By Harald Niederreiter, Alina Ostafe, Daniel Panario, Arne Winterhof

This publication collects the result of the workshops on purposes of Algebraic Curves and purposes of Finite Fieldsat the RICAMin 2013. those workshops introduced jointly the main fashionable researchers within the region of finite fields and their functions all over the world, addressing outdated and new difficulties on curves and different features of finite fields, with emphasis on their assorted functions to many parts of natural and utilized arithmetic.

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Dk 1 Introduction The question of how many rational points a curve of genus ???? defined over a finite field ???????? can have, has been a central and important one in number theory. One of the landmark results in the theory of curves defined over finite fields was the theorem of Hasse and Weil, which is the congruence function field analogue of the Riemann hypothesis. As an immediate consequence of this theorem one obtains an upper bound for the number of rational points on such a curve in terms of its genus and the cardinality of the finite field.

1. Given ???? and ????, does every symmetric admissible Newton polygon of height 2???? occur for the Jacobian of a smooth projective curve of genus ????? The first open case of this question is when ???? = 4, for Newton polygons ????4 , ????30 ⊕ ????1 , and ????11 ⊕????3 . 2, it is not hard to see that each of these Newton polygons is realized by a singular curve of compact type. 3, will resolve this question for all ????. For example, if ???? is inert in ℚ(????3 ), then no Abelian variety with ????-rank 1 has an action by ℤ[????3 ].

Padova 113 (2005), 129–177. Rachel Pries and Hui June Zhu, The ????-rank stratification of Artin-Schreier curves, Ann. Inst. Fourier (Grenoble) 62 (2012), 707–726. Jasper Scholten and Hui June Zhu, Hyperelliptic curves in characteristic 2, Int. Math. Res. Not. 17 (2002), 905–917. Doré Subrao, The ????-rank of Artin–Schreier curves, Manuscripta Math. 16 (1975), 169–193. Jacob Tsimerman, The existence of an Abelian variety over ℚ isogenous to no Jacobian, Ann. Math. 176(2) (2012), 637–650. Gerard van der Geer and Marcel van der Vlugt, Reed–Muller codes and supersingular curves.