Here Algebra and Number Theory (ANT) are interpreted broadly. If you work in related aspects of ANT and like to give a talk here, feel free to contact us: Thomas Bitoun (thomas.bitoun@ucalgary.ca) or Dang-Khoa Nguyen (dangkhoa.nguyen@ucalgary.ca).
For talks in previous years, click here.
October 5 at 11am in BI 211 (special time and place, this is in Biological Sciences Building):
$K$-rational points of curves by Andrew Granville (UdeM).
Abstract (compiled tex-pdf file): Mazur and Rubin's ``Diophantine stability'' program suggests asking, for a given curve $C$, over what fields $K$ does $C$ have rational points, or at least to study the degrees of such $K$. We study this question for planar curves $C$ from various perspectives and relate solvability to the shape of $C$'s Newton polygon (the real original one that Newton worked with, not a $p$-adic one which are frequently used in arithmetic geometry research). This is joint work with Lea Beneish .
November 29 at 2pm in MS 427:
Complex $K$-theory of dual Hitchin system by Michael Groechenig (UofT).
Abstract: I will report on joint work with Shiyu Shen. Moduli spaces of SL(n) and PGL(n)-Higgs bundles are conjecturally related by a derived equivalence and (up to HK rotation) mirror symmetry. This talk will be devoted to a shadow of these equivalences in complex K-theory.
January 23 at 2pm in MS 427:
Arithmetic statistics meets arithmetic topology (in the context of Iwasawa theory) by Anwesh Ray (CRM, UdeM)
Abstract: There are interesting analogies between number theory and topology. Such analogies were systematically observed by Mazur who observed parallels between the arithmetic of primes in number rings and the geometry of knots embedded in 3 manifolds. With respect to this analogy, we may formulate certain questions in arithmetic statistics and then consider their topological analogues. Iwasawa theory is the study of the asymptotic growth of class numbers in certain infinite abelian towers of number fields. This asymptotic growth is determined by Iwasawa invariants associated to the number field. Given a knot in a 3 sphere, a fundamental polynomial invariant is the Alexander polynomial; which is regarded as the analogue of the Iwasawa polynomial from number theory. We consider certain natural families of links embedded in the 3-sphere for which we prove results about the distribution of Iwasawa invariants. In greater detail, we consider distribution questions for the family of 2-bridge links. In time permits we shall also talk about the family of links associated to certain braids on a fixed number of strings.
February 6 at 2pm in MS 427:
Archimedean invariants and number field asymptotics by Erik Holmes (Calgary)
Abstract: Arithmetic statistics is a subfield of number theory which attempts to understand arithmetic objects by studying them in families. In this talk we will briefly introduce some classical questions in arithmetic statistics focusing first on the asymptotics of number fields and Malle’s conjecture. We then introduce an invariant of number fields, called the shape, which comes from the additive structure of the field and the geometry of numbers. We highlight some natural questions concerning this invariant then discuss some recent work in this area in which we are trying to relate the study of these invariants to the asymptotics of number fields: we will focus on explicit examples such as cubic number fields and a family of non-Galois sextic fields which were the first counterexample to Malle’s conjecture. Time permitting, we will discuss another (multiplicative) invariant and some recent work in the family of dihedral number fields.
February 13 at 2pm in MS 427:
A refinement of Christol’s theorem by Seda Albayrak (Calgary)
Abstract (compiled tex-pdf file): Christol's theorem is one of the fundamental results in the theory of finite-state automata. It says that a formal power series $F(x)=\sum_n a_n x^n$ with coefficients in a finite field $\mathbb{F}_q$, $q$ a power of a prime $p$, is algebraic over the field of rational functions $\mathbb{F}_q(x)$ if and only if the sequence $\{a_n\}$ is $p$-automatic. The support of an algebraic power series, i.e.the set of $n$ for which $a_n \neq 0$, is an automatic subset of $\mathbb{N}$. There is a dichotomy for automatic sets that says automatic sets are either sparse, having at most ${\rm O}((\log n)^d)$ elements of size at most $n$ for some $d \ge 1$ and all $n$; or they are non-sparse, have at least $n^{\alpha}$ elements of size at most $n$ for some positive number $\alpha$ and all $n$ sufficiently large. In a joint work with Jason Bell, we characterize algebraic power series with sparse support, giving a refinement of Christol’s theorem. In fact we are able to prove our result in a more general setting, that is for generalized power series, studied and characterized by Kedlaya.
March 6 at 2pm in MS 427:
L-Functions and Arithmetic Ranks of Elliptic Curves over Function Fields by Félix Baril Boudreau (Lethbridge)
Abstract: Elliptic curves are a central object of study in number theory. In this talk, we focus on those defined over function fields and with nonconstant j-invariant. The L-function of such an elliptic curve E/K is polynomial with integer coefficients. We present new results on the L-function of E/K and on the rank of the Abelian group E(K) of K-rational points, two objects which appear in the Birch and Swinnerton-Dyer conjecture.
Inspired by Schoof's algorithm, we study the reduction modulo integers of the L-function. More precisely, when E(K) has nontrivial N-torsion, we give formulas for the reductions modulo 2 and N for any quadratic twist of E/K. This generalizes a formula obtained by Hall for E/K. We give examples where we can compute the global root number of the quadratic twists, the order of vanishing of the L-function at a special value and even the whole L-function from these reductions. However, the group E(K) is finitely generated and in particular has finite torsion. Given a prime ell different from char(K), we provide, in absence of nontrivial ell-torsion and in a quite general context, expressions for the reduction modulo ell of the L-function. We will briefly discuss the algorithmic interest of these formulas.
Time permitting, we present part of our work in progress with Jean Gillibert (Université de Toulouse) and Aaron Levin (Michigan State University), where we give upper bounds, depending on K, on the rank of the group of K-rational points of an Abelian varieties with trivial K/k-trace. Here, K is a function field of arbitrary characteristic and perfect constant field k. Our result generalizes in various ways a previous theorem by Gillibert and Levin on elliptic curves over K and is moreover stated under weaker assumptions.
March 20 (online talk broadcasted in room MS 337, Zoom information is sent by email on March 18 and March 20)
Computing Hilbert modular forms by Ben Breen (Simons Collaboration on Arithmetic Geometry, Number Theory, and Computation)
Abstract: In this talk, we present ongoing work for computing spaces of Hilbert modular forms using a trace formula. We provide a brief description of our method and highlight the key algorithmic obstacles that we faced when developing an explicit implementation. We discuss the advantages and limitations of our method in relation to current implementations for computing spaces of Hilbert modular forms. We conclude with some applications and computations. Joint work with John Voight.
April 24 (online talk broadcasted in room MS 337, Zoom information is sent by email on April 23 and April 24)
Sums of arithmetic functions over values of polynomials and applications to the Loughran-Smeets conjecture by Kevin Destagnol (Paris-Sud)
Abstract: I will explain how one can estimate sums of arithmetic functions over values of polynomials provided that the arithmetic functions is well-behaved in arithmetic progressions and that the number of variables of the polynomials is big enough. I will then give a few applications of this result to the Loughran-Smeets problem regarding the probability for a random algebraic variety among a family to admit a rational point. This is joint work with Efthymios Sofos and Lehonard Hochfilzer.
May 1 at 2pm in MS 427:
Parabolically induced representations of the p-adic group G_2 distinguished by SO_4 by Sarah Dijols (Calgary)
Abstract: After a brief introduction to motivate the study of distinguished representations, I will explain how Mackey's theory for p-adic groups allows us to identify this type of representations and the specificities and difficulties appearing when one wants to apply this method to the exceptional group G_2. I will present a first description of some of the distinguished representations for the pair (G_2, SO_4), and a new approach, in progress, to obtain a more complete classification of these representations where the octonions' structure plays a central role.
May 29 at 1pm (online talk, Zoom information is sent by email on May 27 and May 29)
Hecke algebras for p-adic groups, the explicit Local Langlands Correspondence and stability (e.g. for G_2, GSp(4), etc.) by Yujie Xu (MIT)
Abstract: I will talk about my joint work with Aubert where we prove the Local Langlands Conjecture for G_2 (explicitly). This uses our earlier results on Hecke algebras attached to Bernstein components of (arbitrary) reductive p-adic groups, as well as an expected property on cuspidal support, along with a list of characterizing properties (including stability). In particular, we obtain "mixed" L-packets containing F-singular supercuspidals and non-supercuspidals. Our methods are inspired by the Langlands-Shahidi method, Deligne-Lusztig and Lusztig theories etc.
If time permits, I will explain how to uniquely characterize our correspondence using stability of L-packets, by computing coefficients of local character expansions for Harish-Chandra characters; moreover, I will mention how to adapt our general strategy to construct LLC for other reductive groups, such as GSp(4), Sp(4), etc. The latter parts are based on recent joint work with Suzuki.