4 research outputs found

    On the Zeta Functions of Supersingular Curves

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    In general, the L-polynomial of a curve of genus gg is determined by gg coefficients. We show that the L-polynomial of a supersingular curve of genus gg is determined by fewer than gg coefficients

    Divisibility of L-Polynomials for a Family of Artin-Schreier Curves

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    In this paper we consider the curves Ck(p,a):yp−y=xpk+1+axC_k^{(p,a)} : y^p-y=x^{p^k+1}+ax defined over Fp\mathbb F_p and give a positive answer to a conjecture about a divisibility condition on LL-polynomials of the curves Ck(p,a)C_k^{(p,a)}. Our proof involves finding an exact formula for the number of Fpn\mathbb F_{p^n}-rational points on Ck(p,a)C_k^{(p,a)} for all nn, and uses a result we proved elsewhere about the number of rational points on supersingular curves

    Number of Rational points of the Generalized Hermitian Curves over Fpn\mathbb F_{p^n}

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    In this paper we consider the curves Hk,t(p):ypk+y=xpkt+1H_{k,t}^{(p)} : y^{p^k}+y=x^{p^{kt}+1} over Fp\mathbb F_p and and find an exact formula for the number of Fpn\mathbb F_{p^n}-rational points on Hk,t(p)H_{k,t}^{(p)} for all integers n≥1n\ge 1. We also give the condition when the LL-polynomial of a Hermitian curve divides the LL-polynomial of another over Fp\mathbb F_p

    On the Zeta function and the automorphism group of the generalized Suzuki curve

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    For pp an odd prime number, q0=ptq_{0}=p^{t}, and q=p2t−1q=p^{2t-1}, let XGS\mathcal{X}_{\mathcal{G}_{\mathcal{S}}} be the nonsingular model of Yq−Y=Xq0(Xq−X). Y^{q}-Y=X^{q_{0}}(X^{q}-X). In the present work, the number of Fqn\mathbb{F}_{q^{n}}-rational points and the full automorphism group of XGS\mathcal{X}_{\mathcal{G}_{\mathcal{S}}} are determined. In addition, the L-polynomial of this curve is provided, and the number of Fqn\mathbb{F}_{q^{n}}-rational points on the Jacobian JXGSJ_{\mathcal{X}_{\mathcal{G}_{\mathcal{S}}}} is used to construct \'{e}tale covers of XGS\mathcal{X}_{\mathcal{G}_{\mathcal{S}}}, some with many rational points
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