On the topological aspects of arithmetic elliptic curves

TL;DR

This paper constructs a topological family encompassing all elliptic curves over Q, offering geometric insights into their Hasse-Weil L-functions, especially for those with Mordell-Weil rank ≤ 1.

Contribution

It introduces a novel topological framework that includes all elliptic curves over Q and connects this to geometric interpretations of their L-functions.

Findings

Provides a topological family containing all elliptic curves over Q

Offers geometric interpretations of Hasse-Weil L-functions for rank ≤ 1

Establishes a new perspective linking topology and elliptic curve L-functions

Abstract

In this short note, we shall construct a certain topological family which contains all elliptic curves over Q and, as an application, show that this family provides some geometric interpretations of the Hasse-Weil L-function of an elliptic curve over Q whose Mordell-Weil group is of rank $\leq1$.