Effective construction of Hilbert modular forms of half-integral weight

TL;DR

This paper constructs a family of half-integral weight Hilbert modular forms over totally real fields, with Fourier coefficients encoding central L-values, using generalized theta series and quadratic conditions on primes dividing the level.

Contribution

It provides an effective method to construct such forms over arbitrary totally real fields, extending previous work to a broader class of number fields.

Findings

Constructed modular forms whose coefficients encode central L-values.

Formed a family parametrized by quadratic conditions on primes dividing the level.

Applicable over all totally real fields except for odd degree with square levels.

Abstract

Given a Hilbert cuspidal newform $g$ we construct a family of modular forms of half-integral weight whose Fourier coefficients give the central values of the twisted $L$-series of $g$ by fundamental discriminants. The family is parametrized by quadratic conditions on the primes dividing the level of $g$, where each form has coefficients supported on the discriminants satisfying the conditions. These modular forms are given as generalized theta series and thus their coefficients can be effectively computed. Our construction works over arbitrary totally real number fields, except that in the case of odd degree the square levels are excluded. It includes all discriminants except those divisible by primes whose square divides the level.