OUTLINE OF A COURSE ON

ELLIPTIC CURVES AND GROSS-ZAGIER THEOREMS OVER FUNCTION FIELDS

ARIZONA WINTER SCHOOL 2000

DOUGLAS ULMER

1. Review of elliptic curves over function fields

• Definitions and examples. Constant, isotrivial, and non-constant curves.

• The Mordell-Weil theorem.

• Constant curves. The lattices of Elkies, Shioda, et. al.

• Torsion is uniformly bounded. Ranks are unbounded.

• L-functions.

• Grothendieck’s analysis of L-functions gives analytic continuation, functional equation.

• L-functions should be viewed as functions of characters of the idèle class group.

• Zarhin’s theorem: χ ↦ L(E, χ) determines E up to isogeny.

• The conjecture of Birch and Swinnerton-Dyer.

• Work of Tate and Milne: ord_{s=1} L(E, s) ≥ Rank_{ℤ} E(F) with equality if and only if is finite.

• Outline of the proof:

– The elliptic surface E/F_{q} corresponding to E/F.

– L(E, s) = det(1 − q^{−s}Fr|H) for a certain H ⊆ H^{2}(\overline{E}, Q_{ℓ})

– Points on E correspond to curves on E. Heights are essentially intersection numbers.

– Cycle classes of curves give rise to zeroes of the L-function.

– Finiteness of ⇔ weak BSD comes from the Kummer se-

quence on E and = Br.

• Other work: Brown, Rück-Tipp, Longhi, Pàl.

• References: [Gross], [Zarhin], [Groth], [Milne80], [Tate66], [Milne75], [C-Z], Gross in [Storrs].

Date: March 15, 2000.

2. Automorphic forms and analytic modularity

• Additive characters, multiplicative characters, conductors and real parts.

• Definition of A(K, ϕ), automorphic forms of level K and central character ϕ.

• Analogue with functions on upper half plane. The double coset space X where automorphic forms live.

• X is the set of isomorphism classes of rank 2 vector bundles with level structure (up to twisting by a line bundle).

• Structure of X. (Riemann-Roch and stability.)

• Petersson inner product.

• Cusp forms.

• Hecke operators, new and old forms.

• Fourier expansions.

• L-functions.

• Functional equations.

• Harmonic forms.

• Constructions of forms, classically and in terms of vector bundles: – Eisenstein series

– Poincaré series

– Theta functions

– Converse theorems

– Deligne’s theorem: there is a form f such that L(E, s) = L(f, s)

– Drinfeld’s geometric Langlands construction

• Interesting linear functionals on A(K, ϕ) are represented as PIP with interesting forms f ∈ A(K, ϕ).

• Half of the Gross-Zagier computation is to find the Fourier expansion of the form representing f ↦ L′_{K}(f, 1).

• References: [Weil], [Serre], [Gek], [Del], [Drin83].

3. Drinfeld modular curves and geometric modularity

• The ring A of functions regular outside ∞

• For k of characteristic p, End_{k}(G_{a}) is the twisted polynomial ring k{τ}, τa = a^{p}τ.

• Definition of Drinfeld modules. Rank, characteristic, height.

• Examples.

• Morphisms.

• Division points.

• Isogenies.

• Endomorphisms.

• Complex multiplication.

• Level structures.

• Modular curves.

• Analytic description of Drinfeld modular curves.

• The adelic version of the analytic description.

• The building map.

• Drinfeld reciprocity: relating the cohomology of the modular curve to automorphic forms.

• Geometric modularity via Drinfeld reciprocity, Deligne + converse theorems, and Zarhin.

• References: [Drin74], [D-H], [G-R], [AB], [Ohio].

4. Overview of the Gross-Zagier computation and application to elliptic curves

• Heegner points on X_{0}(𝔫): existence, construction, Galois action. The Heegner point P_{K} ∈ J_{0}(𝔫)(K).

• Goal: L′_{K}(f, 1) = c ht(P_{K,f}) (c a non-zero constant) for new eigenforms f ∈ A(Γ_{0}(𝔫∞), | · |^{−2}).

• Key reduction: do it for all f at once.

– Let h_{an} be the form such that (f, h_{an})_{P IP} = L′_{K}(f, 1).

– Let h_{alg} be the form with Fourier coefficients ⟨P_{K}, T_{𝔪}P_{K}⟩_{ht}.

– A formal Hecke algebra argument shows that the goal is equivalent to the equality h_{an} = c h_{ht}. Prove this coefficient by coefficient.

• The analytic computation.

– Rankin’s method shows that L_{K}(f, s) = (f, h_{s})_{P IP} where h_{s} is the product of a CM form (theta series) and an Eisenstein series which is a function of s.

– Compute a trace to make the level of h_{s} 𝔫∞.

– Take the derivative at s = 1: h = \frac{d}{ds}h|_{s=1}.

– Do a “harmonic projection”: find h_{an} harmonic such that

(f, h_{an})_{P IP} = (f, h)_{P IP} for all harmonic forms f.

• The algebraic computation – Interpret height as a sum of local intersection numbers.

– At finite places, intersection number counts the number of isogenies between certain Drinfeld modules x, y over finite rings O_{v}/(π_{v}^{n}). (Use the moduli interpretation of points.)

– Count these isogenies using the ideal theory of the quaternion ring End(x).

– At ∞ there is no convenient moduli interpretation. Compute the local height using a Green’s function, exactly as in the original G-Z. This is a very analytic way to calculate a rational number, but it meshes well with analytic aspects of the harmonic projection calculation.

• Application to elliptic curves. Show ord_{s=1} L(E, s) ≤ 1 ⇒ BSD for E/F by using G-Z formula and non-vanishing results for Lfunctions. In function field case, non-vanishing results are used for some useful preliminary reductions, and to find a good K/F.

• References: [G-Z].

References

[AB] Gekeler, E.-U. et al., Eds: “Drinfeld Modules, Modular Schemes, and Applications” (Proceedings of a workshop at Alden-Biesen, Belgium 1996) World Scientific, Singapore, 1997

[C-Z] Cox, D. and Zucker, S.: Intersection numbers of sections of elliptic sur-

faces Inventiones Math. 53 (1979), 1–44

[Del] Deligne, P.: Les constantes des équations fonctionnelles des fonctions L In “Modular functions of one variable, II” (Lecture Notes in Math. 349) (1973) 501–597

[D-H] Deligne, P. and Husemoller, D.: Survey of Drinfeld modules. Contempo-

rary Math. 67 (1987), 25–91

[Drin74] Drinfeld, V.G.: Elliptic modules (Russian) Mat. Sb. (N.S.) 94 (136) (1974), 594–627, 656.

[Drin83] Drinfeld, V.G.: Two-dimensional l-adic representations of the fundamental group of a curve over a finite field and automorphic forms on GL(2) Amer. J. Math. 105 (1983), 85–114

[Gek] Gekeler, E.-U.: Automorphe Formen über F_{q}(T) mit kleinem Führer. Abh. Math. Sem. Univ. Hamburg 55 (1985), 111–146

[G-R] Gekeler, E.-U., and Reversat, M.: Jacobians of Drinfeld modular curves.

J. Reine Angew. Math 476 (1996), 27–93

[Gross] Gross, B. H.: Group representations and lattices J. Amer. Math. Soc. 3 (1990), 929–960

[Groth] Grothendieck, A.: Formule de Lefschetz et rationalité des fonctions L Séminaire Bourbaki 1965/66, Exposé 279

[G-Z] Gross, B. and Zagier, D.: Heegner points and derivatives of L-series

Invent. Math. 84 (1986), 225–320

[Milne75] Milne, J.S.: On a conjecture of Artin and Tate Annals of Math. 102 (1975), 517–533

[Milne80] Milne, J.S.: “Etale Cohomology” Princeton University Press, Princeton, 1980

[Ohio] Goss, D. et al. Eds: “The Arithmetic of Function Fields” (Proceedings of a conference at Columbus, OH 1991) de Gruyter, Berlin, 1992

[Serre] Serre, J.-P.: “Trees” Springer, Berlin, 1980

[Storrs] Cornell, G. and Silverman, J. Eds.: “Arithmetic Geometry” (Proceedings of a conference at Storrs, CT 1985), Springer, 1986

[Tate66] Tate, J.: On the conjecture of Birch and Swinnerton-Dyer and a geometric analog. Séminaire Bourbaki 1965/66, Exposé 306

[Tate75] Tate, J.: Algorithm for determining the type of a singular fiber in an elliptic pencil. In “Modular Forms of One Variable IV” (Lecture Notes in Math. 476) (1975), 33–52

[Weil] Weil, A.: “Dirichlet series and automorphic forms” (Lecture Notes in Math. 189) Springer, Berlin, 1971

[Zarhin] Zarhin, Ju. G.: A finiteness theorem for isogenies of abelian varieties over function fields of finite characteristic (Russian) Funkcional. Anal. i Priložen. 8 (1974), 31–34