a rigid analytic structure on H. To every congruence subgroup Γ of GL 2 (O) one associates a Drinfeld modular curve M Γ. It is a (totally split) affine curve over C whose associated analytic space is isomorphic to the union of a finite number of copies of the quotient Γ\H (where Γ acts by Möbius transformations). In analogy with the complex setting, the curve M Γ classifies isomorphism classes of rank-two O-lattices in C with Γ-level structures, and admits a natural compactification M Γ. One can also define a Hecke algebra acting as a ring of correspondences on M Γ. References: Sections , 2.1 and 2.5 of [GR96]. See also Section 6 of [Dri74] and Chapter III of [DH87]. June 15. Drinfeld modules and modular schemes. This talk introduces the Drinfeld moduli scheme M Γ O associated to a congruence subgroup of GL 2 (O), which provides a «good» canonical model of MΓ over O. Recall that M Γ classifies rank-two O-lattices in C with Γ-level structures. In analogy with the complex setting, one would like to interpret M Γ as the solution of a moduli problem on Schemes/C which extends naturally to a moduli problem on Schemes/O. Remarkably Drinfeld pursues this program by considering rank-two Drinfeld modules as the correct analogues in the function field setting of elliptic curves. Roughly speaking a Drinfeld O-module over a O-algebra R is a structure of O-module on the additive group over R. There are notions of rank, division points, isogenies, level structures etc. for Drinfeld O-modules, under which they behave like abelian varieties (of dimension equal to «half the rank»). Notably the category of rank-two Drinfeld modules over C turns out to be equivalent to the category of O-lattices in C, compatibly with Γ-level structures. One then defines M Γ (resp., its

4 4 MASSIMO BERTOLINI AND RODOLFO VENERUCCI compactification M Γ ) as the moduli space of (resp., generalised) rank-two Drinfeld O-modules. More generally one defines Drinfeld moduli schemes M r Γ for rank-r Drinfeld O- modules (hence M Γ = M 2 Γ ). Time permitting, it would be nice to describe M1 Γ and its relation to class field theory as a motivation for the much more difficult Drinfeld reciprocity law for r = 2 discussed in later talks. References: Sections and 2.6 of [GR96] contain a brief account, to be used as a guideline. For more details on Drinfeld moduli schemes see [Dri74, Sections 2, 3, 5 and 9] and [DH87, Chapters I and II]. The relation between M 1 Γ and class field theory is explained in Section 8 of [Dri74]. June 22. Drinfeld modular forms and harmonic cocycles. Drinfeld modular forms for Γ are holomorphic functions on H which satisfy suitable functional equations under the action of Γ. As in the complex setting Drinfeld modular forms of weight two can be described as global sections of the sheaf of differentials on the modular curve M Γ. By exploiting the intimate connection between the analytic structure on M Γ and the quotient graph T Γ = Γ\T, one can give a combinatorial description of weight-two Drinfeld modular forms as suitable C -valued harmonic cocycles on T Γ. The aim of the talk is to explain Hecke-equivariant versions of these results. References: Sections 2 and 3 of [GR96]. June 29. Drinfeld reciprocity law I. This talk and the next one are devoted to the Drinfeld reciprocity law. This fundamental result gives an automorphic description of the cohomology of Drinfeld modular varieties. More precisely, let l p be a rational prime and denote by H ( M O k, Ql ) the direct limit of the étale cohomology groups H ( M Γ k, Z l ) Zl Ql, where Γ runs over the congruence subgroups of GL 2 (O). Let V (k, l) be the space generated by the Q l -valued automorphic forms on GL 2 (k) which are special at, and let sp l be a special Galois representation of G = Gal( k /k ) (viz. a non-split extension of the trivial G -representation Q l by its twist Q l (1)). Drinfeld reciprocity law states that (2) H 1 ( M O k, Ql ) = V (k, l) Ql sp l as GL 2 (A f k ) Gal( k/k)-modules, where A f k is the ring of finite adeles of k. The proof of (2) has two main steps. The cohomology H ( M Γ O k, Ql ) is isomorphic to the (rigid-étale) cohomology H ( M Γ, Q l ) of the analytic modular curve M Γ over C, hence one reduces to the computation of the latter. In the first step one shows that H 1 ( M Γ, Q l ) is essentially isomorphic to the tensor product of sp l with the space of Q l -valued harmonic cocycles on T Γ = Γ\T. In the second step one describes the special representation of GL 2 (k ) in terms of harmonic cocycles on T, and combines this description with Step 1 to deduce (2). This talk states Drinfeld reciprocity law and (roughly) covers the first step in the proof. The next talk then outlines the second step in the proof (and time permitting the applications to the Jacquet Langlands conjecture). References: The isomorphism (2) is proved in Section 10 of [Dri74], and the applications to the Jacquet Langlands conjecture are given in Section 11. The exposition in [DH87] is probably less demanding: the first step of the proof is explained in Chapter 4 and the second step in the first two sections of Chapter 5. July 6: Drinfeld reciprocity law II. Cf. the previous talk. July 13: Geometric modularity of elliptic curves. From the talk «Analytic modularity of elliptic curves» we known that the L-function of an elliptic curve E/k agrees with that an automorphic form ϕ E on GL 2 (k). Under the additional

5 ARITHMETIC OF ELLIPTIC CURVES OVER FUNCTION FIELDS 5 assumption that E/k has split multiplicative reduction, the form ϕ E is special at, hence appears in the cohomology of M Γ by the Drinfeld reciprocity law. As explained in the talk, this can used to prove that (up to isogeny) E/k appears in the Jacobian of M Γ. References: Section 8 of [GR96]. July 20: Gross Zagier formula and BSD in rank one. As in the complex setting one can define Drinfeld Heegner points in the Jacobian of Drinfeld modular curves and use them to construct k-rational points on elliptic curves via a modular parametrisation. Analogues of the Gross Zagier formula in this setting have been proved by Rück Tipp, Ulmer et alii, and can be used to complete the proof of the BSD conjecture in analytic rank one. The aim of this talk is to discuss Gross Zagier type formulae and their application to the BSD conjecture. References: Section 3 of [Ulm04] and [RT00]. References [DH87] Pierre Deligne and Dale Husemoller. Survey of Drinfel d modules. In Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), volume 67 of Contemp. Math., pages Amer. Math. Soc., Providence, RI, , 4 [Dri74] V. G. Drinfel d. Elliptic modules. Mat. Sb. (N.S.), 94(136): , 656, , 4 [FK88] Eberhard Freitag and Reinhardt Kiehl. Étale cohomology and the Weil conjecture, volume 13 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, Translated from the German by Betty S. Waterhouse and William C. Waterhouse, With an historical introduction by J. A. Dieudonné. 2 [Gor79] W. J. Gordon. Linking the conjectures of Artin-Tate and Birch-Swinnerton-Dyer. Compositio Math., 38(2): , [GR96] E.-U. Gekeler and M. Reversat. Jacobians of Drinfeld modular curves. J. Reine Angew. Math., 476:27 93, , 4, 5 [Gro68] Alexander Grothendieck. Le groupe de Brauer. III. exemples et compléments. In Dix exposés sur la cohomologie des schémas, volume 3 of Adv. Stud. Pure Math., pages North-Holland, Amsterdam, [Mil80] James S. Milne. Étale cohomology, volume 33 of Princeton Mathematical Series. Princeton University Press, Princeton, N.J., [RT00] Hans-Georg Rück and Ulrich Tipp. Heegner points and L-series of automorphic cusp forms of Drinfeld type. Doc. Math., 5: , [Tam16] M. Tamiozzo. A shioda-tate formula for arithmetic surfaces and applications. Preprint, [Tat95] John Tate. On the conjectures of Birch and Swinnerton-Dyer and a geometric analog. In Séminaire Bourbaki, Vol. 9, pages Exp. No. 306, Soc. Math. France, Paris, [Ulm04] Douglas Ulmer. Elliptic curves and analogies between number fields and function fields. In Heegner points and Rankin L-series, volume 49 of Math. Sci. Res. Inst. Publ., pages Cambridge Univ. Press, Cambridge, , 5 [Ulm11] Douglas Ulmer. Elliptic curves over function fields. In Arithmetic of L-functions, volume 18 of IAS/Park City Math. Ser., pages Amer. Math. Soc., Providence, RI, , 3 [Ulm14] D. Ulmer. CRM lectures on curves and Jacobians over function fields. In Arithmetic Geometry over Global Function Fields, Advanced Courses in Mathematics CRM Barcelona, , 3