at the

American Institute of Mathematics, San Jose, California

organized by

Alina Bucur, Chantal David, and Jordan Ellenberg

In this workshop, we will concentrate on those questions with a special emphasis on what the statistics looks like when we start from the field $\mathbb{F}_{q}(t)$ of rational functions over a finite field (or, more generally, the function field of a curve over $\mathbb{F}_{q}$) instead of the field $\mathbb{Q}$ of rational numbers. Then the questions about number fields become questions about covers of the projective line $\mathbb{P}^1$, the questions about class groups become questions about groups of rational points on Jacobians, and so on. The main topics for the workshop are:

- formulate a probabilistic model for the large genus limit and fixed finite field of definition $\mathbb{F}_{q}$
- understand how the distribution of points on finite curves and zeta zeroes changes with the arithmetic invariant used
- understand which statistics for families over finite fields can be used to actually prove results about statistics over number fields, as in the work of Entin, Roditty-Gershon, and Rudnick

Methods from a number of areas have been brought to bear on the subject, ranging from the Grothendieck ring of motives to the circle method of classical analytic number theory to etale cohomology and random matrix theory. An appealing example is presented by the work of Yu-Ru Liu and Trevor Wooley, and very recently of Siu-Lun Alan Lee and Thibaut Pugin, about Waring's problem and Birch's theorem over function fields, which draws connections between the circle method over function fields and geometric properties of spaces of rational curves on complete intersections. New methods have been developed, like the sieve developed by Bjorn Poonen to study the equivalent of the Bertini theorem in finite fields.

Finally, in the light of the spectacular recent work of Entin, Roditty-Gershon, and Rudnick it seems that statistics for families over finite fields can be used to actually prove results about statistics over number fields in some cases, and not only predict those statistics.

The workshop schedule.

A report on the workshop activities.

A list of open problems.

Papers arising from the workshop:

Rudnick and Soundararajan's theorem for function fields

by Julio Andrade, *Finite Fields Appl. 37 (2016), 311-327 * MR3426592

A heuristic for boundedness of ranks of elliptic curves

by Jennifer Park, Bjorn Poonen, John Voight, and Melanie Matchett Wood

Moments of zeta and correlations of divisor-sums: III

by Brian Conrey and Jonathan P. Keating, *Indag. Math. (N.S.) 26 (2015), no. 5, 736-747 * MR3425874

Moments of zeta and correlations of divisor-sums: II

by Brian Conrey and Jonathan P. Keating, *Advances in the theory of numbers, 75-85, Fields Inst. Commun., 77, Fields Inst. Res. Math. Sci., Toronto, ON, 2015 * MR3409324

Moments of zeta and correlations of divisor-sums: I

by Brian Conrey and Jonathan P. Keating, *Philos. Trans. Roy. Soc. A 373 (2015), no. 2040, 20140313, 11 pp * MR3338122

The distribution of Fq-points on cyclic l-covers of genus g

by Alina Bucur, Chantal David, Brooke Feigon, Nathan Kaplan, Matilde Lalin, Ekin Ozman and Melanie Matchett Wood, *Int. Math. Res. Not. IMRN 2016, no. 14, 4297-4340 * MR3556420

Arithmetic functions at consecutive shifted primes

by Paul Pollack and Lola Thompson, *Int J. Number Theory 11 (2015), no. 5, 1477-1498 * MR3376222

Moments of zeta functions associated to hyperelliptic curves over finite fields

by Michael O. Rubinstein and Kaiyu Wu, *Philos. Trans. A 373 (2015), no. 2040, 20140307, 37 pp * MR3338121

On the spectral distribution of large weighted random regular graphs

by Leo Goldmakher, Cap Khoury, Steven J. Miller and Kesinee Ninsuwan, *Random Matrices Theory Appl. 3 (2014), no. 4, 1450015, 22 pp * MR3279620

Bounded gaps between primes in number fields and function fields

by Abel Castillo, Chris Hall, Robert J. Lemke Oliver, Paul Pollack, and Lola Thompson, *Proc. Amer. Math. Soc. 143 (2015), no. 7, 2841-2856 * MR3336609

A heuristic for the distribution of point counts for random curves over a finite field

by Jeffrey D. Achter, Daniel Erman, Kiran S. Kedlaya, Melanie Matchett Wood and David Zureick-Brown, *Philos. Trans. A 373 (2015), no. 2040, 20140310, 12 pp * MR3338118