# Statistics for ordinary Artin-Schreier covers and other -rank strata

@article{Bucur2013StatisticsFO, title={Statistics for ordinary Artin-Schreier covers and other -rank strata}, author={Alina Bucur and Chantal David and Brooke Feigon and Matilde Lal{\'i}n}, journal={Transactions of the American Mathematical Society}, year={2013}, volume={368}, pages={2371-2413} }

We study the distribution of the number of points and of the zeroes of the zeta function in different p-rank strata of Artin-Schreier covers over Fq when q is fixed and the genus goes to infinity. The p-rank strata considered include the ordinary family, the whole family, and the family of covers with p-rank equal to p − 1. While the zeta zeroes always approach the standard Gaussian distribution, the number of points over Fq has a distribution that varies with the specific family.

## 7 Citations

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## References

SHOWING 1-10 OF 38 REFERENCES

### Distribution of zeta zeroes of Artin--Schreier curves

- Mathematics
- 2011

We study the distribution of the zeroes of the zeta functions of the family of Artin-Schreier covers of the projective line over $\mathbb{F}_q$ when $q$ is fixed and the genus goes to infinity. We…

### On the distribution of zeroes of Artin–Schreier L-functions

- Mathematics
- 2011

We study the distribution of the zeroes of the L-functions of curves in the Artin–Schreier family. We consider the number of zeroes in short intervals and obtain partial results which agree with a…

### The $p$-rank stratification of Artin-Schreier curves

- Mathematics
- 2006

We study a moduli space A S g for Artin-Schreier curves of genus g over an algebraically closed field k of characteristic p. We study the stratification of A S g by p-rank into strata A S g.s of…

### Statistics of the zeros of zeta functions in families of hyperelliptic curves over a finite field

- MathematicsCompositio Mathematica
- 2009

Abstract We study the fluctuations in the distribution of zeros of zeta functions of a family of hyperelliptic curves defined over a fixed finite field, in the limit of large genus. According to the…

### Statistics of the Zeros of Zeta Functions in a Family of Curves over a Finite Field

- Mathematics
- 2010

Abstract. Let Fq be a finite field of cardinality q and l ≥ 2 be a prime number such that q ≡ 1 (mod l). Extending the work of Faifman and Rudnick [6] on hyperelliptic curves, we study the…

### The distribution of points on superelliptic curves over finite fields

- Mathematics
- 2012

We give the distribution of points on smooth superelliptic curves over a fixed finite field, as their degree goes to infinity. We also give the distribution of points on smooth m-fold cyclic covers…

### Distribution of Zeta Zeroes for Abelian Covers of Algebraic Curves Over a Finite Field

- Mathematics
- 2013

### Biased Statistics for Traces of Cyclic p-Fold Covers over Finite Fields

- MathematicsWIN - Women in Numbers
- 2011

Some of the results on the statistics of the trace of the Frobenius endomorphism associated to cyclic p-fold covers of the projective line that were presented in [1] are discussed in more detail.

### Some families of supersingular Artin-Schreier curves in characteristic > 2

- Mathematics
- 2008

In this short paper we prove that the following two 1-dimensional families of Artin-Schreier curves are supersingular: y^7 - y = x^5 + c.x^2 over F_7 y^5 - y = x^7 + c.x over F_5 (for some parameter…

### Random matrices, Frobenius eigenvalues, and monodromy

- Mathematics
- 1998

Statements of the main results Reformulation of the main results Reduction steps in proving the main theorems Test functions Haar measure Tail estimates Large $N$ limits and Fredholm determinants…