Tamagawa products of elliptic curves over $\mathbb{Q}$

TL;DR

This paper constructs a Dirichlet series for the distribution of Tamagawa products of elliptic curves over rationals, revealing that about half have trivial Tamagawa product and computing the average Tamagawa product.

Contribution

It explicitly formulates the Dirichlet series for Tamagawa products and determines the proportion of elliptic curves with trivial Tamagawa product, providing new statistical insights.

Findings

Proportion of elliptic curves with trivial Tamagawa product is approximately 50.53%.

The average Tamagawa product over all elliptic curves is approximately 1.8193.

The paper connects Tamagawa products to heights, offering applications in height theory.

Abstract

We explicitly construct the Dirichlet series $$L_{\mathrm{Tam}}(s):=\sum_{m=1}^{\infty}\frac{P_{\mathrm{Tam}}(m)}{m^s},$$ where $P_{\mathrm{Tam}}(m)$ is the proportion of elliptic curves $E/\mathbb{Q}$ in short Weierstrass form with Tamagawa product $m.$ Although there are no $E/\mathbb{Q}$ with everywhere good reduction, we prove that the proportion with trivial Tamagawa product is $P_{\mathrm{Tam}}(1)=0.5053\dots.$ As a corollary, we find that $L_{\mathrm{Tam}}(-1)=1.8193\dots$ is the average Tamagawa product for elliptic curves over $\mathbb{Q}.$ We give an application of these results to canonical and Weil heights.