We compute Hecke eigenvalues by numerical analytic methods.

We give an algebraic-geometric construction of a differential operator on Siegel modular forms mod $p$.

We give conditions for lifting Siegel modular forms (mod $p$) to characteristic zero.

We present an implementation in the functional programming language Haskell of the PLE decomposition of matrices over division rings.

We describe the effect of differential operators on the corresponding Galois representations.

We study the number of initial Fourier coefficients necessary to distinguish newforms of fixed level and weight.

Given two Siegel eigenforms of different weights, we determine explicit sets of Hecke eigenvalues for the two forms that must be distinct.

We study the number of initial Fourier coefficients necessary to distinguish eigenforms modulo a prime ideal.

We describe a computational approach to the verification of Maeda's conjecture and verify this conjecture for all weights less than 12,000.

We carry out explicit computations of vector-valued Siegel modular forms of degree two and level one and verify some conjectural congruences between these and classical modular forms.

We describe an algorithm for enumerating the set of level one systems of Hecke eigenvalues arising from modular forms (mod p).

We give an algorithm for enumerating all odd semisimple two-dimensional Galois representations (mod $p$) unramified outside $p$.

We study the number of initial Fourier coefficients that two cuspidal eigenforms can have in common.

We show that systems of Hecke eigenvalues occurring in the spaces of Siegel modular forms are the same as the systems occurring in the spaces of Siegel cusp forms, of possibly different weight.

We derive an explicit upper bound for the number of systems of Hecke eigenvalues coming from Siegel modular forms (mod $p$).

We show that the systems of Hecke eigenvalues given by Siegel modular forms (mod $p$) are the same as the ones given by algebraic modular forms (mod $p$) on a quaternionic unitary group.

© 2017 Alexandru Ghitza ·