arXiv subjects
Publications and source records attributed to Alexandru Ghitza.
We describe a computational approach to the verification of Maeda's conjecture for the Hecke operator T2 on the space of cusp forms of level one. We provide experimental evidence for all weights less than 12000, as well as some applications of these results. The algorithm was implemented using the mathematical software Sage, and the code and resulting data were made freely available.
We carry out some computations of vector valued Siegel modular forms of degree two, weight (k,2) and level one. Our approach is based on Satoh's description of the module of vector-valued Siegel modular forms of weight (k, 2) and an explicit description of the Hecke action on Fourier expansions. We highlight three experimental results: (1) we identify a rational eigenform in a three dimensional space of cusp forms, (2) we observe that non-cuspidal eigenforms of level one are not always rational and (3) we verify a number of cases of conjectures about congruences between classical modular forms and Siegel modular forms.
We describe an algorithm for enumerating the set of level 1 systems of Hecke eigenvalues arising from modular forms (mod p).
We present an algorithm for enumerating all odd semisimple two-dimensional mod p Galois representations unramified outside p. We also discuss the implementation of this algorithm in Sage and give a summary of the results we obtained.
We revisit a theorem of Ram Murty about the number of initial Fourier coefficients that two cuspidal eigenforms of different weights can have in common. We prove an explicit upper bound on this number, and give better conditional and unconditional asymptotic upper bounds. Finally, we describe a numerical experiment testing the sharpness of the upper bound in the case of forms of level one.
We show that the systems of Hecke eigenvalues occurring in the spaces of Siegel modular forms (mod p) of fixed dimension g, fixed level N, and varying weight, are the same as the systems occurring in the spaces of Siegel cusp forms with the same parameters and varying weight. In particular, in the case g=1, this says that the Hecke eigensystems (mod p) coming from classical modular forms are the same as those coming from cusp forms. The proof uses restriction to the superspecial locus. We also give a comparison of cusp forms on the Satake compactification versus the toroidal compactifications of Siegel modular varieties.
We derive an explicit upper bound for the number of systems of Hecke eigenvalues coming from Siegel modular forms (mod p) of dimension g and level N relatively prime to p. In the special case of elliptic modular forms (g=1), our result agrees with recent work of G. Herrick.
In a letter to Tate (published in Israel J. Math. in 1996), J.-P. Serre proves that the systems of Hecke eigenvalues given by modular forms (mod p) are the same as the ones given by locally constant functions on an adelic double coset space constructed from the endomorphism algebra of a supersingular elliptic curve. We generalize this result to Siegel modular forms, proving 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, as defined by Gross in Israel J. Math. in 1999. The correspondence is obtained by restricting to the superspecial locus of the moduli space of abelian varieties.
In a letter to Tate, Serre proves that the systems of Hecke eigenvalues given by modular forms (mod p) are the same as the ones given by locally constant functions on an adelic double coset space constructed from the endomorphism algebra of a supersingular elliptic curve. We generalize this result to Siegel modular forms (mod p), proving that the systems of Hecke eigenvalues obtained from them are the same as the ones given by algebraic modular forms (mod p) on a quaternionic unitary group, as defined by Gross. The correspondence is obtained by restricting to the superspecial locus of the moduli space of abelian varieties.