Boundedness of denominators of special values of the L-functions for modular forms

TL;DR

This paper refines results on the boundedness of denominators of algebraic parts of special values of L-functions associated with modular forms, extending to products of Hecke L-functions for primitive forms.

Contribution

It provides a more precise version of Boecherer's boundedness result and extends the analysis to products of Hecke L-functions for primitive forms.

Findings

Bounded denominators for algebraic parts of L-values at critical points.

Extension of boundedness results to products of Hecke L-functions.

Refined bounds improve understanding of algebraic properties of L-values.

Abstract

For a cuspidal Hecke eigenform $F$ for $Sp_n(Z)$ and a Dirichlet character $\chi$ let $L(s,F,\chi,St)$ be the standard $L$-function of $F$ twisted by $\chi$. Boecherer showed the boundedness of denominators of the algebraic part of $L(m,F,\chi,St)$ at a critical point $m$ when $\chi$ varies. In this paper, we give a refined version of his result We also prove a similar result for the products of Hecke $L$ functions of primitive forms for $SL_2(Z)$.