# Period relations for Riemann surfaces with many automorphisms

## TL;DR

This paper introduces a new explicit bound on the number of algebraic relations among periods of Riemann surfaces with many automorphisms, improving previous estimates and aiding in understanding their Jacobians.

## Contribution

It develops a novel bound based on vector-valued automorphic forms, surpassing Wolfart's genus-based bound, and demonstrates its application in computing endomorphism algebra dimensions.

## Key findings

- New explicit bound for period relations improves previous estimates.
- Bound can be used to determine Jacobian endomorphism algebra dimensions.
- Examples show the bound's effectiveness in identifying complex multiplication.

## Abstract

By employing the theory of vector-valued automorphic forms for non-unitarizable representations, we provide a new bound for the number of linear relations with algebraic coefficients between the periods of an algebraic Riemann surface with many automorphisms. The previous best-known general bound for this number was the genus of the Riemann surface, a result due to Wolfart. Our new bound significantly improves on this estimate, and it can be computed explicitly from the canonical representation of the Riemann surface. As observed by Shiga and Wolfart, this bound may then be used to estimate the dimension of the endomorphism algebra of the Jacobian of the Riemann surface. We demonstrate with a few examples how this improved bound allows one, in some instances, to actually compute the dimension of this endomorphism algebra, and to determine whether the Jacobian has complex multiplication.