Abstracts

Second Moment of L-functions: There and Back Again

In this talk, I will discuss moments of quadratic Dirichlet L-functions associated with prime numbers and its function field analogue. The function field problem is joint work with J. Keating and the number field question is joint work with R. Heath-Brown, X. Li, M. Radziwill and K. Soundararajan.

Cycles in the supersingular l-Isogeny graph and corresponding Endomorphisms

We study the problem of generating the endomorphism ring of a supersingular elliptic curve by two cycles in l-isogeny graphs. We prove a necessary and sufficient condition for the two endomorphisms corresponding to two cycles to be linearly independent, expanding on the work by Kohel . We also give a criterion under which the order generated by two cycles is not a maximal order. We give some examples in which we compute cycles which generate the full endomorphism ring. This is a joint work with Catalina Camacho-Navarro, Kirsten Eisentrager, Travis Morrison and Jennifer Park.

Quadratic twists of elliptic curve L-functions over function fields

We compute various moments of the quadratic twists of elliptic curve L-functions at the central point over function fields. These in particular lead to some non-trivial results on the simultaneous vanishing of the quadratic twists of two distinct elliptic curve L-functions. This is joint work with Florea, Keating and Roditty-Gershon.

Is there a prime number theorem in algebraic groups?

Consider the following two elementary results for a finite field k: (a) the analogue of the prime number theorem (PNT) for polynomials over k; (b) counting the number of invertible matrices of fixed size over k. Both results are related to the study of the dynamics of the Frobenius endomorphism on the algebraic groups G_a and GL(n) over the algebraic closure K of k. For the first result, we count “prime” orbits; for the second, we count fixed points. In our work, we vary the endomorphism and consider counting orbits and fixed points of general endomorphisms of algebraic groups over K. We show there is a sharp dichotomy: either the associated dynamical zeta function is a rational function (like the Weil zeta function), and an analogue of PNT holds; or the zeta function is transcendental, and the set of limit points in PNT is uncountable. In the latter case, the number of fixed points of the m-th iterate involves p-adic properties of m. The distinction is very similar to dichotomies observed in measurable dynamics (mixing/non-mixing). Sometimes, the dichotomy has a clear geometric interpretation, e.g., on abelian varieties, and on reductive groups, in relation to a famous formula of Steinberg generalizing the count for GL(n). It seems hopeless to study the analogue of the Riemann Hypothesis in the transcendental case, but it turns out there is a way after all. [Joint work with Jakub Byszewski and Marc Houben.]

The first moment of cubic Dirichlet twists over function fields

We present in this talk asymptotic formulae for the mean value of L-functions associated to cubic characters over F_q[T]. We solve this problem in the non-Kummer case (when q is equivalent to 2 modulo 3) and in the Kummer case (when q is equivalent to 1 modulo 3). This rely on obtaining precise asymptotics for averages of cubic Gauss sums over function fields, which can be studied using the theory of metaplectic Eisenstein series. In the non-Kummer setting, we display some explicit cancellation between the main term and the dual term coming from the approximate functional equation of the L-functions. This is joint work with A. Florea (Columbia) and M. Lalin (Montreal).

Properties of the limiting distribution in Chebyshev's bias

Following the framework of Rubinstein and Sarnak for Chebyshev’s bias, and its translation in function fields by Cha, one gets a limiting distribution μ. We will present weak conditions to ensure that µ is continuous and symmetric. We will illustrate these properties using joint work with X. Meng on the bias in the distribution of products of k irreducible polynomials with coefficients in a fixed finite field among different arithmetic progressions. In contrast with the “translation” in the ring of integers, we uncover instances of complete biases in the function field setting.

Monodromy of Linear Systems on Curves and Applications to the Arithmetic of Function Fields

We study the monodromy group associated with a linear system on an algebraic curve. We develop some tools for computing this monodromy that work in a fairly general setting. We apply our results to study the decomposition statistics of divisors in a linear system over a finite field F_q in the large q limit. We deduce as corollaries several function field analogues of conjectures from classical number theory, including a Chebotarev density theorem in short intervals and some results on the distribution of sums of squares. We also apply our methods to study the Galois group of a composition f(g(t)) of polynomials where g has constant coefficients and f depends linearly on several free variables. This generalizes many previously studied special cases of this problem.

The minimal ramification problem for function fields

For a finite group G, the minimal ramification problem asks for the minimal number of ramified primes in Galois extension of the field of rational numbers with Galois group G. Boston-Markin gave a conjectural answer for any group G, but only few special cases are proven. If we replace the field of rational numbers by a rational function field over a field K, complete answers are known in the case of K the complex numbers (by Riemann's Existence Theorem) and the algebraic closure of a finite field (the Abhyankar conjecture, proven by Raynaud-Harbater). I will discuss the minimal ramification problem for rational function fields over finite and other non-algebraically closed fields, with a focus on symmetric groups. Joint work with Lior Bary-Soroker.

Variance of the number of Chebotarev primes in short intervals

Given a Galois extension K/Q, we can consider the number of primes in a short interval [x,x+H], with a fixed Artin symbol. We study the variance of this quantity, where the start point x is drawn uniformly at random between 1 and X. Conditionally on a general pair correlation conjecture for Artin L-functions, we obtain an interesting piecewise-linear behavior for the variance, which depends strongly on the Galois group of K and its representations, and will be explained. We support our conditional result by proving an analogous result in the function field setting.

Chebyshev’s bias for elliptic curves over function fields

Since Chebyshev's observation that there seems to be more primes of the form 4n+3 than of the form 4n+1, many other types of ‘arithmetical biases’ have been found. As was observed by Mazur and explained by Sarnak and Fiorilli, such a bias appears in the count of points on reductions modulo primes of a fixed elliptic curve E/Q, and is created by the analytic rank. We will report on joint work with B. Cha and D. Fiorilli addressing the analogous question for elliptic curves over function fields. We will highlight the occurrence of extreme biases, which originate from different sources than in the number field case. Also, we will discuss what happens to a ‘typical curve’, in relation with linear independence properties of the zeros of the associated L-functions.

Joint Moments

I will discuss some recent results concerning the joint moments of the characteristic polynomial of a random matrix and its derivative, in the context of applications to L-functions defined over function fields.

Multiplicative functions over F_q[x] and the Erd\H{o}s Discrepancy problem.

We discuss analog of several classical results about mean values of multiplicative functions over \mathbb{F}_q[x] explaining some features that are not present in the number field setting. In the first part of the talk, which is based on the joint work with C. Pohoata and K. Soundararajan, we describe spectrum of multiplicative functions over \mathbb{F}_q[x]. In the second part of the talk (based on the joint work with A. Mangerel and J. Teravainen), we will focus on the "corrected" function field analog of the Erd\H{o}s Discrepancy Problem.

Prime and Möbius correlations for very short intervals in F_p[x]

We investigate function field analogs of the distribution of primes, and prime $k$-tuples, in very short intervals'' of the form $I(f) := \{ f(x) + a : a \in \fp \}$ for $f(x) \in \fp[x]$ and $p$ prime, as well as cancellation in sums of function field analogs of the Möbius $\mu$ function and its correlations (similar to sums appearing in Chowla's conjecture). For generic $f$, i.e., for $f$ a "Morse polynomial", we show that error terms are roughly of size $O(\sqrt{p})$ (with typical main terms of order $p$). We also give examples of $f$ for which there is no cancellation at all, and intervals where the heuristic primes are independent'' fails very badly. Time permitting we will discuss the curios fact that (square root) cancellation in Möbius sums is *equivalent* to (square root) cancellation in Chowla type sums.

Some generalizations of Chebyshev’s bias

I will briefly introduce some of my work on the generalization of "Chebyshev's bias" to some restricted integers. Motivated by some ideas for dealing with the case of integers, joint with Lucile Devin, we consider the number of products of $k$ irreducible polynomials over a finite filed among different arithmetic progressions. We unconditionally obtain asymptotic formula for the difference of the counting functions uniformly for $k$ in certain range. Then we derive the existence of the limiting distribution for the difference function. Due to the existence of possible central zeros of the associated $L$-functions, the difference function may behave very differently from the case of integers.

Generic linear independence of roots of exponential sums and applications

In several recent works, Keating, Rodgers, Roditty-Gershon, Rudnick and Waxman obtained unconditional function field results that are analogues of deep arithmetic questions over the integers, based on distribution results from arithmetic geometry due to Katz. For example, Keating--Rudnick (resp. Rudnick--Waxman) study the variance of the number of prime in short intervals (resp. of gaussian primes in sectors) over function fields. In this talk, we will discuss the validity of generic independence of the roots of the L-functions involved in some of these results, and consequences on finer properties of the distributions. Similar matters will be discussed for Kloosterman and Birch sums. The techniques follow that of Kowalski (for L-functions of curves), namely combining the large sieve with information on integral monodromy and Girstmair's method, as well as recent results on Chebyshev's bias over function fields.

The distribution of traces of powers of matrices over finite fields

Consider a random N by N unitary matrix chosen according to Haar measure. A classical result of Diaconis and Shashahani shows that traces of low powers of this matrix tend in distribution to independent centered gaussians as N grows. A result of Johansson shows that this converge is very fast -- superexponential in fact. Similar results hold for other classical compact groups. This talk will discuss analogues of these results for N by N matrices taken from a classical group over a finite field, showing that as N grows traces of powers of these matrices equidistribute superexponentially. A little surprisingly, the proof is connected to the distribution in short intervals of certain arithmetic functions in F_q[T]. This is joint work with O. Gorodetsky.

Sums of singular series and primes in short intervals in algebraic number fields

There is a recent generalization by Gross and Smith of the HardyLittlewood twin prime conjecture for algebraic number fields. Working over algebraic number fields, we investigate the behavior of sums of the singular series involved, estimating these sums up to lower order terms. Based upon the conjecture of Gross and Smith, we use our result to study the variance of counts of prime elements in a random short interval for large enough short intervals in an algebraic number field K. The conjecture over number fields generalizes a classical conjecture of Goldston and Montgomery over the integers and complements work of Tsai and Zaharescu, who considered smaller short intervals for algebraic number fields. This is a joint project with Brad Rodgers