Serre's conjectures

Serre's conjectures
Name	Serre's conjectures
Field	Number theory
Proposed by	Jean-Pierre Serre
Date	1970s
Status	partly proved

Serre's conjectures are a collection of influential predictions by Jean-Pierre Serre about relationships between Galois group representations, modular forms, and cohomological invariants that shaped late 20th‑century number theory and influenced work in algebraic geometry, representation theory, and arithmetic algebraic number theory. They include a precise modularity conjecture for two‑dimensional mod p representations of the absolute Galois group of Q, a conjecture about second cohomology groups for simply connected semisimple groups, and related expectations tying local field behavior to global automorphic phenomena. The conjectures motivated major advances by researchers at institutions such as the Institute for Advanced Study, University of California, Berkeley, Harvard University, Princeton University, Stanford University, and projects associated with the Langlands program.

Statement and historical context

Serre formulated his mod p modularity conjecture in the 1970s and articulated a separate cohomological conjecture about Galois cohomology for simply connected groups; both emerged from Serre's work on Galois representations attached to elliptic curves and modular forms such as those studied by Pierre Deligne, Kenneth Ribet, Goro Shimura, and Gerald Faltings. The mod p conjecture relates continuous two‑dimensional representations of the absolute Galois group of Q unramified outside finitely many primes to reduction modulo p of Hecke eigenforms in spaces studied by Atkin–Lehner theory and motivated by the Taniyama–Shimura conjecture addressed by Andrew Wiles and Richard Taylor. The cohomological conjecture (often called Conjecture II) predicts vanishing of the Galois cohomology group H^2 for certain simply connected semisimple algebraic groups over global fields, a theme connected to work of Claude Chevalley, Armand Borel, Jean-Louis Colliot‑Thélène, and Serre himself. These conjectures connect to the Fontaine–Mazur conjecture, the Langlands–Tunnell theorem, and structural results used by John Coates, Barry Mazur, Mazur–Wiles techniques, and later developments at Imperial College London and Université Paris‑Sud.

Serre's modularity conjecture (mod p)

The modularity conjecture posits that an odd, irreducible, continuous two‑dimensional representation rho: Gal(Qbar/Q) -> GL2(F_pbar) arises from a cuspidal eigenform of a specific weight, level, and character; the precise recipe for weight and level involves Serre's conductor and local restrictions at primes dividing p developed using ideas of Évariste Galois, Hecke operator theory, and congruences studied by Jean‑Louis Nicolas and A. O. L. Atkin. Serre predicted a minimal weight k(rho) determined by the restriction of rho to a decomposition group at p (linking to inertia group structure analyzed by Jean‑Pierre Serre and Jean‑M. Fontaine), and a level N(rho) given by the Artin conductor tied to local ramification invariants studied by Alexander Grothendieck and Henri Carayol. The conjecture implies that reductions of p‑adic Galois representations attached to newforms produce exactly the irreducible odd mod p representations, connecting to the Eichler–Shimura relation and the theory of modular curves such as X_0(N) and X_1(N) explored by Barry Mazur and Kenneth A. Ribet.

Serre's conjecture II and Galois cohomology

Conjecture II concerns the vanishing of H^2(k,G) for a global field k and simply connected semisimple algebraic group G, asserting that principal homogeneous spaces under G satisfy a local‑global principle akin to the Hasse principle studied by Hasse, Helmut Hasse, and L. J. Mordell. This cohomological statement is tied to classification results by Claude Chevalley, structural theory of algebraic groups from Armand Borel and Jacques Tits, and later cohomological techniques of Serre and Jean‑Louis Colliot‑Thélène. Conjecture II has implications for rationality questions explored in work by Serge Lang and Manjul Bhargava, and relates to period–index problems considered by Grothendieck and Alexander Merkurjev.

Proofs and partial results

The mod p modularity conjecture was proved by a sequence of breakthroughs culminating in the work of Chandrashekhar Khare and Jean‑Pierre Wintenberger, building on modularity lifting theorems of Andrew Wiles, Richard Taylor, Fred Diamond, Toby Gee, Mark Kisin, and innovative level‑lowering techniques of Ken Ribet and Barry Mazur. Earlier special cases were handled using the Langlands–Tunnell theorem and the Taylor–Wiles method combined with Iwasawa theory insights by John Coates and Barry Mazur. Conjecture II has been settled for many classes of groups by results of Chernousov, Platonov, Voskresenskii, and work on the Hasse principle by Borovoi and Sansuc; however, full generality over arbitrary global fields remains open in some cases and continues to be explored by researchers at CNRS laboratories and at the Mathematical Sciences Research Institute.

Consequences and applications

Serre's conjectures revolutionized the study of modular forms, elliptic curve arithmetic, and the Langlands program, yielding tools for proving instances of the Birch and Swinnerton‑Dyer conjecture in special cases via modularity results used by Gross–Zagier type formulas and work of Kolyvagin. The mod p results enabled progress on deformation theory of Galois representations developed by Barry Mazur and applied in proofs by Wiles and Taylor; applications include explicit level‑lowering criteria used by Ken Ribet in the proof of the epsilon conjecture and descent arguments for rational points on modular curves used by Mazur and Darren Glass. Conjecture II influences rationality and classification problems for algebraic groups relevant to Noether's problem and central simple algebras studied by Albert and Brauer, and informs computational approaches to modular forms at centers such as Max Planck Institute for Mathematics.

Examples and counterexamples in related settings

Concrete examples illustrating Serre's mod p recipe include reductions of Galois representations attached to classical newforms such as those studied by Hecke, Deligne–Serre, and explicit elliptic curves like the Fermat curve examples and curves used by Wiles in his work on semistable elliptic curves. Counterexamples to naive generalizations appear over totally real fields, CM fields, and function fields where obstructions tied to local deformation rings studied by Brian Conrad and Richard Taylor arise; related pathologies were identified in work on mod l representations over imaginary quadratic fields by Francesc Fité and Mark Kisin. Analogues of Conjecture II have been tested and sometimes fail for exotic groups over fields with complex arithmetic such as function fields over finite fields investigated by Vladimir Drinfeld and Laurent Lafforgue.

Category:Conjectures in number theory