Infinitely many pairs of non-isomorphic elliptic curves sharing the same BSD invariants

TL;DR

This paper proves the existence of infinitely many pairs of non-isomorphic elliptic curves over the rationals that share all BSD invariants and various other invariants, yet have different j-invariants.

Contribution

It establishes the existence of infinitely many such pairs, highlighting a surprising level of invariance among non-isomorphic elliptic curves.

Findings

Existence of infinitely many pairs of elliptic curves sharing BSD invariants.

Shared invariants include Mordell--Weil groups, Tate--Shafarevich groups, Tamagawa numbers, regulators, and real periods.

Pairs have distinct j-invariants, showing invariants do not determine isomorphism class.

Abstract

We prove that there exist infinitely many pairs of non-isomorphic elliptic curves over $\mathbb{Q}$ sharing the same BSD invariants -- including their Mordell--Weil groups, Tate--Shafarevich groups, Tamagawa numbers, regulators, and real periods -- and their Kodaira symbols and minimal discriminants, while having distinct $j$-invariants.