Modular Forms Database: Papers of William Stein

William A. Stein

Algorithms for modular abelian varieties

• An introduction to computing modular forms using modular symbols
• Modular/Manin symbols
• Component groups of optimal quotients of Jacobians
• Component groups of quotients of J_0(N) (ANTS IV)
• Generating the Hecke algebra
• Congruences between modular forms
• Computing characteristic polynomials of Hecke operators when the weight is large

Birch and Swinnerton-Dyer conjecture

• Some Abelian Varieties with Visible Shafarevich-Tate Groups
• Table of Special Values of Modular L-functions
• Visibility of invisible Shafarevich-Tate groups (Utrecht notes, July 2000)
• Visibility of Shafarevich-Tate groups (talk notes)
• Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves

Artin's conjecture

• A mod five approach to modularity of icosahedral Galois representations

Serre's conjecture

• Serre's conjecture modulo pq?

Tables

• Table of Invariants of Modular Abelian Varieties
• Table of nonmaximal orders attached to newforms
• Bounds on discriminants
• Ordinary reduction

Ph.D. Thesis

• Explicit approaches to modular abelian varieties

T-shirts

• Coming soon.

Investigations

• Visualizing Mordell-Weil Groups of elliptic curves using Shafarevich-Tate groups. This also involves constructing Sha of surprising orders.
• There are genus one curves over Q of every odd index
• The field generated by the points of small prime order on an elliptic curve (with Loic Merel)
• X₀(389)
• The first newform such that Q(a_n) =/= Q(a₁,a₂,...) for all n

Combinatorics

• Parity structures and generating functions from Boolean rings

Undergraduate

• Factoring Primes