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

@article{Faifman2009StatisticsOT, title={Statistics of the zeros of zeta functions in families of hyperelliptic curves over a finite field}, author={Dmitry Faifman and Ze{\'e}v Rudnick}, journal={Compositio Mathematica}, year={2009}, volume={146}, pages={81 - 101} }

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 Riemann hypothesis for curves, the zeros all lie on a circle. Their angles are uniformly distributed, so for a curve of genus g a fixed interval ℐ will contain asymptotically 2g∣ℐ∣ angles as the genus grows. We show that for the variance of number of angles in ℐ is asymptotically (2/π2)log (2g…

## 39 Citations

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…

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

- Mathematics
- 2013

Abstract For a function field k over a finite field with F q as the field of constants, and a finite abelian group G whose exponent divides q − 1 , we study the distribution of zeta zeroes for a…

Traces of high powers of the Frobenius class in the hyperelliptic ensemble

- Mathematics
- 2008

The zeta function of a curve over a finite field may be expressed in terms of the characteristic polynomial of a unitary symplectic matrix, called the Frobenius class of the curve. We compute the…

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…

Moments of zeta functions associated to hyperelliptic curves over finite fields

- Mathematics, MedicinePhilosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2015

Using techniques that were originally developed for studying moments of L-functions over number fields, Andrade and Keating conjectured an asymptotic formula for the moments for q fixed and .

Conjectures for the integral moments and ratios of L-functions over function fields

- Mathematics
- 2014

Abstract We extend to the function field setting the heuristic previously developed, by Conrey, Farmer, Keating, Rubinstein and Snaith, for the integral moments and ratios of L-functions defined over…

The mean value of L(12,χ) in the hyperelliptic ensemble

- Mathematics
- 2012

Abstract We obtain an asymptotic formula for the first moment of quadratic Dirichlet L-functions over the rational function field at the central point s = 1 2 . Specifically, we compute the expected…

CURVES AND ZETA FUNCTIONS OVER FINITE FIELDS ARIZONA WINTER SCHOOL 2014: ARITHMETIC STATISTICS

- Mathematics
- 2014

The lectures will be concerned with statistics for the zeroes of L-functions in natural families. This will include discussions of the number field and the function field case (the latter case being…

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

- Mathematics
- 2013

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.…

Traces, high powers and one level density for families of curves over finite fields

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 2017

Abstract The zeta function of a curve C over a finite field may be expressed in terms of the characteristic polynomial of a unitary matrix ΘC. We develop and present a new technique to compute the…

## References

SHOWING 1-10 OF 48 REFERENCES

Zeroes of zeta functions and symmetry

- Mathematics
- 1999

Hilbert and Polya suggested that there might be a natural spectral interpretation of the zeroes of the Riemann Zeta function. While at the time there was little evidence for this, today the evidence…

Linear functionals of eigenvalues of random matrices

- Mathematics
- 2000

Let Mn be a random n×n unitary matrix with distribution given by Haar measure on the unitary group. Using explicit moment calculations, a general criterion is given for linear combinations of traces…

Number Theory in Function Fields

- Mathematics
- 2002

Polynomials over Finite Fields.- Primes, Arithmetic Functions, and the Zeta Function.- The Reciprocity Law.- Dirichlet L-series and Primes in an Arithmetic Progression.- Algebraic Function Fields and…

On the Characteristic Polynomial¶ of a Random Unitary Matrix

- Mathematics
- 2001

Abstract: We present a range of fluctuation and large deviations results for the logarithm of the characteristic polynomial Z of a random N×N unitary matrix, as N→∞. First we show that , evaluated at…

Random Matrix Theory and ζ(1/2+it)

- Mathematics
- 2000

Abstract: We study the characteristic polynomials Z(U, θ) of matrices U in the Circular Unitary Ensemble (CUE) of Random Matrix Theory. Exact expressions for any matrix size N are derived for the…

Ten lectures on the interface between analytic number theory and harmonic analysis

- Mathematics
- 1994

Uniform distribution van der Corput sets Exponential sums I: The methods of Weyl and van der Corput Exponential sums II: Vinogradov's method An introduction to Turan's method Irregularities of…

Wieand Eigenvalue distributions of random unitary matrices

Abstract. Let U be an n × n random matrix chosen from Haar measure on the unitary group. For a fixed arc of the unit circle, let X be the number of eigenvalues of M which lie in the specified arc. We…

Eigenvalue distributions of random unitary matrices

- Mathematics
- 2002

Abstract. Let U be an n × n random matrix chosen from Haar measure on the unitary group. For a fixed arc of the unit circle, let X be the number of eigenvalues of M which lie in the specified arc. We…

Finite-N fluctuation formulas for random matrices

- Mathematics, Physics
- 1997

For the Gaussian and Laguerre random matrix ensembles, the probability density function (p.d.f.) for the linear statistic ΣjN=1 (xj − 〈x〉) is computed exactly and shown to satisfy a central limit…

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…