Organizers
Alina Carmen Cojocaru (UIC)
and
Francesc Fité (MIT)
Confirmed speakers
Registration
In order to participate, fill in the registration form.
Note that registration is free, but required in order to be admitted in the conference.
Schedule of talks
| Boston time |
Monday, June 28
|
Tuesday, June 29
|
Paris time |
| 9:30am-10:20am |
Sawin |
Bilu |
3:30pm-4:20pm |
| 10:20am-11:10am |
Cantoral Farfán |
Poonen |
4:20pm-5:10pm |
| 11:10am-12:00pm |
Shankar |
Smith |
5:10pm-6:00pm |
| 12:00pm-12:50pm |
Lorenzo García |
BREAK / VANTAGE |
6:00pm-6:50pm |
| 12:50pm-2:00pm |
BREAK |
BREAK / VANTAGE |
6:50pm - 8:00pm |
| 2:00pm-2:50pm |
Sutherland |
Wang |
8:00pm-8:50pm |
| 2:50pm-3:40pm |
Fiorilli |
Kowalski |
8:50pm-9:40pm |
| 3:40pm-4:30pm |
Kim |
Papikian |
9:40pm-10:30pm |
Titles and abstracts
-
Margaret Bilu (Institute of Science and Technology, Austria)
Zeta statistics
(slides)
In this talk, we will introduce several different topologies in which a
sequence of zeta functions of varieties over a finite field can be taken
to converge. These topologies will be defined in terms of the sizes of
the coefficients of the power series expansions at zero or in terms of
the zeros and poles. We will explain how these types of convergence can
be interpreted arithmetically and/or geometrically, and how this leads
to a conjectural way of unifying arithmetic and motivic statistics. As
evidence for our conjectures we will mention some convergence results
for spaces of zero-cycles. This is joint work with Ronno Das and Sean Howe.
-
Victoria Cantoral Farfán (Georg-August-Universität Göttingen, Germany)
Towards the motivic Nagao's conjecture and its connections with the Tate conjectures
(slides)
In 1997, Nagao conjectured that the rank of an elliptic surface could be given by a limit formula arising from a weighted average of Frobenius traces from each fiber.
During this talk, I would like to report on a joint work with S. Kim where we introduced, for the first time, the Motivic Nagao conjecture for pure motives. In addition, I will highlight as well its links with some well-known conjectures in arithmetic geometry.
-
Daniel Fiorilli (CNRS Université Paris-Saclay, France)
Distribution of Frobenius elements in families of Galois extensions
This is joint work with Florent Jouve. I will discuss three types of results: Linnik type questions on the prime ideal of least norm with prescribed Frobenius, the generic order of magnitude of the error term in the Chebotarev density theorem, and unconditional instances of Chebyshev's bias in number fields.
-
Seoyoung Kim (Queen's University, Canada)
From the Birch and Swinnerton-Dyer conjecture to Nagao's conjecture
(slides)
Let E be an elliptic curve over Q, and let a_p be the Frobenius trace for each prime p. In 1965, Birch and Swinnerton-Dyer formulated a conjecture which implies the convergence of the Nagao-Mestre sum
lim_{x -> infty} (1/log x) sum_{p < x} (a_p log p)/p = -r+1/2,
where r is the order of the zero of the L-function of E at s=1, which is predicted to be the Mordell-Weil rank of E(Q).
We show that if the above limit exists, then the limit equals -r+1/2, and study the connections to the Riemann hypothesis for E.
We also relate this to Nagao's conjecture for elliptic curves.
Furthermore, we discuss a generalization of the above results for the Selberg classes
and hence (conjecturally) for the L-function of abelian varieties,
and their relations to the generalized Nagao's conjecture.
This is a joint work with M. Ram Murty.
-
Emmanuel Kowalski (ETH Zürich, Switzerland)
Fourier analysis over commutative algebraic groups and Frobenius
distribution
(slides)
In ongoing joint work with A. Forey and J. Fres‡n, we generalize to any
connected commutative algebraic group the convolution approach to
equidistribution problems pioneered by Katz for the multiplicative
group.
The lecture will survey the general statements before focusing on
concrete examples, including a special case related to lines on cubic
threefolds, where the exceptional group E_6 appears.
-
Elisa Lorenzo García (Université de Neuchâtel, Switzerland, and Université de Rennes, France)
Sato-Tate distributions of twists of the Fermat and the Klein quartics
(slides)
I will start by reviewing the Sato-Tate conjecture and its generalisations. I will focus on the Sato-Tate distributions and computational aspects. After reviewing the elliptic curves case and the genus 2 case, I will move to my results on genus 3 with F. FitŽ and A. Sutherland. In this common work we determine the Sato-Tate groups and the Sato-Tate distributions of the twists of the Fermat and Klein quartics, the two quartics with the largest automorphism group. This produces 60 different Sato-Tate distributions in genus 3, which are already enough to see new phenomenons: for instance in genus 3 the individual distribution of the coefficients of the normalized Euler factor do not determine the Sato-Tate distribution.
-
Mihran Papikian (Pennsylvania State University, USA)
Computing endomorphism rings and Frobenius matrices of Drinfeld modules
(slides)
Let F_q[T] be the polynomial ring over a finite field F_q.
We study the endomorphism rings of Drinfeld F_q[T]-modules of arbitrary rank over finite fields.
We compare the endomorphism rings to their subrings generated by the Frobenius endomorphism and deduce from this a reciprocity law for the division fields of Drinfeld modules. We then use these results to give an efficient algorithm for computing the endomorphism rings and discuss some interesting examples produced by our algorithm. This is a joint work with Sumita Garai.
-
Bjorn Poonen (Massachusetts Institute of Technology, USA)
Abelian varieties of prescribed order over finite fields
(slides)
We give several new constructions of Weil polynomials to show that given a prime power q and n >> 1, every integer in a large subinterval of the Hasse-Weil interval is realized as #A(F_q) for some n-dimensional abelian variety A over F_q. Moreover, we can make A geometrically simple, ordinary, and principally polarized. On the one hand, our work generalizes a theorem of Howe and Kedlaya for F_2. On the other hand, it improves upon theorems of DiPippo and Howe; Aubry, Haloui, and Lachaud; and Kadets. This talk will focus on one construction that leads to explicit (and nearly best possible) bounds, in terms of q, on the largest integer that is not A(F_q) for any A. This is joint work with Raymond van Bommel, Edgar Costa, Wanlin Li, and Alexander Smith.
-
Will Sawin (Columbia University, USA)
Frobenius distribution in number theory over function fields
There exists a natural analogue of the Chebotarev density
theorem for the field of rational functions in one variable over a
finite field, or extensions of it. Because of the additional geometric
flexibility of that setting, this theorem can be used to prove
number-theoretic statements over that field which have little or no
apparent relationship to Chebotarev. I will explain an example of this
phenomenon in my work with Michael Lipnowski and Jacob Tsimerman on
the Cohen-Lenstra heuristics over function fields.
-
Ananth Shankar (University of Wisconsin, USA)
Abelian varieties not isogenous to Jacobians over global fields
Let K be the algebraic closure of a global field of any characteristic.
For every g > 3, we prove that there exists a g-dimensional abelian variety over K which is not isogenous to a Jacobian. This is joint work with Jacob Tsimerman.
-
Alexander Smith (Massachusetts Institute of Technology, USA)
Totally positive integers of small trace and extreme orders of abelian varieties over finite fields
(slides)
Outside of finitely many exceptions, we show that the average real valuation of a totally positive algebraic integer
is at least 1.80, improving the prior best of 1.7919.
As a consequence, for a sufficiently large square prime power q, we show that all but finitely many simple abelian varieties A/F_q satisfy
(q - 2q^{1/2} + 2.8)^{dim A} < #A(F_q) < (q + 2q^{1/2} - 0.8)^{dim A},
and we explain how our approach can be adapted to other q.
We will also give some evidence that there are infinitely many totally positive algebraic integers whose average valuation is less than 1.82 and explain the implications of such a result for abelian varieties over finite fields.
Our starting point is the fact that the discriminant of a rational integer polynomial must be a rational integer.
We are able to take advantage of this fact in our computational approach by using logarithmic potential theory.
-
Andrew V. Sutherland (Massachusetts Institute of Technology, USA)
Stronger arithmetic equivalence
( slides)
Number fields K1 and K2 with the same Dedekind zeta function
are said to be arithmetically equivalent. Such number fields
necessarily have the same degree, signature, unit group, discriminant,
and Galois closure, and the distributions of their Frobenius elements
are compatible in a strong sense: for every unramified prime p the base
change of the Q-algebras K1 and K2 to Qp are isomorphic. This need not
hold at ramified primes, so the adele rings of K1 and K2 need not be
isomorphic, and global invariants such as the regulator and class number
may differ.
Motivated by a recent result of Prasad, I will discuss three stronger
notions of arithmetic equivalence that force isomorphisms of some or all
of these invariants without forcing an isomorphism of number fields, and
present examples that address questions of Scott and of Guralnick and
Weiss, and shed some light on a question of Prasad. These results also
have applications to the construction of curves with the same L-function
(due to Prasad), isospectral Riemannian manifolds (due to Sunada), and
isospectral graphs (due to Halbeisen and Hungerbuhler).
-
Tian Wang (University of Illinois at Chicago, USA)
Bounds for the distribution of the Frobenius traces associated to products of non-CM elliptic curves
Let A/Q be an abelian variety that is isogenous over the algebraic closure of Q to the product
E_1 x ... x E_g of elliptic curves E_1/Q, ..., E_g/Q
without complex multiplication and pairwise non-isogenous over the algebraic closure of Q.
For an integer t and a positive real number x, denote by pi_A(x, t) the number of primes p < x,
of good reduction for the abelian variety A, for which the Frobenius trace associated to the reduction of A modulo p equals t. Based on prior approaches to the Lang-Trotter Conjecture for the Frobenius traces associated to the reductions of an elliptic curve, under RH and GRH we prove a non-trivial upper bound for pi_A(x, t).
This is joint work with A.C. Cojocaru.
Sponsors
Support for the conference comes from
the Massachusetts Institute of Technology,
the University of Illinois at Chicago,
and the Simons Foundation.
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