Schedule for: 23w2006 - Alberta Number Theory Days XIV

We investigate the distribution of values of cubic Dirichlet $L$-functions at $s=1$. Following ideas of Granville and Soundararajan, and Dahl and Lamzouri for quadratic $L$-functions, we model values of $L(1,\chi)$ with the distribution of random Euler products $L(1,\mathbb{X})$ for certain family of random variables $\mathbb{X}(p)$ attached to each prime. We obtain a description of the proportion of $|L(1,\chi)|$ that are larger or that are smaller than a given bound, and yield more light into the Littlewood bounds. Unlike the quadratic case, there is a clear asymmetry between lower and upper bounds for the cubic case. This is joint work with Darbar, David, and Lumley.

In this talk, we will discuss an estimate for a discrete mean value of the Riemann zeta function and its derivatives multiplied by Dirichlet polynomials. Assuming the Riemann Hypothesis, we obtain a lower bound for the $2k^{th}$ moment of all the derivatives of the Riemann zeta function evaluated at its nontrivial zeros. This is based on a joint work with Kübra Benli and Nathan Ng.

Let $\bG$ be a simply-connected (split) Chevalley group, and write $G:= \bG(\R)$ be its group of real points. Let $\hG^e$ be the extended loop group, \textit{i.e.} the semi-direct product of the Kac-Moody central extension $\hG$ by $\R^*$ of $G$ with the automorphism of loop rotation $\eta(s)$. For $0 < s < 1$, so that the reduction theory of H. Garland is applicable. Let us also write $\hK \subset \hG$ and $\hGam \subset \hG$ for the analogues of the maximal compact subgroup and `arithmetic' subgroups in $\hG.$ The main objects of this note are the extended `loop' symmetric space $X^+:= \hK \setminus \hG ^+ $ and its arithmetic quotient $Y^+:= \hK \setminus \hG^+ / \hGam$, where $\displaystyle \hG^+ = \bigcup_{0 (Kiguli Room)

Let $K$ be a function field with perfect constant field $k$ of arbitrary characteristic. We give upper bounds, depending on $K$, on the rank of the Mordell-Weil group over $K$ of any Abelian variety which has trivial $K/k$-trace. Our result generalizes in various ways a previous theorem by Jean Gillibert (Université de Toulouse) and Aaron Levin (Michigan State University) on elliptic curves over functions fields of arbitrary characteristic and is moreover stated under weaker assumptions. We also explore some consequences of our result. This is a joint work with Jean Gillibert and Aaron Levin.

A guiding question in number theory, specifically in arithmetic statistics, is counting number fields of fixed degree and Galois group as their discriminants grow to infinity. We will discuss the history of this question and take a closer look at the story in the case of quartic fields. In joint work with Florea, Serrano Lopez, and Varma, we extend and make explicit the counts of extensions of an arbitrary number field that was done over the rationals by Cohen, Diaz y Diaz, and Olivier.

Local Arthur packets, or A-packets, are sets of representations of p-adic groups that help us realize an automorphic form, a global object. The local Langlands correspondence for a connected reductive p-adic group G partitions the set of equivalence classes of smooth irreducible representations of G(F) into L-packets using Langlands parameters. Arthur’s work introduces the notion of local A-packets, which are sets of smooth irreducible representations of G(F) attached to Arthur parameters. On the other hand, Vogan’s geometric perspective on the Langlands correspondence establishes a bijection between equivalence classes of smooth irreducible representations of G(F) and simple equivariant perverse sheaves on a moduli space of Langlands parameters. This gives us the notion of an ABV-packet, which is also a set of smooth irreducible representations of G(F), but now attached to any Langlands parameter. Conjecturally, ABV-packets generalize A-packets; we call this Vogan’s conjecture. In recent work joint with Clifton Cunningham, we prove this conjecture for p-adic $\text{GL}_n$. In this talk, we will explore the geometry of the moduli space of Langlands parameters for $\text{GL}_n$ via examples. We will provide comments on the proof of Vogan’s conjecture in this setting and discuss the scope of further generalizations.

Unicritical polynomials, typically written in the form $z^d+c$, have been widely studied in arithmetic and complex dynamics and are characterized by their one finite critical point. The behavior of a map’s critical points under iteration often determines the dynamics of the entire map. Rational maps where the critical points have a finite forward orbit under iteration are called postcritically finite (PCF), and these are of great interest in arithmetic dynamics. The family of (single-cycle normalized) dynamical Belyi polynomials have two fixed critical points, so they are PCF by construction. These maps provide a new testing ground for conjectures and ideas related to post-critically finite polynomials and using this family, we can begin to explore properties of polynomials with two critical points. In this talk we will discuss how the family of dynamical Belyi polynomials connects to the more general setting of bicritical polynomials and how we can use it classify PCF polynomials, to answer a question of Silverman pertaining to the height of critically fixed maps, and to determine reduction properties of PCF maps. In particular, we will demonstrate that considering maps with only one additional critical point may be enough to provide complete answers to questions in arithmetic dynamics. Much of the work discussed is joint with Michelle Manes and Jacqueline Anderson.

The study of fusion categories was born in the mathematical physics literature of the 1980's, but came into its own at the turn of the millennium. Many of the deeper results in the field are simply consequences of numerical invariants associated with these objects. In this brief talk we will outline some of these numerical invariants and demonstrate the impact even a modest understanding of number theory can have on the study of fusion categories.

In this talk, I will discuss a nonarchimedean integral geometry formula for the action of a compact K-analytic group on a homogeneous space. This formula is analogous to a result over the reals obtained by Howard. Some applications will be discussed. Joint work with Antonio Lerario and Peter Burgisser.

Computing the class group and unit group of a number field is a fundamental problem in number theory. In practice the fastest algorithm for doing this in quadratic fields is the self-initializing quadratic sieve. This algorithm works by sieving a carefully chosen sequence of norm forms of ideals. Despite its success, the algorithm has never been adapted to fields of higher degree. We discuss an adaptation of this approach to quartic fields.

We computed the theta liftings of a “cusp form” on the loop group $GL_n$ induced from a classical cusp form for the loop group “dual pair” $(GL_n,GL_n)$, and explained the result an Eisenstein series.

In 1979, Erdos conjectured that for $k \ge 9$, $2^k$ is not the sum of distinct powers of $3$. That is, the set of powers of two (which is $2$-automatic) and the $3$-automatic set consisting of numbers whose ternary expansions omit $2$ has finite intersection. In the theory of automata, a theorem of Cobham (1969) says that if $k$ and $\ell$ are two multiplicatively independent natural numbers then a subset of the natural numbers that is both $k$- and $\ell$-automatic is eventually periodic. A multidimensional extension was later given by Semenov (1977). Motivated by Erdos’ conjecture and in light of Cobham’s theorem, we give a quantitative version of the Cobham-Semenov theorem for sparse automatic sets, showing that the intersection of a sparse $k$-automatic subset of $\mathbb{N}^d$ and a sparse $\ell$-automatic subset of $\mathbb{N}^d$ is finite. Moreover, we give effectively computable upper bounds on the size of the intersection in terms of data from the automata that accept these sets.

Let the triple $(a,b,c)$ of integers be such that $\gcd(a,b,c)=1$ and $a+b=c$. We call such a triple an $ABC$-solution. The quality of an $ABC$-solution $(a,b,c)$ is defined as $$q(a,b,c) = \frac{\max \{\log |a|, \log|b|, \log|c|\}}{\log {\text{rad}}(|abc|)},$$ where $\text{rad}(|abc|)$ is the product of distinct prime factors of $|abc|$. The $ABC$-conjecture states that given $\epsilon>0$ the number of the $ABC$-solutions $(a,b,c)$ with $q(a,b,c) \geq 1+\epsilon$ is finite. In the first part of this talk, under the $ABC$-conjecture, we explore the quality of certain families of the $ABC$-solutions formed by terms in Lucas and associated Lucas sequences. We also introduce, unconditionally, a new family of $ABC$-solutions that has quality $> 1$. In the remainder of the talk, we provide a result on a conjecture of Erd\H{o}s on the solutions of the Brocard-Ramanujan equation $$n!+1=m^2$$ by assuming an explicit version of the $ABC$-conjecture proposed by Baker. We will also discuss a generalization of this problem in Lucas sequences inspired by the work of Pink and Szikszai.