Applications of Curves Over Finite Fields: (Contemporary Mathematics)

This volume presents the results of the AMS-IMS-SIAM Joint Summer Research Conference held at the University of Washington (Seattle). The talks were devoted to various aspects of the theory of algebraic curves over finite fields and its numerous applications. The three basic themes are the following: Curves with many rational points. Several articles describe main approaches to the construction of such curves: the Drinfeld modules and fiber product methods, the moduli space approach, and the constructions using classical curves; Monodromy groups of characteristic $p$ covers. A number of authors presented the results and conjectures related to the study of the monodromy groups of curves over finite fields. In particular, they study the monodromy groups from genus $0$ covers, reductions of covers, and explicit computation of monodromy groups over finite fields; and, Zeta functions and trace formulas.To a large extent, papers devoted to this topic reflect the contributions of Professor Bernard Dwork and his students. This conference was the last attended by Professor Dwork before his death, and several papers inspired by his presence include commentaries about the applications of trace formulas and $L$-function. The volume also contains a detailed introduction paper by Professor Michael Fried, which helps the reader to navigate in the material presented in the book.

Table of Contents:
Beyond Weil bounds; Curves with many rational points: Curve sequences with asymptotically many rational points by H. Niederreiter and C. Xing Shimura curves over finite fields and their rational points by Y. Ihara Distribution of minimal ideals in imaginary quadratic function fields by D. R. Hayes Division points of Drinfeld modules and special values of Weil $L$-functions by Z. Chen Constructing curves over finite fields with many points by solving linear equations by G. van der Geer and M. van der Vlugt On maximal curves having classical Weierstrass gaps by A. Garcia and F. Torres Monodromy groups of characteristic $p$ curves: Twice more nice equations for nice groups by S. S. Abhyankar and P. A. Loomis Linearized algebra and finite groups of Lie type: I: Linear and symplectic groups by N. D. Elkies Regular realization of abelian groups with controlled ramification by P. Debes On reduction of covers of arithmetic surfaces by M. Emsalem Function field sieve method for discrete logarithms over finite fields by L. M. Adleman and M.-D. Huang Zeta functions and trace formulas: A quick introduction to Dwork's conjecture by D. Wan On the degree of the zeta function of a complete intersection by A. Adolphson and S. Sperber The modular points of a genus 2 quotient of $X_0(67)$ by F. Leprevost Function fields: Arithmetic applications by C.-L. Chai and W.-C. W. Li Spanning trees in subgraphs of lattices by F. Chung Average rank for elliptic curves and a conjecture of Nagao by M. Rosen.