Several natural complex configuration spaces admit surprising uniformizations as arithmetic ball quotients, by identifying each parametrized object with the periods of some auxiliary object. In each case, the theory of canonical models of Shimura varieties gives the ball quotient the structure of a variety over the ring of integers of a cyclotomic field. I will show that these (transcendentally-defined) period maps respect these algebraic structures, and thus are actually arithmetic.

For the introductory slides on this topic, please see: http://www.fields.utoronto.ca/sites/default/files/uploads/achter_warmup0421.pdf

For non-negative integers $a$, $b$, and $n$, let $t(a,b;n)$ be the number of representations of $n$ as a sum of triangular numbers with coefficients $1$ and $3$ ($a$ of ones and $b$ of threes) and let $r(a, b; n)$ be the number of representations of $n$ as a sum of squares with coefficients $1$ and $3$ ($a$ of ones and $b$ of threes). It is known that for $a$ and $b$ satisfying $1\leq a+3b \leq 7$, we have $$ t(a,b;{n}) = \frac{2}{2+{a\choose4}+ab} r(a,b;8n+a+3b) $$ and for $a$ and $b$ satisfying $a+3b=8$, we have $$ t(a,b;{n}) = \frac{2}{2+{a\choose4}+ab} \left( r(a,b;8n+a+3b) - r(a,b; (8n+a+3b)/4) \right). $$ Such identities are not known for $a+3b>8$.

Such identities are not known for $a+3b>8$. We report on our joint work with Zafer Selcuk Aygin (University of Calgary) in which, for general $a$ and $b$ with $a+b$ even, we prove asymptotic equivalence of formulas similar to the above, as $n\rightarrow\infty$. One of our main results extends a theorem of Bateman, Datskovsky, and Knopp where the case $b=0$ and general $a$ was considered. Our approach is different from Bateman-Datskovsky-Knopp's proof where the circle method and singular series were used. We achieve our results by explicitly computing the Eisenstein components of the generating functions of $t(a,b;n)$ and $r(a,b;n)$.

If a is an odd positive integer, then a result of Murty, Murty and Shorey implies that there are at most finitely many positive integers n for which tau(n)=a, where tau(n) is the Ramanujan tau-function. In this talk, I will discuss non-archimidean analogues of this result and show how the machinery of Frey curves and their associated Galois representations can be employed to make such results explicit, at least in certain situations. Much of what I will discuss generalizes readily to the more general situation of coefficients of cuspidal newforms of weight at least 4, under natural arithmetic conditions. This is joint work with Adela Gherga, Vandita Patel and Samir Siksek.

For the introductory slides on this topic, please see: http://www.fields.utoronto.ca/sites/default/files/uploads/Fields-2020.pdf

We start computing the periods and the dimension of the motivic Galois group of 1-motives. Then we apply Grothendieck period conjecture to 1-motives and we will see some consequences of this conjecture.

Prep material:

- Périodes de 1-motifs et transcendence

J. Number Theory 97 (2002), no. 2, pp. 204--221.

http://www.fields.utoronto.ca/sites/default/files/uploads/JournalOfNumberTheory1.pdf

- (with an appendix by M. Waldschmidt and a letter of Y. André) Third kind elliptic integrals and 1-motives

J. Pure Appl. Algebra 224 (2020), no.10, 106396, 28 pp.

http://www.fields.utoronto.ca/sites/default/files/uploads/JPAA.pdf

Let $E_1/\Q, \ldots, E_g/\Q$ be elliptic curves over $\Q$, without complex multiplication and pairwise non-isogenous over $\overline{\Q}$. For an integer $t$ and a positive real number $x$, denote by $\pi_A(x, t)$ the number of primes $p \leq x$, of good reduction for the abelian variety $A := E_1 \times \ldots \times E_g$, for which the Frobenius trace associated to the reduction of $A$ modulo $p$ equals $t$. We present unconditional and conditional upper bounds for $\pi_A(x, t)$. This is joint work with Tian Wang (University of Illinois at Chicago).

Abstract: This talk concerns work in progress on a generalization of the notion of local Arthur packets from Arthur-type representations of classical groups over $p$-adic fields to all admissible representations of all connected reductive algebraic groups over p-adic fields. In this talk our goal is much more modest: to report on this project for unipotent representations of the exceptional group G_2(F) for a p-adic field F. We will explain how to use the microlocal geometry of the moduli space of unramified Langlands parameters to compute what we call Adams-Barbasch-Vogan packets, or ABV-packets for short, for all unipotent representations of G_2(F) and how to find the packet coefficients that are required to build stable distributions from ABV-packets. This talk will focus on the case that is the more interesting geometrically and will include a discussion of unipotent representations that are not of Arthur type. We will argue that ABV-packets provide the correct extension of the notion of Arthur packets by explaining that the packet coefficients satisfy expected conditions coming from endoscopic character identities.

Joint work with Andrew Fiori and Qing Zhang, based on earlier joint work with Andrew Fiori, Ahmed Moussaoui, James Mracek and Bin Xu, which in turn is based on earlier work by David Vogan.

For the introductory slides on this topic, please see: http://www.fields.utoronto.ca/sites/default/files/uploads/Introduction_1.pdf

For an introductory lecture on this topic, please see: https://youtu.be/ad85GSAVAK4

Please register here: https://zoom.us/meeting/register/tJEtcuuvqTksHdfsLV9ZiYeWaHDdqnotA5-5 .

I will describe two ongoing works whose unifying theme is to establish the algebraicity of the RM values of rigid meromorphic cocycles, by realizing these invariants as the fourier coefficients of certain p-adic modular generating series, thereby obtaining the desired properties from the theory of deformations of Galois representations and from global class field theory. The first project, in collaboration with Alice Pozzi and Jan Vonk, considers the RM values of the Dedekind-Rademacher cocycle and its relation to the diagonal restrictions of certain first order deformations of Hilbert modular Eisenstein series. The second, in collaboration with Yingkun Li and Jan Vonk, considers the RM values of certain rigid meromorphic cocycles and p-adic modular forms of weight 3/2 that arise as the fourier coefficients of Zagier’s holomorphic kernel for the Shimura-Shintani correspondence.

For an introductory lecture on this topic, please see: https://www.youtube.com/watch?v=JsH1yMBwt_I .

In this talk we will discuss two central problems in algebraic number theory and their interconnections: explicit class field theory (also known as Hilbert's 12th Problem), and the special values of L-functions. The goal of explicit class field theory is to describe the abelian extensions of a ground number field via analytic means intrinsic to the ground field. Meanwhile, there is an abundance of conjectures on the special values of L-functions at certain integer points. Of these, Stark's Conjecture has special relevance toward explicit class field theory. I will describe my recent proof, joint with Mahesh Kakde, of the Brumer-Stark conjecture away from p=2. This conjecture states the existence of certain canonical elements in CM abelian extensions of totally real fields. Next I will state a conjectural exact formula for these Brumer-Stark units that has been developed over the last 15 years. I will conclude with a description of work in progress that aims to prove this conjecture and thereby give a solution to Hilbert's 12th problem.

Slides for the introductory lecture can be found here: http://www.fields.utoronto.ca/sites/default/files/uploads/Note%20Aug%2022%2C%202020.pdf

For an introductory lecture on this topic, please see: https://www.youtube.com/watch?v=_z_3uv5yE3A

We shall discuss a recent joint work with Makoto Kawashima and Noriko Hirata-Kohno. For any set of algebraic numbers in a fixed number field K satisfying standard metric conditions in the theory (close enough to zero), we prove that the values of polylogarithms are linearly independent over K. This is done via a construction of an explicit system of Padé approximation. We shall also discuss extensions to lerch functions

Introductory reading:

http://www.fields.utoronto.ca/sites/default/files/uploads/transcend.pdf

http://www.fields.utoronto.ca/sites/default/files/uploads/HP-ankara.pdf

http://www.fields.utoronto.ca/sites/default/files/uploads/CMI122009.pdf

http://www.fields.utoronto.ca/sites/default/files/uploads/ASENS_1899_3_16__395_0.pdf

A set of natural numbers $\{a_1, a_2, \cdots, a_m\}$ is said to be a Diophantine $m$-tuple with property $D(n)$ if $a_ia_j +n$ is a perfect square for $i \neq j$. One may ask, what is the largest $m$ for which such a tuple exists. This problem has a long history, attracting the attention of many, including Fermat, Baker, Davenport etc, with significant progress made in recent times due to Dujella and others. In this talk, we consider a similar question by replacing the condition $a_ia_j+n$ from being a square to $k$-th powers. This is joint work with Ram Murty and Seoyoung Kim.

This talk will be accessible to graduate students!

The following link gives a quick introduction to the topic. It should help bring students up to speed on the history and recent developments on the problem:

https://web.math.pmf.unizg.hr/~duje/dtuples.html

Additional introductory reading material:

http://www.fields.utoronto.ca/sites/default/files/uploads/Paley%20graph_Diophantine%20tuples-RM_AG.pdf

http://www.fields.utoronto.ca/sites/default/files/uploads/Diophantine%20tuples-RM_RB.pdf

http://www.fields.utoronto.ca/sites/default/files/uploads/Diophantine%20tuples-Dujella.pdf

We will show that the Ceresa cycles of Fermat curves of prime degree greater than 7 are of infinite order modulo rational equivalence (i.e. in the Chow group). The proof combines several results due to B. Harris, Pulte, Kaenders and Darmon-Rotger-Sols on the arithmetic and geometry of the mixed Hodge structure on the space of quadratic iterated integrals on an algebraic curve with a result of Gross and Rohrlich on points of infinite order on Jacobians of Fermat curves. This is a joint work with V. Kumar Murty.

For the introductory slides on this topic, please see: http://www.fields.utoronto.ca/sites/default/files/uploads/pre-talk.pdf

The fundamental difference between pure motives (roughly speaking coming from cohomology of smooth projective varieties) and mixed motives (coming from cohomology of arbitrary varieties) is existence of nontrivial extensions in the latter setting. The unipotent radical of the motivic Galois group of a mixed motive M is intimately related to the extension data in the category generated by M. A fairly recent result of Deligne describes this unipotent radical in terms of the extensions $0\longrightarrow W_pM\longrightarrow M\longrightarrow M/W_pM\longrightarrow 0$ collectively ($W_\cdot$ being the weight filtration). We shall recall this result and then discuss what information each of these extensions individually contains. We will end the talk with an application to motives whose unipotent radical of the motivic Galois group is as large as possible, and discuss some examples in the category of mixed Tate motives over $\mathbb{Q}$. This is a joint work with Kumar Murty.

The classical theta correspondence establishes a relationship between automorphic representations on special orthogonal groups and automorphic representations on symplectic groups or their double covers. This correspondence is achieved by using as integral kernel a theta series that is constructed from the Weil representation. In this talk I will briefly survey earlier work on (local and global, classical and other) theta correspondences and then present an extension of the classical theta correspondence to higher degree metaplectic covers. The key issue here is that for higher degree covers there is no analogue of the Weil representation (or even a minimal representation), so additional ingredients are needed. Joint work with David Ginzburg.

For an introductory lecture on this topic, please see: https://youtu.be/p_jNha9u7DQ

For the introductory slides on this topic, please see: http://www.fields.utoronto.ca/sites/default/files/uploads/friedberg-background.pdf

A remarkable theorem due to Khovanskii asserts that for any finite subset A of an abelian semigroup, the cardinality of the h-fold sumset hA grows like a polynomial for all sufficiently large h. However, neither the polynomial nor what sufficiently large means are understood in general. In joint work with Michael Curran (Oxford), we obtain an effective version of Khovanskii's theorem for any subset of $\mathbb{Z}^d$ whose convex hull is a simplex; previously such results were only available for d=1. Our approach also gives information about the structure of hA, answering a recent question posed by Granville and Shakan. The talk will be broadly accessible; interested mathematicians from any field are encouraged to attend.

In this talk, we will discuss about omega results of Fourier-coefficients of half-integer weight cusp forms which are not necessarily eigenforms. This is a joint work with Kohnen and Soudararajan.

Our goal is to review the existing quantum algorithms to compute class groups of number fields. First we briefly survey results for this problem on a standard “classical” computer. After a short introduction to quantum computing, we look at the hidden subgroup problem and quantum algorithms for it, as a fundamental tool. Finally we present the polynomial-time quantum algorithm for computing the ideal class group (under the Generalized Riemann Hypothesis) introduced by Biasse and Song (2016).

Selmer groups of elliptic curves encodes various aspects of the arithmetic of elliptic curves. In this talk, we will discuss a duality result for Selmer groups, over a p-adic Lie extension of a number field. This duality result can be thought of as an algebraic functional equation.

This talk is based on joint works with T. Ochiai and G. Zabradi.

Introductory reading:

1. Introduction to Iwasawa Theory of elliptic curves, Notes by. Ralph Greenberg: http://www.fields.utoronto.ca/sites/default/files/uploads/Introduction-IwT.pdf

2. The following paper by Perrin-Riou: Groupes de Selmer et Accouplements; Cas Particulier des Courbes Elliptiques (French): http://www.fields.utoronto.ca/sites/default/files/uploads/perrin-riou.dm.pdf

3. Iwasawa theory for p-adic representations by Ralph Greenberg. Algebraic number theory, 97–137, Adv. Stud. Pure Math., 17, Academic Press, Boston, MA, 1989.

Inspired by the work of Iwasawa on growth of class groups in $\mathbb{Z}_p$-extensions, Mazur developed an analogous theory to study the growth of Selmer groups of Abelian varieties in such extensions. In my pre-talk, I will introduce Iwasawa theory and briefly explain the work of Iwasawa and Mazur. I will finally introduce the fine Selmer group whose systematic study was initiated by Coates-Sujatha in 2005. This is a subgroup of the Selmer group obtained by imposing stronger conditions at primes above $p$.

In the main talk, I will explain the growth of the fine Selmer group in towers of number fields and relate it to the growth of class groups in such towers. This will show the close relationship between the conjectures in classical Iwasawa theory and the Iwasawa theory of Abelian varieties. I will report on some modest progress made towards some of these conjectures. If time permits, I will also talk about Control Theorems of Fine Selmer Groups

For an introductory lecture on this topic, please see: https://youtu.be/CiwR-YcEetI

For Introductory slides on this topic, please see: http://www.fields.utoronto.ca/sites/default/files/uploads/pre-talk_0.pdf

We prove a semi-global arithmetic Siegel-Weil formula as conjectured by Kudla, which relates the arithmetic intersection numbers of special cycles on GSpin Shimura varieties at a place of good reduction and the central derivatives of nonsingular Fourier coefficients of Siegel Eisenstein series. We will motivate this conjecture and discuss some aspects of the proof.

This is joint work with Wei Zhang.

In this talk, we shall study the growth of the etale wild kernels in various p-adic Lie extensions. The etale wild kernels (coined by Banaszak, Kolster, Nguyen Quand Do etc) are related to the special values of the Dedekind zeta function. In this talk, we reinterpret the etale wild kernel as an appropriate fine Selmer group in the sense of Coates-Sujatha. This viewpoint brings us to the problem of studying a control theorem of the said fine Selmer groups, which in turn allows us to minic the strategies developed by Greenberg. However, this improvisation is not a direct procedure, as one needs to estimate the growth of cohomology groups of open subgroups of p-adic Lie groups which is not accessible directly from the lie algebraic approach of Greenberg (one of the main issue is that the open subgroups have the same lie algebra and so the cohomology of the lie algebra cannot distinguish the cohomology groups of the subgroups). Among the tools used in estimatiing these cohomology groups, one notable ingredient is Tate's lemma which asserts the vanishing of the first $\Gamma$-cohomology groups of nonzero Tate twist of Qp/Zp.

Once we have such a control theorem, we apply them to obtain asymptotic growth formulas for the etale wild kernels in various said p-adic Lie extensions. The leading terms of the growth formulas are related to a certain Galois group, and an appropriate noncommutative variant of Greenberg's conjecture predicts that this said Galois group is not "too big". In particular, we shall see that Greenberg conjecture gives an asymptotic upper bound on the growth of the etale wild kernels. These upper bound are not necessarily always optimal. Indeed, building on calculations of Sharifi, we can give some examples which show that the etale wild kernels can grow much slower than the predicted estimate of Greenberg.

Finally, if time permits, we shall mention briefly on a joint work with Debanajana Kundu on the fine Selmer groups of elliptic curves which builds on a natural analogue/generalization of Tate's lemma in the elliptic situation.

Introductory reading:

Greenberg - Iwasawa Theory—Past and Present: http://www.fields.utoronto.ca/sites/default/files/uploads/Greenberg.pdf

Kolster, Movahhedi - Galois co-descent of étale wild kernels and capitulation: http://www.fields.utoronto.ca/sites/default/files/uploads/Kolster-Movahhedi.pdf

Quang Do: Analogues supérieurs du noyau sauvage: http://www.fields.utoronto.ca/sites/default/files/uploads/NQD.pdf

In this talk, I will explain how to use the circle method and the recently proved Vinogradov mean-value conjecture in the Waring's problem, to obtain power saving results for a class of shifted convolution problems involving automorphic forms on GL(n).

Recording of the talk available upon request.

In 1984, Kumar Murty and I studied the normal number of prime factors of Fourier coefficients of modular forms. In recent joint work with Kumar Murty and Sudhir Pujahari, we study the normal number of prime factors of sums of Fourier coefficients of Hecke eigenforms using recent advances in the theory of $\ell$-adic Galois representations.

Fix a positive integer $N \geq 2$. In this talk, we will focus on the problem of determining all rational valued arithmetic functions, periodic with period $N$ such that $L(1,f) := \sum_{n \geq 1} f(n)/n = 0$. This study was initiated by S. Chowla in the 1960s, drawing inspiration from Dirichlet's theorem that $L(1,\chi)\neq 0$ for a non-principal character $\chi$. We will discuss recent joint work with M. Ram Murty, wherein we use a vanishing criterion of Okada to construct an explicit basis for the $\mathbb{Q}$-vector space of functions $f \pmod N$ such that $L(1,f)=0$. This enables us to extend previous works of Baker-Birch-Wirsing and Murty-Saradha, with the arithmetic nature of Euler's constant $\gamma$ emerging as an important theme.

This talk will be accessible to graduate students.

I will begin by reviewing the notion of an algebraic Hecke character over a number field in some depth. This part of the talk should be accessible to a wide audience; the only prerequisites being basic algebraic number theory and some sheaf theory. The main goal of the talk will be to discuss the critical values of L-functions attached to algebraic Hecke characters. I will especially discuss certain signs that appear in reciprocity laws satisfied by these special values. Towards the end, I will talk about how this generalizes to L-functions for GL(n) over a CM field.

Introductory reading:

Read the first 5 sections of "On the arithmetic of Hecke characters": http://www.fields.utoronto.ca/sites/default/files/uploads/arithmetic_hecke_character.pdf

This is a report on joint work with Philippe Michel and Liyang Yang. Let f, resp. g, denote a holomorphic cusp form on U(2,1), resp. U(1,1), of weight (-k, k/2), resp. k, for an even integer k , taken to be at least 260 for technical reasons. We consider the Rankin-Selberg L-function L(s, f x g), which is defined by base changing to GL over the relevant imaginary quadratic field. We average over a family of f with varying square-free levels, and establish a non-vanishing result at the central value, as well as a hybrid subconvexity in the level aspect.

The Cohen--Lenstra--Martinet conjectures have been verified in only two cases. Davenport--Heilbronn compute the average size of the 3-torsion subgroups in the class group of quadratic fields and Bhargava computes the average size of the 2-torsion subgroups in the class groups of cubic fields. The values computed in the above two results are very stable. In particular, work of Bhargava--Varma shows that they do not change if one instead averages over the family of quadratic or cubic fields satisfying any finite set of splitting conditions.

However for certain "thin" families of cubic fields, namely, families of monogenic and n-monogenic cubic fields, the story is very different. In this talk, we will determine the average size of the 2-torsion subgroups of the class groups of fields in these thin families. Surprisingly, these values differ from the Cohen--Lenstra--Martinet predictions! We will also provide an explanation for this difference in terms of the Tamagawa numbers of naturally arising reductive groups. This is joint work with Manjul Bhargava and Jon Hanke.

Let $k$ be a number field and $G$ be a finite group, and let $\mathfrak{F}_{k}^{G}$ be a family of number fields $K$ such that $K/k$ is normal with Galois group isomorphic to $G$. Together with Robert Lemke Oliver and Jesse Thorner, we prove for many families that for almost all $K \in \mathfrak{F}_k^G$, all of the $L$-functions associated to Artin representations whose kernel does not contain a fixed normal subgroup are holomorphic and non-vanishing in a wide region.

These results have several arithmetic applications. For example, we prove a strong effective prime ideal theorem that holds for almost all fields in several natural large degree families, including the family of degree $n$ $S_n$-extensions for any $n \geq 2$ and the family of prime degree $p$ extensions (with any Galois structure) for any prime $p \geq 2$. I will discuss this result, describe the main ideas of the proof, and share some applications to bounds on $\ell$-torsion subgroups of class groups, to the extremal order of class numbers, and to the subconvexity problem for Dedekind zeta functions.

Prep material:

1) If you are unfamiliar with the Chebotarev density theorem, watch Lillian’s Pierce 1 hour lecture on the motivating problem: https://www.youtube.com/watch?v=heK2pYlMi3U

2) If you are familiar with the Chebotarev density theorem, read Sections 1 and 2 of the paper. https://arxiv.org/pdf/2012.14422.pdf

I will construct several cohomology theories for pairs (X,f) consisting of an algebraic variety X and a regular function f on X, and explain how to produce a universal cohomology theory for such pairs. This universal cohomology theory takes its values in the tannakian category of Exponential Motives. I will then show by means of concrete examples how to compute some interesting tannakian fundamental groups of exponential motives, and explain their relevance for transcendence questions.

Prep material:

Read the introduction and the examples in ch. 12.

http://www.fields.utoronto.ca/sites/default/files/uploads/Exponential_motives.pdf