Applications of Curves over Finite Fields
About this Title
Michael D. Fried, Editor
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:
1. 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.
2. 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.
3. 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.
Graduate students and research mathematicians interested in number theory, specifically arithmetic of function fields and finite field applications, such as coding theory; computer scientists.
Table of Contents
- Harald Niederreiter and Chaoping Xing – Curve sequences with asymptotically many rational points [MR 1732223]
- Yasutaka Ihara – Shimura curves over finite fields and their rational points [MR 1732224]
- David R. Hayes – Distribution of minimal ideals in imaginary quadratic function fields [MR 1732225]
- Zesen Chen – Division points of Drinfeld modules and special values of Weil $L$-functions [MR 1732226]
- Gerard van der Geer and Marcel van der Vlugt – Constructing curves over finite fields with many points by solving linear equations [MR 1732227]
- Arnaldo Garcia and Fernando Torres – On maximal curves having classical Weierstrass gaps [MR 1732228]
- Shreeram S. Abhyankar and Paul A. Loomis – Twice more nice equations for nice groups [MR 1732229]
- Noam D. Elkies – Linearized algebra and finite groups of Lie type. I. Linear and symplectic groups [MR 1732230]
- Pierre Dèbes – Regular realization of abelian groups with controlled ramification [MR 1732231]
- Michel Emsalem – On reduction of covers of arithmetic surfaces [MR 1732232]
- Leonard M. Adleman and Ming-Deh Huang – Function field sieve method for discrete logarithms over finite fields [MR 1732233]
- Daqing Wan – A quick introduction to Dwork’s conjecture [MR 1732234]
- Alan Adolphson and Steven Sperber – On the degree of the zeta function of a complete intersection [MR 1732235]
- Franck Leprévost – The modular points of a genus $2$ quotient of $X_0(67)$ [MR 1732236]
- Ching-Li Chai and Wen-Ching Winnie Li – Function fields: arithmetic and applications [MR 1732102]
- Fan Chung – Spanning trees in subgraphs of lattices [MR 1732103]
- Michael Rosen – Average rank for elliptic curves and a conjecture of Nagao [MR 1732104]