On Modular Computation

On Modular Computation of Standard Basis

In this article I want to report about modular methods to compute standard bases and the implementation in Singular.

On the Computation of Algebraic Modular Forms

In this paper, we revisit the theory of Brandt matrices for algebraic modular forms. We then apply this to the computation of Hilbert-Siegel modular forms of genus 2 over totally real number fields of narrow class number one. Some numerical examples of eigensystems of Hilbert-Siegel modular forms of genus 2 over Q( √ 2) are provided. Introduction Let G/Q be a connected reductive group such that...

On the Computation of the Coefficients of a Modular Form

On modular computation of Groebner bases with integer coefficients

A as a subset of Q[X], so, if I is an ideal of A, then QI is the ideal of Q[X] generated by I. Given a ring R, we denote the natural mapping A → A ⊗ R = R[X] by ι R. This note is devoted to the following algorithmic problem (see [1] for a definition and properties of Gröbner bases of ideals in polynomial rings over Z). Problem (P). Suppose that we have an infinite sequence f 1 , f 2 ,. .. of el...

Computation of ATR Darmon Points on Nongeometrically Modular Elliptic Curves

ATR points were introduced by Darmon as a conjectural construction of algebraic points on certain elliptic curves for which in general the Heegner point method is not available. So far the only numerical evidence, provided by Darmon–Logan and Gärtner, concerned curves arising as quotients of Shimura curves. In those special cases the ATR points can be obtained from the already existing Heegner ...