The maximum number of points on a curve of genus 4 over F_8 is 25
David Savitt, Kristin Lauter
备注：
16 pages, 5 page appendix; replaced July 7, 2002. Paper by David Savitt with an Appendix by Kristin Lauter. Includes a collection of examples by David Savitt that will not be included in the print version of the paper
This paper proves that the maximum number of rational points on a smooth, absolutely irreducible genus 4 curve over the field of 8 elements is 25. The body of the paper shows that 27 points is not possible using techniques from algebraic geometry. The appendix shows that 26 points is not possible by examining the zeta functions.
We revisit and generalize some geometric techniques behind deterministic primality testing for some integer sequences using curves of genus 1 over finite rings. Subsequently we develop a similar primality test using the Jacobian of a genus 2 curve.
In this article, we study the Mahler measures of more than 500 families of reciprocal polynomials defining genus 2 and genus 3 curves. We numerically find relations between the Mahler measures of these polynomials with special values of $L$-functions. We also numerically discover more than 100 identities between Mahler measures involving different families of polynomials defining genus 2 and genus 3 curves. Furthermore, we study the Mahler measures of several families of nonreciprocal polynomials defining genus 2 curves and numerically find relations between the Mahler measures of these famili…
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Initially motivated by the relations between Anabelian Geometry and Artin's L-functions of the associated Galois-representations, here we study the list of zeta-functions of genus two abelian coverings of elliptic curves over finite fields. Our goal is to provide a complete description of such a list.
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We explain how we computed equations for all genus 4 curves defined of the field with 2 elements, up-to-isomorphism, and some of the data we obtained. We give descriptions also of nice models for genus 4 curves over characteristic 2 fields, in both the hyperelliptic case and the trigonal case.
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Abelian surfaces over finite fields containing no curves of genus $3$ or less
CNRS（法国国家科学研究中心；波尔多大学）；IMB, Université de Bordeaux（阿松森国立大学工程学院）；Facultad de Ingeniería, Universidad Nacional de Asunción（乌得勒支大学数学研究所）；Mathematisch Instituut, Universiteit Utrecht（法属波利尼西亚大学GAATI实验室）；Laboratoire GAATI, Université de la Polynésie française（蔚蓝海岸大学让·亚历山大·迪厄多内实验室）；Laboratoire Jean Alexandre Dieudonnè, Universitè Côte d’Azur
We prove that there exists a supersingular nonsingular curve of genus $4$ in arbitrary characteristic $p$. For $p>3$ we shall prove that the desingularization of a certain fiber product over $\mathbb{P}^1$ of two supersingular elliptic curves is supersingular.
We describe a method that allows, under some hypotheses, to compute all the rational points of some genus 5 curves defined over a number field. This method is used to solve some arithmetic problems that remained open.
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We give bounds on the primes of geometric bad reduction for curves of genus three of primitive CM type in terms of the CM orders. In the case of genus one, there are no primes of geometric bad reduction because CM elliptic curves are CM abelian varieties, which have potential good reduction everywhere. However, for genus at least two, the curve can have bad reduction at a prime although the Jacobian has good reduction. Goren and Lauter gave the first bound in the case of genus two. In the cases of hyperelliptic and Picard curves, our results imply bounds on primes appearing in the denominators…
In this paper, we consider the hyperelliptic analogue of the Frobenius endomorphism generator and show that it produces sequences with large linear complexity on the Jacobian of genus 2 curves.
