Non-Isomorphic Abelian Varieties with the Same Arithmetic

TL;DR

This paper constructs two non-isomorphic abelian varieties over the rationals that share identical arithmetic invariants across all number fields, challenging assumptions about the uniqueness of such invariants.

Contribution

It introduces explicit examples of non-isomorphic abelian varieties with identical arithmetic properties over all number fields, revealing limitations of current invariants in distinguishing them.

Findings

Constructed non-isomorphic abelian varieties with identical invariants

Showed invariants like Mordell--Weil groups and Tate modules are insufficient for differentiation

Highlighted gaps in the discriminative power of arithmetic invariants

Abstract

We construct two abelian varieties over $\mathbb{Q}$ which are not isomorphic, but have isomorphic Mordell--Weil groups over every number field, isomorphic Tate modules and equal values for several other invariants.