Integral Frobenius for Abelian Varieties with Real Multiplication

TL;DR

This paper introduces an integral Frobenius concept for abelian varieties with real multiplication, providing a method to construct it when the field has class number one and applying it to specific modular abelian surfaces.

Contribution

It develops an integral analogue of Frobenius for abelian varieties with real multiplication and offers a construction method for class number one fields, with practical computations.

Findings

Constructed integral Frobenius for certain abelian varieties

Applied the method to three modular abelian surfaces

Demonstrated compatibility with existing algorithms

Abstract

In this paper we introduce the concept of $\mathit{integral}$ $\mathit{Frobenius}$ to formulate an integral analogue of the classical compatibility condition linking the collection of rational Tate modules $V_\lambda(A)$ arising from abelian varieties over number fields with real multiplication. Our main result gives a recipe for constructing an integral Frobenius when the real multiplication field has class number one. By exploiting algorithms already existing in the literature, we investigate this construction for three modular abelian surfaces over $\mathbf{Q}$.