# Arithmetic statistics over finite fields and function fields

January 27 to January 31, 2014

at the

American Institute of Mathematics, San Jose, California

organized by

Alina Bucur, Chantal David, and Jordan Ellenberg

## Original Announcement

This workshop will be devoted to the study of statistical questions about objects of arithmetic geometry, especially algebraic varieties over function fields and finite fields. Some typical questions in arithmetic statistics are the following. What is the probability that a random integer is square free, or is prime? How many number fields of degree $d$ are there with discriminant of absolute value at most $X$? What does the class group of a random quadratic field look like? Many aspects of the subject are well-understood, but many more remain the subject of conjectures, by Cohen-Lenstra, Malle, Bhargava, Batyrev-Manin, and others.

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
The analogy between function fields and number fields has been a rich source of insights throughout the modern history of number theory. In this setting, three active directions of number theory and algebraic geometry meet. The first direction is techniques imported from classical analytic number theory, which often must be modified and improved in the function field setting (sieve methods, Hardy-Littlewood method, etc.) The second direction is the theory of random matrices, which in this setting involves not only the much- studied properties of random complex matrices but those of random $l$-adic matrices as well. Finally, following Katz-Sarnak, arithmetic statistical questions over function fields can often be phrased as problems of counting $\mathbb{F}_{q}$-rational points on moduli schemes defined over $\mathbb{Z}$: such problems are deeply connected with the algebraic geometry and algebraic topology of the manifolds formed by the complex points on these moduli spaces.

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.

## Material from the workshop

A list of participants.

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