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# Class Note for MATH 105 at UA

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Date Created: 02/06/15

OUTLINE OF A COURSE ON ELLIPTIC CURVES AND GROSSZAGIER THEOREMS OVER FUNCTION FIELDS ARIZONA WINTER SCHOOL 2000 DOUGLAS ULMER 1 REVIEW OF ELLIPTIC CURVES OVER FUNCTION FIELDS 0 De nitions and examples Constant7 isotriVial7 and non constant curves 0 The Mordell Weil theorem 0 Constant curves The lattices of Elkies7 Shioda7 et al 0 Torsion is uniformly bounded Ranks are unbounded o L functions o Grothendieck7s analysis of L functions gives analytic continuation7 functional equation 0 L functions should be Viewed as functions of characters of the idele class group 0 Zarhin7s theorem X gt gt LEX determines E up to isogeny o The conjecture of Birch and Swinnerton Dyer 0 Work of Tate and Milne ords1LEs 2 Rank with equality if and only if M is nite 0 Outline of the proof 7 The elliptic surface SFq corresponding to LEs det17 q sFrlH for a certain H Q H26 Oz 7 Points on E correspond to curves on S Heights are essentially intersection numbers 7 Cycle classes of curves give rise to zeroes of the L function Finiteness of M gt weak BSD comes from the Kummer se quence on S and M Br 0 Other work Brown7 Riick Tipp7 Longhi7 Pal 0 References Gross7 Zarhin7 Groth7 Milne807 Tate667 Milne757 C Z7 Cross in Storrs Date March 157 2000i DOUGLAS ULMER 2 AUTOMORPHIC FORMS AND ANALYTIC MODULARITY o Additive characters7 multiplicative characters7 conductors and real parts 0 De nition of x4K7 i57 automorphic forms of level K and central character 15 o Analogue with functions on upper half plane The double coset space X where automorphic forms live 0 X is the set of isomorphism classes of rank 2 vector bundles with level structure up to twisting by a line bundle 0 Structure of X Riemann Roch and stability 0 Petersson inner product 0 Cusp forms 0 Hecke operators7 new and old forms 0 Fourier expansions o L functions 0 Functional equations 0 Harmonic forms 0 Constructions of forms7 classically and in terms of vector bundles Eisenstein series Poincare series Theta functions Converse theorems Deligne7s theorem there is a form f such that LEs LU 5 Drinfeld7s geometric Langlands construction 0 Interesting linear functionals on AK are represented as PIP with interesting forms f E x4K7 b 0 Half of the Gross Zagier computation is to nd the Fourier expan sion of the form representing f gt gt L Kf7 1 0 References Weill7 Serre7 Gek7 Del7 Drin83 GROSS ZAGIER OVER FUNCTION FIELDS 3 3 DRINFELD MODULAR CURVES AND GEOMETRIC MODULARITY o The ring A of functions regular outside 00 o For k of characteristic p7 EndkGa is the twisted polynomial ring k7 7 Ta apT 0 De nition of Drinfeld modules Rank7 characteristic7 height 0 Examples 0 Morphisms 0 Division points 0 lsogenies o Endomorphisms 0 Complex multiplication 0 Level structures 0 Modular curves 0 Analytic description of Drinfeld modular curves 0 The adelic version of the analytic description 0 The building map 0 Drinfeld reciprocity relating the cohomology of the modular curve to automorphic forms 0 Geometric modularity via Drinfeld reciprocity7 Deligne converse theorems7 and Zarhin 0 References Drin747 D H7 G R7 AB7 Ohio DOUGLAS ULMER 4 OVERVIEW OF THE GROSS ZAGIER COMPUTATION AND APPLICATION TO ELLIPTIC CURVES o Heegner points on X001 existence7 construction Galois action The Heegner point PK 6 J0n 0 Goal L Kf7 1 c htPKf c a non zero constant for new eigen forms f E AP0nool l 0 Key reduction do it for all f at once i Let hm be the form such that f7 hmp1p L Kf7 1 i Let hazy be the form with Fourier coef cients ltPK7TmPKgtht A formal Hecke algebra argument shows that the goal is equiv alent to the equality hm chm Prove this coef cient by coef cient 0 The analytic computation Rankin7s method shows that LKfs f7hsp1p where hS is the product of a CM form theta series and an Eisenstein series which is a function of s 7 Compute a trace to make the level of hS noo Take the derivative at s 1 h ih 51 Do a harmonic projection nd hm harmonic such that f7 hmp1p f7 hp1p for all harmonic forms f o The algebraic computation lnterpret height as a sum of local intersection numbers 7 At nite places7 intersection number counts the number of isogenies between certain Drinfeld modules L y over nite rings OvWZ Use the moduli interpretation of points 7 Count these isogenies using the ideal theory of the quaternion ring End At 00 there is no convenient moduli interpretation Compute the local height using a Green7s function7 exactly as in the original G Z This is a very analytic way to calculate a ratio nal number7 but it meshes well with analytic aspects of the harmonic projection calculation 0 Application to elliptic curves Show ords1 LEs S 1 i BSD for EF by using G Z formula and non vanishing results for L functions In function eld case7 non vanishing results are used for some useful preliminary reductions7 and to nd a good 0 References G Z AB CZl Del lDHl Drin74 Drin83 Gek GR Gross Groth GZ Milne75 MilneSO Ohio Serre Storrs Tate66 Tate75 Weill Zarhin GROSS ZAGIER OVER FUNCTION FIELDS 5 REFERENCES Gekeler EiUi et all Eds Drinfeld Modules Modular Schemes and Ap plications Proceedings of a workshop at AldenBiesen Belgium 1996 World Scienti c Singapore 1997 Cox Di and Zucker Si Intersection numbers of sections of elliptic sur faces lnventiones Math 53 1979 1744 Deligne Pi Les constantes des equations fonctionnelles des functions L In Modular functions of one variable 11 Lecture Notes in Math 349 1973 5017597 Deligne Pi and Husemoller Di Survey of Drinfeld modules Contempo rary Math 67 1987 25791 Drinfeld ViGi Elliptic modules Russian Mat Sb NS 94 136 1974 5947627 656 Drinfeld ViGi Twodimensional ladic representations of the fundamen tal group of a curve over a nite eld and automorphic forms on GL2 Amer 1 Math 105 1983 857114 Gekeler EiUi Automorphe Formen uber FqT mit kleinem Fuhrer Abhi Mathi Semi UniVi Hamburg 55 1985 1117146 Gekeler EiUi and Reversat Mi Jacobians of Drinfeld modular curves 1 Reine AngeWi Math 476 1996 27793 Gross Bi Hi Group representations and lattices J Amer Math Soc 3 1990 9297960 Grothendieck Al Formule de Lefschetz et rationalit des fonctions L Seminaire Bourbaki 196566 Expose 279 Gross Bi and Zagier Di Heegner points and derivatives of Lseries lnventi Math 84 1986 2257320 Milne JiSi On a conjecture of Artin and Tate Annals of Math 102 1975 5177533 Milne JiSi Etale Cohomology Princeton University Press Princeton 1980 Goss Di et al Eds The Arithmetic of Function Fields Proceedings of a conference at Columbus OH 1991 de Gruyter Berlin 1992 Serre JiPi Trees Springer Berlin 1980 Cornell Cl and Silverman Ji Eds Arithmetic Geometry Proceedings of a conference at Storrs CT 1985 Springer 1986 Tate 1 On the conjecture of Birch and SwinnertonDyer and a geomet ric analog Seminaire Bourbaki 196566 Expose 306 Tate 1 Algorithm for determining the type of a singular ber in an elliptic pencil 1n Modular Forms of One Variable 1V Lecture Notes in Math 476 1975 33752 Weil A Dirichlet series and automorphic forms Lecture Notes in Math 189 Springer Berlin 1971 Zarhin Jul Gi A niteness theorem for isogenies of abelian varieties over function elds of nite characteristic Russian Funkcionali Anali i PriloZeni 8 1974 31734

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