# Counting curves over finite fields

@article{Geer2014CountingCO,
title={Counting curves over finite fields},
author={Gerard van der Geer},
journal={Finite Fields Their Appl.},
year={2014},
volume={32},
pages={207-232}
}
• G. Geer
• Published 22 September 2014
• Mathematics
• Finite Fields Their Appl.
10 Citations

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## References

SHOWING 1-10 OF 55 REFERENCES

### Genus bounds for curves with fixed Frobenius eigenvalues

• Mathematics
• 2013
For every finite collection C of abelian varieties over F_q, we produce an explicit upper bound on the genus of curves over F_q whose Jacobians are isogenous to a product of powers of elements of C.

### Rational Points on Curves Over Finite Fields: Theory and Applications

• Mathematics, Computer Science
• 2001
This chapter discusses function fields with many rational places, applications to algebraic coding theory, and applications to low-discrepancy sequences.

### ON THE NUMBER OF POINTS OF A HYPERELLIPTIC CURVE OVER A FINITE PRIME FIELD

A new method is proposed in this paper for investigating algebraic congruences with prime modulus, leading in the case of hyperelliptic curves to estimates of the same order of strength as the

### The genus of maximal function fields over finite fields

• Mathematics
• 1995
AbstractWe prove that if there exists a maximal function field of one variable of genusg over $$\mathbb{F}_{q^2 }$$ , theng≤(q−1)2/4 org=qr/2 withq−1/2≤r≤q−1.

### Tables of curves with many points

• Mathematics
Math. Comput.
• 2000
These tables record results on curves with many points over finite fields by giving in two tables the best presently known bounds for N q (g), the maximum number of rational points on a smooth absolutely irreducible projective curve of genus g over a field F q of cardinality q.

### Towers of Function Fields over Non-prime Finite Fields

• Mathematics
• 2012
Over all non-prime finite fields, we construct some recursive towers of function fields with many rational places. Thus we obtain a substantial improvement on all known lower bounds for Ihara's

### Algebraic Stacks Whose Number of Points over Finite Fields is a Polynomial

• Mathematics
• 2005
The aim of this article is to investigate the cohomology (l-adic as well as Betti) of schemes, and more generally of certain algebraic stacks, that are proper and smooth over the integers and have

### The genus of curves over finite fields with many rational points

• Mathematics
• 1996
AbstractWe prove the following result which was conjectured by Stichtenoth and Xing: letg be the genus of a projective, irreducible non-singular algebraic curve over the finite field

### Cohomology of local systems on the moduli of principally polarized abelian surfaces

Let A2 be the moduli stack of principally polarized abelian surfaces. Let V be a smooth ‘-adic sheaf on A2 associated to an irreducible rational finitedimensional representation of Sp.4/. We give an

### The trace of Hecke operators on the space of classical holomorphic Siegel modular forms of genus two

We prove multiplicity one for vector valued holomorphic Siegel modular forms of weights greater or equal to 3 and the full Siegel modular group and give a trace formula for the action of the Hecke