 School of Mathematics and Statistics  Research Publications
School of Mathematics and Statistics  Research Publications
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ItemNo Preview AvailableDifferential operators mod p: analytic continuation and consequencesEischen, E ; Flander, M ; Ghitza, A ; Mantovan, E ; McAndrew, A (MATHEMATICAL SCIENCE PUBL, 20210101)

ItemDifferential operators on modular forms mod pGhitza, A (RIMS, 2019)We give a survey of recent work on the construction of differential operators on various types of modular forms (mod p) . We also discuss a framework for determining the effect of such operators on the mod p Galois representations attached to Hecke eigenforms.

ItemAnalytic evaluation of Hecke eigenvalues for Siegel modular forms of degree twoGhitza, A ; Colman, O ; Ryan, NC (Mathematical Sciences Publishers, 2019)The standard approach to evaluate Hecke eigenvalues of a Siegel modular eigenform F is to determine a large number of Fourier coefficients of F and then compute the Hecke action on those coefficients. We present a new method based on the numerical evaluation of F at explicit points in the upper halfspace and of its image under the Hecke operators. The approach is more efficient than the standard method and has the potential for further optimization by identifying good candidates for the points of evaluation, or finding ways of lowering the truncation bound. A limitation of the algorithm is that it returns floating point numbers for the eigenvalues; however, the working precision can be adjusted at will to yield as close an approximation as needed.

ItemComputational Mathematics with SageMathZimmermann, P ; Casamayou, A ; Cohen, N ; Connan, G ; Dumont, T ; Fousse, L ; Maltey, F ; Meulien, M ; Mezzarobba, M ; Pernet, C ; Thiéry, NM ; Bray, E ; Cremona, J ; Forets, M ; Ghitza, A ; Thomas, H (Society for Industrial and Applied Mathematics, 20181210)Flip to almost any random page in this amazing book, and you will learn how to play with and visualize some beautiful part of mathematics.

ItemSOME MODULAR ABELIAN SURFACESCalegari, F ; Chidambaram, S ; Ghitza, A (AMER MATHEMATICAL SOC, 20200101)In this note, we use the main theorem of Boxer, Calegari, Gee, and Pilloni in Abelian surfaces over totally real fields are potentially modular (arXiv:1812.09269, 2018) to give explicit examples of modular abelian surfaces A with Endc A = Z and A smooth outside 2, 3, 5, and 7.