**Contemporary Mathematics**

Volume: 245;
1999;
226 pp;
Softcover

MSC: Primary 11;
Secondary 12

**Print ISBN: 978-0-8218-0925-9
Product Code: CONM/245**

List Price: $51.00

AMS Member Price: $40.80

MAA Member Price: $45.90

**Electronic ISBN: 978-0-8218-7835-4
Product Code: CONM/245.E**

List Price: $48.00

AMS Member Price: $38.40

MAA Member Price: $43.20

# Applications of Curves over Finite Fields

Share this page *Edited by *
*Michael D. Fried*

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

## Applications of Curves over Finite Fields

- Contents vii8 free
- Applications of curves over finite fields ix10 free
- 1. Beyond Weil bounds; curves with many rational points 126 free
- Curve sequences with asymptotically many rational points 328
- Shimura curves over finite fields and their rational points 1540
- Distribution of minimal ideals in imaginary quadratic function fields 2550
- Division points of Drinfeld modules and special values of Weil L-functions 3156
- Constructing curves over finite fields with many points by solving linear equations 4166
- On maximal curves having classical Weierstrass gaps 4974

- 2. Monodromy groups of characteristic p covers 6186
- Twice more nice equations for nice groups 6388
- Linearized algebra and finite groups of Lie type: I: Linear and symplectic groups 77102
- Regular realization of Abelian groups with controlled ramification 109134
- On reduction of covers of arithmetic surfaces 117142
- Function field sieve method for discrete logarithms over finite fields 133158

- 3. Zeta functions and trace formulas 145170
- A quick introduction to Dwork's conjecture 147172
- On the degree of the zeta function of a complete intersection 165190
- The modular points of a genus 2 quotient of X0 (67) 181206
- Function fields: Arithmetic and applications 189214
- Spanning trees in subgraphs of lattices 201226
- Average rank for elliptic curves and a conjecture of Nagao 221246