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Applications of Curves over Finite Fields
Edited by: Michael D. Fried, University of California, Irvine, CA
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Contemporary Mathematics
1999; 226 pp; softcover
Volume: 245
ISBN-10: 0-8218-0925-3
ISBN-13: 978-0-8218-0925-9
List Price: US$45
Member Price: US$36
Order Code: CONM/245
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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.

Readership

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

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