Mazur's torsion theorem

Mazur's torsion theorem
Name	Mazur's torsion theorem
Field	Number theory
Discovered by	Barry Mazur
Year	1977
Key results	classification of torsion subgroups of elliptic curves over the rationals

Mazur's torsion theorem provides a complete classification of possible torsion subgroups of elliptic curves defined over the field of rational numbers. The theorem, proven by Barry Mazur in 1977, restricts torsion to a finite list of cyclic groups and product groups, linking deep techniques from Galois theory, algebraic geometry, modular curves, and the arithmetic of elliptic curves. It has ramifications across work by scholars such as Andrew Wiles, Gerhard Frey, Joseph Oesterlé, Ken Ribet, and institutions like Princeton University and Harvard University.

Statement of the theorem

Mazur's torsion theorem states that for an elliptic curve E over Q the torsion subgroup E(Q)tors is isomorphic to one of fifteen groups: cyclic groups Z/NZ for N = 1, 2, ..., 10, 12, or one of the four groups Z/2Z × Z/2N Z for N = 1, 2, 3, 4. The statement connects the arithmetic of elliptic curves studied by John Tate and André Weil with the theory of rational points on modular curves associated to congruence subgroups like Γ0(N) and Γ1(N), involving contributions from Pierre Deligne, Alexander Grothendieck, Jean-Pierre Serre, and Nicholas Katz.

Historical context and motivation

Mazur proved the theorem in the context of a surge of interest following work by Yuri Manin, Heegner, Brauer and Hecke on modular parametrizations and by Gerd Faltings on rational points. The result was motivated by questions raised by Louis Mordell and the Mordell–Weil theorem about the structure of the rational points of elliptic curves, continuing threads from André Weil and Serre about Galois representations attached to torsion points. The pursuit of modular methods that later underpinned the proof of the Taniyama–Shimura–Weil conjecture and the work of Andrew Wiles and Richard Taylor drew on techniques that Mazur refined, influencing research at places like Cambridge University and Princeton University.

Proof overview and key ideas

Mazur's proof combines the study of rational points on modular curves X1(N) with methods from algebraic number theory and deformation theory of Galois representations as developed in the milieu of Iwasawa theory and the Langlands program. He analyzed the cuspidal subgroup and Eisenstein ideals in the Jacobians J1(N), building on ideas of Barry Mazur himself and predecessors including Ken Ribet and Glenn Stevens. The proof uses the modular interpretation of X1(N) via elliptic curves with level structure, reduction properties at primes studied by Jean-Pierre Serre and John Tate, and an application of the method of Mazur's Eisenstein ideal to rule out rational points on X1(N) for large N by relating them to the structure of Hecke algebras studied by Atkin and Lehner.

Classification of torsion subgroups over Q

The classification enumerates the allowed torsion structures explicitly: cyclic Z/NZ for N ∈ {1,2,3,4,5,6,7,8,9,10,12} and the four products Z/2Z × Z/2N Z for N ∈ {1,2,3,4}. Examples realizing each case were constructed by explicit elliptic curves exhibited in tables by researchers including John Cremona and computational databases maintained by Cremona's tables projects and later by researchers at Millennium Prize-linked efforts. The classification interlocks with work on rational parametrizations of modular curves X1(N) for small N studied by Dawson and Ogg, and with explicit rational models obtained by Igusa and Shimura.

Consequences and applications

Mazur's theorem constrains possible torsion in algorithms for computing the rank and the structure of E(Q) used in implementations by John Cremona and William Stein; it impacts the search strategies for integer points studied by Baker and Bilu. It feeds into the proof strategies for results about rational isogenies used by Ken Ribet in context with the Frey curve and the modularity theorem central to Andrew Wiles's proof of Fermat's Last Theorem. The theorem also informs classification problems in arithmetic geometry pursued at institutions such as Institut des Hautes Études Scientifiques and University of Cambridge and underlies constraints used in computational projects by SageMath developers and researchers like William Stein.

Generalizations and related results

Generalizations examine torsion of elliptic curves over number fields beyond Q: theorems by Loïc Merel provide uniform bounds for torsion over degree d extensions, while work by Kamienny, Parent, Derickx, Najman, Clark and Sutherland refine classifications over quadratic, cubic, and higher degree fields. Related topics include the Uniform Boundedness Conjecture for torsion points, modular curves X0(N) and X1(N) studied by Kenku and Momose, and the arithmetic of abelian varieties investigated by Faltings and Zarhin. Contemporary research connects Mazur's insights with explicit modularity lifting theorems by Richard Taylor and Christophe Breuil and computational advances influenced by projects at Boston University and research groups linked to Simons Foundation.

Category:Elliptic curves