# Selmer groups of families of elliptic curves with an $\ell$-isogeny

## TL;DR

This paper proves a central limit theorem for Tamagawa ratios in families of elliptic curves with a prime degree isogeny, and shows the existence of elliptic curves with arbitrarily large -Selmer groups for specific primes.

## Contribution

It establishes a probabilistic limit theorem for Tamagawa ratios and links it to the size of -Selmer groups, providing new bounds and existence results.

## Key findings

- Central limit theorem for Tamagawa ratios
- Bounds on average Tamagawa ratios
- Existence of elliptic curves with arbitrarily large -Selmer groups for certain primes

## Abstract

For certain families of elliptic curves admitting a rational isogeny of prime degree $\ell$, we establish a central limit theorem for the Tamagawa ratio and derive bounds on its average value. By using the Tamagawa ratio to bound the size of the $\ell$-isogeny Selmer group from below, we show that for $\ell \in\{ 2, 3, 5, 7, 13\}$, there exist elliptic curves with arbitrarily large $\ell$-Selmer groups.