Stefan Patrikis

COMPUTATIONAL VERIFICATION OF THE BIRCH AND SWINNERTON-DYER CONJECTURE FOR INDIVIDUAL ELLIPTIC CURVES

We describe theorems and computational methods for verifying the Birch and Swinnerton-Dyer conjectural formula for specific elliptic curves over Q of analytic ranks 0 and 1. We apply our techniques to show that if E is a non-CM elliptic curve over Q of conductor < 1000 and rank 0 or 1, then the Birch and Swinnerton-Dyer conjectural formula for the leading coefficient of the L-series is true for E, up to odd primes that divide either Tamagawa numbers of E or the degree of some rational cyclic isogeny with domain E. Since the rank part of the Birch and Swinnerton-Dyer conjecture is a theorem for curves of analytic rank 0 or 1, this completely verifies the full conjecture for these curves up to the primes excluded above.

Automorphy and irreducibility of some l -adic representations

In this paper we prove that a pure, regular, totally odd, polarizable weakly compatible system of

Deformations of Galois representations and exceptional monodromy

l

Anabelian geometry and descent obstructions on moduli spaces

-adic representations is potentially automorphic. The innovation is that we make no irreducibility assumption, but we make a purity assumption instead. For compatible systems coming from geometry, purity is often easier to check than irreducibility. We use Katzs theory of rigid local systems to construct many examples of motives to which our theorem applies. We also show that if

Deformations of Galois representations and exceptional monodromy, II: raising the level

F

Variations on a theorem of Tate

is a CM or totally real field and if

Verification of the Birch and Swinnerton - Dyer Conjecture for Specific Elliptic Curves

\pi

Generalized Kuga-Satake theory and rigid local systems, I: the middle convolution

is a polarizable, regular algebraic, cuspidal automorphic representation of

Mumford-Tate groups of polarizable Hodge structures

GL_n(\A_F)

Residual Irreducibility of Compatible Systems

, then for a positive Dirichlet density set of rational primes