Eichler–Shimura relation

Eichler–Shimura relation
Name	Eichler–Shimura relation
Field	Number theory, Algebraic geometry
Introduced	1950s
Key figures	Martin Eichler, Goro Shimura, André Weil, Jean-Pierre Serre

Eichler–Shimura relation The Eichler–Shimura relation is a foundational theorem connecting Hecke operators on modular forms with Frobenius endomorphisms on Jacobians of modular curves. It links the work of Martin Eichler, Goro Shimura, André Weil, Jean-Pierre Serre and influences research directions at Institute for Advanced Study, Princeton University, Harvard University and University of Tokyo. The relation underpins deep connections exploited in the proofs of the Modularity theorem, the Taniyama–Shimura conjecture and in the study of Galois group representations attached to modular forms.

Background and Definitions

The statement rests on objects defined in the theories developed by Ernst Kummer, Bernhard Riemann, Felix Klein, Heinrich Weber and modernized by Atle Selberg, Hermann Weyl, Erich Hecke and Robert Langlands. One considers modular curves such as X0(N) and X1(N) studied at Hiroshima University and University of Göttingen and their Jacobian varieties first systematized by Carl Friedrich Gauss and Niels Henrik Abel. Hecke operators T_p introduced by Erich Hecke act on spaces S_k(Γ0(N)) of cusp forms constructed in the tradition of Tom M. Apostol and Serge Lang. Over finite fields one has Frobenius endomorphisms studied by Évariste Galois and formalized by Alexander Grothendieck and Jean-Pierre Serre; moreover, the notion of l-adic cohomology crucial to modern formulations was developed by Alexander Grothendieck, Pierre Deligne and Gérard Laumon.

Statement of the Eichler–Shimura Relation

Let f be a normalized newform of weight k = 2 (or more generally a Hecke eigenform) on Γ0(N) as in work of Atkin–Lehner and William F. Osgood; let J0(N) denote the Jacobian of X0(N) as treated by Armand Borel and Haruzo Hida. For a prime p ∤ N, the Hecke operator T_p acts on the étale cohomology H^1_et(X0(N)_{/Qbar}, Q_l) studied by Jean-Pierre Serre and Pierre Deligne. The Eichler–Shimura relation asserts that on H^1_et the characteristic polynomial of Frobenius Frob_p equals X^2 − T_p X + p, mirroring the Hecke polynomial X^2 − a_p X + p where a_p is the eigenvalue of T_p on f; this relation intertwines the action of T_p and the geometric correspondences on J0(N) developed by Yutaka Ihara and Robert Coleman.

Proofs and Key Ideas

Proofs combine algebraic geometry from Alexander Grothendieck and analytic theory from Goro Shimura, exploiting correspondences introduced by Martin Eichler and the theory of l-adic representations crafted by Jean-Pierre Serre and Pierre Deligne. One constructs an algebraic correspondence C_p on X0(N) following methods of Armand Borel and interprets T_p geometrically; using comparison theorems of Alexander Grothendieck and Jean-Pierre Serre between étale cohomology and singular cohomology, one identifies the action of Frobenius with the transpose of the correspondence. Deligne’s purity results and the trace formulas of Atle Selberg and Jacques Tits enter to control eigenvalues; an alternative approach uses p-adic Hodge theory developed by Jean-Marc Fontaine and deformations of Galois representations studied by Barry Mazur.

Consequences and Applications

The relation enables the construction of two-dimensional l-adic Galois representations attached to modular forms, a direction crucial to the work of Andrew Wiles and Richard Taylor on the Taniyama–Shimura conjecture and the proof of Fermat's Last Theorem. It informs the study of L-functions linked to Robert Langlands's reciprocity conjectures and the formulation of the Langlands program at Institute for Advanced Study. In arithmetic geometry it yields results on rational points on modular curves as pursued by Mazur, Barry and Ken Ribet, and it contributes to algorithms in computational number theory developed at Carnegie Mellon University and Massachusetts Institute of Technology for computing coefficients a_p used by John Cremona.

Examples and Computations

In weight 2 one may compute for Δ, the Ramanujan form studied by Srinivasa Ramanujan, and for forms associated to elliptic curves cataloged by John Cremona and Andrew Sutherland; the relation predicts the equality of the trace of Frob_p on H^1 with a_p. For the elliptic curve E: y^2 + y = x^3 − x^2 studied by Gerhard Frey and Noam Elkies one checks the polynomial X^2 − a_p X + p equals the characteristic polynomial of Frobenius in implementations by William Stein and computational packages originating from PARI/GP projects at Université Bordeaux. Explicit examples appear in tables compiled by John Cremona and in data used by Nicolas Katz.

Generalizations and Related Results

Generalizations extend to Hilbert modular forms studied by Goro Shimura and Haruzo Hida, Siegel modular forms treated by Igusa, Jun-ichi and Franz Oesterlé, and to automorphic forms on reductive groups in the framework of Robert Langlands and James Arthur. The Eichler–Shimura paradigm influences the Eichler–Shimura isomorphism in Hodge theory developed by Pierre Deligne and the compatibility of local Langlands correspondences proven by Michael Harris and Richard Taylor. Related results include the work of Ken Ribet on level raising, the Fontaine–Mazur conjecture formulated by Jean-Marc Fontaine and Barry Mazur, and the modularity lifting theorems advanced by Christophe Breuil and Fred Diamond.

Category:Number theory