Computing level one Hecke eigensystems (mod p)

We describe an algorithm for enumerating the set of level 1 systems of Hecke eigenvalues arising from modular forms (mod p).Comment: 19 pages, 3 figures, 1 tabl

Resonance equals reducibility for A-hypergeometric systems

Classical theorems of Gel'fand et al., and recent results of Beukers, show that non-confluent Cohen-Macaulay A-hypergeometric systems have reducible monodromy representation if and only if the continuous parameter is A-resonant. We remove both the confluence and Cohen-Macaulayness conditions while simplifying the proof.Comment: 9 pages, final versio

On Birch and Swinnerton-Dyer's cubic surfaces

In a 1975 paper of Birch and Swinnerton-Dyer, a number of explicit norm form cubic surfaces are shown to fail the Hasse Principle. They make a correspondence between this failure and the Brauer--Manin obstruction, recently discovered by Manin. We generalize their work, making use of modern computer algebra software to show that a larger set of cubic surfaces have a Brauer--Manin obstruction to the Hasse principle, thus verifying the Colliot-Th\'el\`ene--Sansuc conjecture for infinitely many cubic surfaces

Two-divisibility of the coefficients of certain weakly holomorphic modular forms

We study a canonical basis for spaces of weakly holomorphic modular forms of weights 12, 16, 18, 20, 22, and 26 on the full modular group. We prove a relation between the Fourier coefficients of modular forms in this canonical basis and a generalized Ramanujan tau-function, and use this to prove that these Fourier coefficients are often highly divisible by 2.Comment: Corrected typos. To appear in the Ramanujan Journa

Effectivity of Brauer-Manin obstructions on surfaces

We study Brauer-Manin obstructions to the Hasse principle and to weak approximation on algebraic surfaces over number fields.Comment: 13 page