182 research outputs found
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
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