The list of talks below is tentative and subject to updates; a final version will be published without this notice in early August.

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Alp Bassa
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Explicit recursive towers

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TBD
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Barry Fagin
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Idempotent Integers: The complete class of numbers n with bipartite factorizations that work correctly with RSA

*
Semiprimes (n=p*q with p and q prime) are used as the basis of the RSA
cryptosystem due to the difficulty of factoring large integers. Public and
private keys (e,d) are chosen such that they are congruent to 1 mod phi(n) =
(p-1)(q-1), with encryption and decryption achieved through modular
exponentiation mod e and d.
What is often misunderstood in presenting public key cryptography is that the
use of semiprimes is required not because of correctness, but because of
security. The class of positive integers n with at least one bipartite
multiplicative partition n=pq such that p and q can correctly encrypt and
decrypt, even if one or both of (p,q) is composite, was firrst thought to
include only the semiprimes, and then later the Carmichael numbers. Recent
work has shown that characterization to also be incomplete. The complete class
consists exactly of those square-free numbers n=pq such that lambda(n) |
(p-1)(q-1), where lambda is the Carmichael function. Note that one or both of
(p,q) may be composite. We refer to such n as idempotent integers, and the
multiplicative partition n=pq as an idempotent factorization. Even though only
the semiprimes are useful cryptographically, idempotent integers are worthy of
study on their own as their discovery and construction lie at the intersection
of hard problems in computer science and number theory.
We discuss what it known about these integers, including:
Smallest examples with 2,3,4,5,6 and 7 factors;
Fully composite and semi-composite factorizations;
Less efficient and more efficient search techniques;
Constructive techniques;
Theorems.
In the process of identifying idempotent integers, we discovered that certain
idempotent integers are maximal in that all their bipartite multiplicative
partitions are idempotent. We show how to construct maximally idempotent
integers of arbitrary size using specially constructed number-theoretic graphs.
The largest maximally idempotent integer built by the author has 2081 digits,
with approximately 10^43 smaller maximally idempotent integers as divisors. We
will present numerous examples of idempotent and maximally idempotent integers
and go through examples of how to find and construct them.
*

**
Daniel Fiorilli
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Omega results for cubic field counts via the Katz-Sarnak philosophy

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I will discuss recent joint work with P. Cho, Y. Lee and A. Södergren. Since
the work of Davenport-Heilbronn, much work has been done to obtain a precise
estimate for the number of cubic fields of discriminant at most X. This
includes work of Belabas-Bhargava-Pomerance, Bhargava-Shankar-Tsimerman and
Taniguchi-Thorne. In this talk I will present a negative result, which states
that the GRH implies that the error term in this estimate cannot be too small.
Our approach involves low-lying zeros of Dedekind zeta functions of cubic
fields (first studied by Yang), and is strongly related to the Katz-Sarnak
conjectures and the ratios conjecture of Conrey, Farmer and Zirnbauer.
*

**
Olivier Fouquet
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The Birch and Swinnerton-Dyer Conjecture for elliptic curves of rank 0 for almost all primes

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The Birch and Swinnerton-Dyer Conjecture predicts the p-adic valuation of the
first non-zero term in the Taylor expansion of the L-function of an elliptic
curve about s=0 divided by a suitable period and regulator. In joint work with
Xin Wan, we prove it holds for all rational elliptic curves of rank 0 when
p≥167.
*

**
Richard Griffon
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New cases of the generalised Brauer-Siegel theorem

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Tsfasman and Vladuts made a conjecture concerning families of number fields
which, if true, would be a vast generalisation of the classical Brauer—Siegel
theorem. Their conjecture, call it BS-TV, is known to hold conditionally to
GRH, but also unconditionally in a number of cases. For instance, Lebacque and
Zykin have shown that BS-TV holds for asymptotically exact families of stepwise
Galois number fields. In this talk, I will report on recent joint work with
Philippe Lebacque, where we exhibit new families of number fields where the
BS-TV is true. Among other examples, we prove that any infinite global field
contained in a p-class field tower satisfies the generalised form of the
Brauer—Siegel theorem.
*

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Marc Hindry
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Arithmetic of algebraic surfaces over finite fields

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TBD
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Louis Ioos
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Asymptotics of relative Poincaré series over Hermitian symmetric spaces

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The method of Poincaré series and its generalizations are classical methods to
construct automorphic forms, and knowing when those produce non-trivial
examples constitutes a basic problem in this theory. In particular, the method
of relative Poincaré series allows for a computation of the periods of
automorphic forms along closed geodesics, which constitute a natural
replacement of the Fourier coefficients of modular forms. In this talk, I will
explain how one can compute their asymptotics as the weight tends to infinity,
using the so-called semiclassical expansion of the Bergman kernel.
*

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Nicolina Istrati
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An introduction to Oeljeklaus-Toma manifolds

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Oeljeklaus-Toma manifolds are compact quotients of C^t x H^s by a discrete
group arising from a choice of a number field K and a group of units U of K.
They have interesting metric and cohomological properties, closely related to
the arithmetic properties of K and U. I will first present their construction
and outline their main geometrical properties. Then I will describe how one can
compute their de Rham cohomology. This talk is based on joint work with
Alexandra Otiman.
*

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Kiran Kedlaya
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Every positive integer is the order of an ordinary abelian variety over F_2

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We prove the theorem stated in the title, from our joint paper with Everett
Howe of that title. By Honda-Tate, this reduces to generating suitable Weil
polynomials, which we do by strategically applying a criterion of di
Pippo-Howe. We then report on some related developments and open problems.
*

**
Elisa Lorenzo García
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Decomposing Jacobians via Galois covers

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Let ϕ:X→Y be a cover between two algebraic curves of positive genus. We develop
tools that may identify the Prym variety of ϕ, up to isogeny, as the Jacobian
of a quotient curve C in the Galois closure of the composition of ϕ with a
well-chosen map Y→ℙ¹. This method allows us to recover all previously obtained
descriptions of a Prym variety in terms of a Jacobian that are known to us. We
also find algebraic equations for some of these new cases, including one where
X has genus 3, Y has genus 1 and ϕ is a degree 3 map totally ramified over 2
points. This is joint work with Davide Lombardo, Christophe Ritzenthaler and
Jeroen Sijsling.
*

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Fabien Pazuki
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Isogeny estimates - the tryptic mirror

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We will state and compare three results concerning j-invariants in isogeny
classes. We begin by studying the case of elliptic curves over number fields.
In joint work with Griffon, we turn to elliptic curves over function fields.
The third part will focus on Drinfeld modules of rank 2 and is joint work with
Breuer and Razafinjatovo.
*

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Renate Scheidler
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Computing modular polynomials of rank-2 Drinfeld modules

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Drinfeld modules represent the function field analogue of the theory of complex
multiplication. They were introduced as "elliptic modules" by Vladimir Drinfeld
in the 1970s in the course of proving the Langlands conjectures for GL(2) over
global function fields. Drinfeld modules of rank 2 exhibit very similar
behaviour to elliptic curves: they are classified as ordinary or supersingular,
support isogenies and their duals, and their endomorphism rings have an
analogous structure. Their isomorphism classes are parameterized by
j-invariants, and the n-th Drinfeld modular polynomial parameterizes pairs of
n-isogenous rank-2 Drinfeld modules, visualized by their isogeny graph whose
ordinary connected components take the shape of volcanos when n is prime. While
the rich analytic and algebraic theory of Drinfeld modules has undergone
extensive investigation, very little has been explored from a computational
perspective. This research represents a first foray in this direction,
introducing an algorithm for computing Drinfeld modular polynomials. This is
joint work with Perlas Caranay and Matthew Greenberg, as well as ongoing
research with Edgar Pacheco Castan. Some familiarity with elliptic curves is
expected for this talk, but no prior knowledge of Drinfeld modules is assumed.
*

**
Alexander Schmitt
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A moduli space for singular principal bundles over the moduli space of stable curves

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In the study of moduli spaces of vector or principal bundles over smooth
projective curves and their properties, one may use degenerations to singular
curves. An example for this approach is Gieseker's proof of a conjecture by
Newstead and Ramanan in the rank two case. Motivated by this and using prior
work by Bhosle, the speaker, and Muñoz Castañeda, the speaker and Muñoz
Castañeda constructed a moduli space for singular principal bundles over the
moduli space of stable curves, generalizing Pandharipande's construction for
the structure group GL_n. Recent work of Wilson establishes an interesting link
between this moduli space and the algebra of conformal blocks. In the talk, we
will briefly review the formalism of singular principal bundles, outline the
construction of the moduli space, and make some remarks concerning conformal
blocks.
*

**
Claudia Schoemann
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The twisted forms of a semisimple group over the integral domain of a global function field

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Let K=F_q(C) be the global field of rational functions on a smooth and
projective curve C defined over a finite field F_q. Any finite but non-empty
set S of closed points on C gives rise to an integral domain O_S=F_q[C-S] of K.
Given a semisimple and almost-simple group scheme G defined over Spec O_s with
a smooth fundamental group F(G), we describe the set of (O_S-classes of)
twisted forms of G in terms of the geometric invariants of F(G) and the
absolute type of the Dynkin diagram of G. This finite set is given by
H_fl^1(O_S,Aut(G)) seen as the disjoint union over P of H_fl^1(O_S,{}^P G^{ad})
modulo the Out(G)-action, where P are the G^{ad}-torsors, and turns out in most
cases to biject to a disjoint union of finite abelian groups. This is joint
work with Rony A. Bitan and Ralf Köhl.
*

**
René Schoof
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Finite flat group schemes over Z

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TBD
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Nicolas Thériault
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Index Calculus Algorithm for Non-Planar Curves

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The vast majority of previous works on the discrete logarithm problem for
algebraic curves assume a planar (2-dimensional) model for the curve, and
this has so far been the case for all previous results on index-calculus
techniques. However, for curves of genus 4 or higher, non-planar models are
more flexible and in some contexts may be more natural.
In this talk we describe how the index calculus algorithm can be adapted to
work in non-planar models of the curve, building up from the works of Diem
for non-hyperelliptic curves and the variation by Lane and Lauter. We show
that non-planar models allow us to always reach the best (heuristic) bound
for discrete log computation on planar models of non-hyperelliptic curves,
thus removing the need for a heuristic assumption. In fact, working in
higher dimensions gives more flexibility to the index calculus algorithm
which allows us to gain some constant-factor speedups compared to planar
models.
*

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Robin Zhang
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A Galois–dynamics correspondence for unicritical polynomials

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A universal fact about abelian varieties that was used in the proof of the
Mordell–Lang conjecture is that their torsion points always satisfy the Galois
homothety property. We describe a dynamical analogue with connections to the
irreducibility of dynatomic polynomials and to quadratic points on dynatomic
curves. Using existing results on explicit Hilbert irreducibility, we deduce
the non-existence (outside of a specified finite set) of quadratic polynomials
with periodic points of exact period 5 or 6 in quadratic number fields.
*