A note on Jacobians of quasiplatonic Riemann surfaces with complex multiplication

TL;DR

This paper proves that certain quasiplatonic Riemann surfaces have Jacobian varieties with complex multiplication, and provides a criterion for when this occurs, advancing understanding of the interplay between surface symmetries and complex multiplication.

Contribution

It establishes that Jacobians of quasiplatonic Riemann surfaces with specific automorphism groups admit complex multiplication and offers a general criterion for this property.

Findings

Jacobian of surfaces with automorphism group C_2^2 ⋊ C_m admits complex multiplication.

Extension of results to a broader criterion for complex multiplication.

Advances understanding of automorphism groups and Jacobian properties.

Abstract

Let $m \geqslant 6$ be an even integer. In this short note we prove that the Jacobian variety of a quasiplatonic Riemann surface with associated group of automorphisms isomorphic to $C_2^2 \rtimes_2 C_m$ admits complex multiplication. We then extend this result to provide a criterion under which the Jacobian variety of a quasiplatonic Riemann surface admits complex multiplication.