ON THE ARITHMETIC OF A FAMILY OF TWISTED CONSTANT ELLIPTIC CURVES

Let 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 y² = x³ + t^q - t over K = 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 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)| Reg(E) has size comparable to r^q/6 as q → ∞, regardless of p (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.