On the arithmetic of a family of twisted constant elliptic curves

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 Néron--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 $p(\mathrm{mod}6)$. Our approach relies on the BSD conjecture, an explicit expression for the $L$-function of $E$, and a geometric analysis of the Néron model of $E$.

38 pages. v2: minor changes following referee report