A note on the torsion of the Jacobians of superelliptic curves $y^{q}=x^{p}+a$

Abstract

This article is a short version of the paper published in J. Number Theory 145 (2014) but we add new results and a brief discussion about the Torsion Conjecture. Consider the family of superelliptic curves (over Q) Cq,p,a : y q = x p + a, and its Jacobians Jq,p,a, where 2 < q < p are primes. We give the full (resp. partial) characterization of the torsion part of J3,5,a(Q) (resp. Jq,p,a(Q)). The main tools are computations of the zeta function of C3,5,a (resp. Cq,p,a) over F l for primes l ≡ 1, 2, 4, 8, 11 (mod 15) (resp. for primes l ≡ −1 (mod qp)) and applications of the Chebotarev Density Theorem.

On torsion of superelliptic Jacobians

“…Section 2 is devoted to the proof of Proposition 1. In Section 3 we look at the curves of the form y q = x p − x + a over a field of characteristic p. We show under suitable conditions they have no q-torsion over F p using methods similar to ones used in [7], involving Gauss sums and Hasse-Weil zeta functions. Lastly, in Section 4, we use those results and methods based on those in [8] to finish the proof of Theorem 2.…”

“…The methods described above can be applied to curves which have the form y q = x p + a after reduction modulo some prime ≡ 1 (mod pq). Torsion on Jacobians of such curves is studied at length in [7], see for instance Lemma 4 of that paper, where one can find a different proof of the analogue of Theorem 7 for the case q = 3.…”

On the ℓ-adic valuation of certain Jacobi sums

“…They appear as Frobenius eigenvalues for the Fermat curve in [9]. The Jacobi sums of the type studied in this paper are Frobenius eigenvalues of the diagonal curve y = x f + 1, which are studied in [1,7,8].…”

On torsion of superelliptic Jacobians

“…Section 2 is devoted to the proof of Theorem 1, first in the case d = 1, then in the general case. In Section 3 we look at the curves of the form y q = x p − x + a over a field of characteristic p. We show under suitable conditions they have no q-torsion over F p using methods similar to ones used in [6], involving Gauss sums and Hasse-Weil zeta functions. Lastly, in Section 4, we use those results and methods based on those in [7] to finish the proof of Theorem 2.…”

“…The methods described above can be applied to curves which have the form y q = x p + a after reduction modulo some prime ℓ ≡ 1 (mod pq). Torsion on Jacobians of such curves is studied at length in [6], see for instance lemma 4 of that paper, where one can find a different proof of the analogue of Theorem 3 for the case q = 3.…”