On the arithmetic of a family of twisted constant elliptic curves

Abstract

Let ‫ކ‬ 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 = ‫ކ‬ 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 X(E) (which we prove is finite). These invariants have radically different behaviors depending on the congruence class of p modulo 6. For instance X(E) either has trivial p-part or is a p-group. On the other hand, we show that the product |X(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.

On the arithmetic of a family of superelliptic curves

“…In particular, we use the minimal proper regular SNC model of C to prove that J has totally unipotent reduction at each place of bad reduction. For more specific information about the reduction type in the elliptic curve case, see [GU20]. We also compute the height of J, and prove that it is K-simple for when both a and b are prime.…”

“…We note that there are several sequences of elliptic curves for which similar behaviour has been described. See [HP16,Gri16,Gri18,Gri19,GU20] For a fixed pair (a, b), the genus g of C = C a,b,q is constant as q varies. Hence the term log r g / log H(J) is o(1) as q → ∞.…”

“…In this article, we generalize the work of [GU20], showing that the rank of J is sometimes 0 and sometimes unbounded as q varies, depending on r, a and b. To state our results, we define o p (n) to be the order of p in Z/nZ and recall that an integer n is said to be supersingular for p if some power of p is congruent to −1 modulo n. Note that if n is supersingular for p, then o p (n) is automatically even.…”

“…In fact, [PU16] shows that many families of superelliptic curves over F r (t) have Jacobians with bounded rank as q varies. More recently, [GU20] studied the family of elliptic (and superelliptic) curves with affine model y 2 = x 3 + t q − t. In this case, they show that the behavior of the rank is either 0 or unbounded as q varies, depending only on the congruence class of p modulo 6.…”

“…The proof of Theorem 1.5 contains several statements which are of interest in their own right. For instance, in Section 7, we describe the asymptotics of the special value of the L-function as q → ∞ via analytic methods, generalizing results from [GU20] in the elliptic curve case. We prove: Theorem 1.6.…”