Shafarevich conjecture

Shafarevich conjecture
Name	Shafarevich conjecture
Named after	Igor Shafarevich
Field	Number theory; Algebraic geometry
Introduced	1962
Status	Partially proved; variants resolved
Related	Mordell conjecture; Faltings theorem; Fontaine–Mazur conjecture

Shafarevich conjecture

The Shafarevich conjecture is a set of influential conjectures formulated by Igor Shafarevich connecting arithmetic properties of varieties over number fields with finiteness statements. It originally predicted finiteness of isomorphism classes of certain algebraic objects over a fixed global field with prescribed ramification, inspiring work by John Tate, Gerd Faltings, Jean-Pierre Serre, and others across Mordell conjecture, Taniyama–Shimura conjecture, Grothendieck's anabelian program, and Fontaine–Mazur conjecture developments.

History and statement

Shafarevich proposed finiteness statements in letters and lectures in the early 1960s while collaborating with Alexander Grothendieck, Andrei Suslin, and contemporaries influenced by problems raised at Steklov Institute of Mathematics and conferences in Moscow. The original articulation concerned finiteness of abelian varieties of fixed dimension over a number field with bounded bad reduction outside a finite set of places, echoing earlier finiteness themes in work by Helmut Hasse and Ernst Witt. The conjecture bifurcates into formulations for abelian varieties and for algebraic curves tied to étale fundamental groups and Galois representations, leading to crosslinks with Shinichi Mochizuki, Friedrich Hirzebruch, Serge Lang, and the research programs at Institut des Hautes Études Scientifiques and Institute for Advanced Study.

Shafarevich conjecture for abelian varieties

The abelian variety version predicts that for a fixed number field K, a finite set S of places of K, and a fixed dimension g, there are only finitely many K-isomorphism classes of principally polarized abelian varietys of dimension g with good reduction outside S. This formulation motivated work by Gerd Faltings whose proof of the Mordell conjecture and results on Tate conjecture instances relied on reduction techniques developed in collaboration with ideas from Jean-Pierre Serre, Barry Mazur, and John Tate. The abelian case ties into Néron model theory developed partly by André Néron and later refinements by Alexander Grothendieck and Raynaud; it also connects to moduli spaces studied by David Mumford, Igor Dolgachev, and Pierre Deligne in the context of level structures and arithmetic compactifications investigated at C.E.M. de Picard conferences and by research groups at Princeton University and Harvard University.

Shafarevich conjecture for algebraic curves and fundamental groups

An anabelian variant predicts finiteness for isomorphism classes of smooth projective curves of genus g over K with good reduction outside S, framed via the structure of the étale fundamental group and outer Galois actions developed by Alexander Grothendieck in his Esquisse d'un Programme and pursued by Shinichi Mochizuki and Akio Tamagawa. The curve version links to Grothendieck's conjectures on reconstructions of varieties from étale fundamental groups and to results by Friedrich Pop and Hideo Tamagawa about Galois sections and birational anabelian geometry investigated at workshops at Max Planck Institute for Mathematics and Université Paris-Sud. It further connects with work on mapping class groups by William Thurston, moduli stack geometry by Deligne–Mumford, and arithmetic of coverings studied by André Weil.

Proofs and partial results

Faltings proved the abelian variety case using methods culminating in his proof of the Mordell conjecture, building on earlier work by Serre on Galois representations and Tate on endomorphisms of abelian varieties. Subsequent refinements and alternative proofs involved contributions from Richard Taylor, Kazuya Kato, Ken Ribet, and Loïc Merel within the Langlands program context and modularity lifting techniques emerging from research at Institute for Advanced Study, Princeton University, and Harvard University. Partial progress on curve and anabelian versions came from work by Mochizuki, Tamagawa, Fedor Bogomolov, and Yuri Manin using geometric and p-adic Hodge theoretic tools related to Fontaine and Jannsen theories. Counterexamples to naive generalizations and obstacles were identified in research by Peter Scholze, Aise Johan de Jong, and Matthew Emerton exploring limits of p-adic uniformization and perfectoid techniques developed at Institute for Advanced Study and University of Bonn.

Related conjectures and consequences

The conjecture interlaces with Mordell conjecture (proved by Faltings), the Tate conjecture, and the Fontaine–Mazur conjecture on p-adic Galois representations, as well as with predictions from the Birch and Swinnerton-Dyer conjecture in arithmetic of abelian varieties. It influenced developments in anabelian geometry and motivated research programs at IHES, MPI, MSRI, and Clay Mathematics Institute. Consequences include finiteness theorems used in diophantine geometry by Joseph Silverman, constraints on moduli stacks studied by Deligne–Mumford and Ching-Li Chai, and implications for automorphic lifting results pursued by Langlands, Clozel, and Harris.

Examples and counterexamples

Concrete cases verifying the conjecture include lists of elliptic curves over Q with prescribed conductor studied by John Cremona and classification results for abelian surfaces over quadratic fields by researchers at University of Cambridge and Imperial College London. Explicit finiteness computations used algorithms developed by Andrew Wiles collaborators and databases compiled by LMFDB contributors, drawing on computational projects at University of Warwick and Boston University. Known counterexamples apply only to overly broad generalizations; specific pathological families identified by Igor Shafarevich discussions and later analyzed by Fedor Bogomolov and Yuri Zarhin illustrate limits where naive extension to higher-dimensional varieties or non-proper models fails.

Category:Conjectures in number theory