Divisibility of L-polynomials for a family of curves

Chacón, Iván Blanco; Chapman, Robin; Fordham, S. Blake; McGuire, Gary

doi:10.1142/9789814719261_0001

Cited by 2 publications

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“…This theorem is sometimes used to show divisibility. The p = 2 case of Conjecture 1 was proved in [4] by finding a map C…”

Section: A Divisibilty Theoremmentioning

confidence: 99%

“…where a ∈ F p , which are defined over F p and have genus p k (p − 1)/2. We will prove the following conjecture, which is stated in [4]. The L-polynomials in the conjecture are over F p .…”

Section: Introductionmentioning

confidence: 98%

“…The L-polynomials in the conjecture are over F p . This conjecture was proved for p = 2 in [4], so we will assume that p is odd for this paper. In Section 10 we explain why we can assume a = 1 without loss of generality.…”

Section: Introductionmentioning

confidence: 99%

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Divisibility of L-polynomials for a family of Artin–Schreier curves

McGuire

Yılmaz

2019

Journal of Pure and Applied Algebra

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In this paper we consider the curves C (p,a) k : y p − y = x p k +1 + ax defined over F p and give a positive answer to a conjecture about a divisibility condition on L-polynomials of the curves C (p,a) k . Our proof involves finding an exact formula for the number of F p n -rational points on C (p,a) k for all n, and uses a result we proved elsewhere about the number of rational points on supersingular curves.2. If X(F q ) is minimal then X(F q n ) is minimal for all n.Proof. We use Corollaries 3 and 4 which give the period of C k .Case II: If p ∤ kand p | ℓ, then the period of C k is 4kp and the period of C ℓ is 4ℓ. Since 4kp ∤ 4ℓ, L(C k ) does not divide L(C ℓ ) by Proposition 5.Case III: If p ∤ kand p ∤ ℓ, then the period of C k is 4kp and the period of C ℓ is 4ℓp. Since 4kp ∤ 4ℓp, L(C k ) does not divide L(C ℓ ) by Proposition 5.Case IV -A: If p | k, p ∤ ℓ and (k/p) ∤ ℓ, then the period of C k is 4k and the period of C ℓ is 4ℓp. Since 4k ∤ 4ℓp, L(C k ) does not divide L(C ℓ ) by Proposition 5.Case IV -B: If p | k, p ∤ ℓ and (k/p) | ℓ, then the period of C k is 4k and the period of C ℓ is 4ℓp. Since 4k | 4ℓp, we cannot use the Proposition 5.which gives a contradiction by Lemma 26.Corollary 8. Let k and ℓ be positive integers such that k < ℓ and k does not divide ℓ. Then there is no map from C ℓ to C k .Proof. By the Kleiman-Serre theorem (Theorem 2) and Theorem 27.

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“…This theorem is sometimes used to show divisibility. The p = 2 case of Conjecture 1 was proved in [4] by finding a map C…”

Section: A Divisibilty Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%