Small Tamagawa numbers of elliptic curves with isogenies or torsion

TL;DR

This paper investigates the Tamagawa numbers of elliptic curves over rationals with isogenies or torsion points, aiming to bound prime divisors or identify subfamilies with minimal Tamagawa numbers, and explores their behavior in elliptic surface specializations.

Contribution

It provides new bounds on primes dividing Tamagawa numbers and identifies infinite subfamilies with minimal Tamagawa numbers, extending understanding of elliptic curve invariants.

Findings

Bounded the set of primes dividing Tamagawa numbers for certain elliptic curves.

Identified infinite subfamilies with minimal Tamagawa numbers.

Analyzed Tamagawa numbers in specializations of elliptic surfaces.

Abstract

In this article with study Tamagawa numbers of elliptic curves defined over $\mathbb{Q}$ that have isogenies or torsion points. More precisely, our aim is either to bound the set of primes primes that can divide their Tamagawa numbers or, when such a bound is not possible, to find infinite subfamilies whose Tamagawa numbers are as small as possible. Finally, we also investigate Tamagawa numbers of specializations of elliptic surfaces.