# On the arithmetic of a family of twisted constant elliptic curves

## TL;DR

This paper investigates the arithmetic invariants of a family of twisted constant elliptic curves over function fields, revealing how these invariants vary with the characteristic of the base field and providing explicit growth estimates.

## Contribution

It provides explicit descriptions of the rank, regulator, and Tate-Shafarevich group of these elliptic curves, including their dependence on the characteristic modulo 6, and establishes growth bounds for their product.

## Key findings

- The Tate-Shafarevich group is finite for these curves.
- The invariants exhibit different behaviors depending on the prime's congruence class modulo 6.
- The product of the Tate-Shafarevich group order and the regulator grows roughly as r^{q/6}.

## Abstract

Let $\mathbb{F}_r$ be a finite field of characteristic $p>3$. For any power $q$ of $p$, consider the elliptic curve $E=E_{q,r}$ defined by $y^2=x^3 + t^q -t$ over $K=\mathbb{F}_r(t)$. We describe several arithmetic invariants of $E$ such as the rank of its Mordell--Weil group $E(K)$, the size of its N\'eron--Tate regulator $\text{Reg}(E)$, and the order of its Tate--Shafarevich group $III(E)$ (which we prove is finite). These invariants have radically different behaviors depending on the congruence class of $p$ modulo 6. For instance $III(E)$ either has trivial $p$-part or is a $p$-group. On the other hand, we show that the product $|III(E)|\text{Reg}(E)$ has size comparable to $r^{q/6}$ as $q\to\infty$, regardless of $p\pmod{6}$. Our approach relies on the BSD conjecture, an explicit expression for the $L$-function of $E$, and a geometric analysis of the N\'eron model of $E$.