### Session S09 - Number Theory in the Americas

## Talks

Wednesday, July 14, 12:00 ~ 12:30 UTC-3

## Intersections of modular geodesics and quaternary theta series

### Henri Darmon

#### McGill U., Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

I will describe a formula expressing the generating series of topological intersections of real quadratic geodesics on modular or Shimura curves as products of certain indefinite Hecke theta series, which occur in different guises in recent work by Rickards, Vonk, and Rotger-Harris-Venkatesh and the speaker.

Wednesday, July 14, 12:30 ~ 13:00 UTC-3

## When is the ring of integers of a number field coverable?

### Omar Kihel

#### Brock U., Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

It is easy to see that a group cannot be the union of two of its proper subgroups. Scorza showed that a group is a union of three of its proper subgroups if and only if it has a quotient isomorphic to the Klein 4-group $V=C_2^2$. Similar results exist for coverings by four, five, and six proper subgroups. Consideration of a covering by seven proper subgroups yields a result akin to the two proper subgroups case: no group can be written as a union of seven of its proper subgroups. Few authors have considered the problem of covering a ring by its proper subrings. We say that a ring $R$ is coverable if $R$ is equal to a union of its proper subrings. If this can be done using a finite number of proper subrings, then $\sigma(R)$ denotes the \emph{covering number} of $R$, which is the minimum number of subrings required to cover $R$. We set $\sigma(R)=0$ if $R$ is not coverable, and we set $\sigma(R)=\infty$ if $R$ is coverable but not by a finite number of proper subrings. In this talk, among other results, we will answer to the two following questions:\\ \indent (1) When is the ring R of algebraic integers of a number field finitely coverable?\\ \indent (2) Calculate $\sigma(R)$?

Wednesday, July 14, 13:00 ~ 13:30 UTC-3

## On some exponential Diophantine equations involving sequences

### Alain Togbe

#### Purdue U. Northwest, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $(F_n)_{n\geq 0}$ be the Fibonacci sequence given by $F_0=0,~F_1=1$ and $F_{n}=F_{n-1}+F_{n-2}$ for all $n\geq 2$. We consider the Diophantine equation $$ F_n^x+F_{n+1}^x+\cdots+F_{n+k-1}^x=F_m $$ and determine all nonnegative integer solutions $(x, m, n, k)$ of this equation. Moreover, if we replace the Fibonacci sequence by the Pell sequence and the Padovan sequence, we solve the corresponding equations.

Joint work with E. Tchammou (IMSP, Benin) and F. Luca (Wits, South Africa and UNAM, Mexico).

Wednesday, July 14, 13:30 ~ 14:00 UTC-3

## Rational approximation and extension of scalars

### Damien Roy

#### U. Ottawa, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

We establish a new transference principle which, by extending scalars from $\mathbb{Q}$ to a number field, and by combination with a result of P. Bel, allows us to construct algebraic curves defined over $\mathbb{Q}$, of arbitrarily large degree, containing points that are very singular with respect to approximation by rational points. The results also admit interpretation in the setting of parametric geometry of numbers.

Joint work with Anthony Poels (U. Ottawa, Canada).

Wednesday, July 14, 14:00 ~ 14:30 UTC-3

## The conorm code of an AG-code

### Maria Chara

#### U. N. Litoral, Santa Fe, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $\mathbb{F}_q$ be a finite field with $q$ elements. For a given trascendental element $x$ over $\mathbb{F}_q$, the field of fractions of the ring $\mathbb{F}_q[x]$ is denoted as $\mathbb{F}_q(x)$ and it is called a rational function field over $\mathbb{F}_q$. An (algebraic) function field $F$ of one variable over $\mathbb{F}_q$ is a field extension $F/\mathbb{F}_q(x)$ of finite degree. The \textit{Riemann-Roch space} associated to a divisor $G$ of $F$ is the vector space over $\mathbb{F}_q$ defined as $$\mathcal{L}(G)=\{x\in F\,:\, (x)\geq G\}\cup \{0\},$$ where $(x)$ denotes the principal divisor of $x$. It turns out that $\mathcal{L}(G)$ is a finite dimensional vector space over $\mathbb{F}_q$ for any divisor $G$ of $F$. Given disjoint divisors $D=P_1+\cdots+P_n$ and $G$ of $F/\mathbb{F}_q$, where $P_1,\ldots,P_n$ are different rational places, the \textit{algebraic geometry code} (AG-code for short) associated to $D$ and $G$ is defined as \begin{equation} \label{CDG} C_\mathcal{L}^F (D,G) = \{(x(P_1),\ldots, x(P_n))\,:\,x\in \mathcal{L}(G)\}\subseteq (\mathbb{F}_q)^n, \end{equation} where $x(P_i)$ denotes the residue class of $x$ modulo $P_i$ for $i=1,\ldots,n$. In this talk the concept of the \textit{conorm code} associated to an AG-code will be introduced. We will show some interesting properties of this new code since some well known families of codes such as repetition codes, Hermitian codes and Reed-Solomon codes can be obtained as conorm codes from other more basic codes. We will see that in some particular cases over geometric Galois extensions of function fields, the conorm code and the original code are different representations of the same algebraic geometry code.

Joint work with R. Podesta (U. N. Cordova, Argentina) and R. Toledano (U.N. Litoral, Argentina).

Wednesday, July 14, 14:30 ~ 15:00 UTC-3

## On a construction of Poincar\'e series for $SU(2,1)$

### Roberto Miatello

#### U. N. Cordova, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

Poincar\'e series are holomorphic automorphic forms on the upper half-plane, defined by Poincar\'e, and studied by Hecke and Petersson who showed in particular that the inner product of a cusp form $f$ and a Poincar\'e series $P_m$ is a nonzero multiple of the $m^{th}$ Fourier coefficient of $f$. As a consequence, the Poincar\'e series span the full space of cusp forms of weight $k$, for each weight $ k> 2$. In the case of a larger group like the group $G=SU(2,1)$, this construction cannot be directly generalized since there exist many non-generic cusp forms, i.e., those having all its Fourier coefficients equal to zero. Jointly with Roelof Bruggeman we have defined a holomorphic family of Poincar\'e series attached to a Stone-von Neumann representation of the maximal unipotent subgroup $N$ of $G$. After carrying out a meromorphic continuation, we study the singularities and the special values, showing in particular that, as in the classical case, the special values of the family span the spaces holomorphic (and antiholomorphic) cusp forms of the corresponding symmetric space $G/K$ (the complex hyperbolic space $CH^2$).

Wednesday, July 14, 15:00 ~ 15:30 UTC-3

## Towards a classification of adelic Galois representations attached to elliptic curves over ${\mathbb Q}$

### Alvaro Lozano-Robledo

#### U. Connecticut, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. The adelic Galois representation attached to $E$ (this object will be defined during the talk) captures all sorts of interesting information about the arithmetic of the points on $E(\overline{\mathbb{Q}})$, including data about the torsion subgroup, isogenies, and other finer invariants of the curve and its isogeny class. In this talk, we will give a summary of recent results towards the classification (up to isomorphism) of the possible adelic Galois representations that arise from elliptic curves over $\mathbb{Q}$, and present some recent results of the author and his collaborators (Garen Chiloyan, Harris Daniels, Jackson Morrow) in this area.

Wednesday, July 14, 15:30 ~ 16:00 UTC-3

## Creating normal numbers using the factorisation of integers

### Jean-Marie De Koninck

#### U. Laval, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

Given an integer $q\ge 2$, we say that an irrational number is {\it $q$-normal} if, as we examine its base $q$ expansion, we find that any sequence of $k$ digits appears with a frequency of $1/q^k$. We will show how one can use the local chaos and global regularity both inherent to the factorization of integers in order to create large families of normal numbers.

Joint work with Imre K\'atai (Eotvos Lorand University, Hungary).

Thursday, July 15, 11:00 ~ 12:00 UTC-3

## Galois groups of random integer polynomials

### Manjul Bhargava

#### Princeton , USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

Of the $(2H+1)^n$ monic integer polynomials $f(x)=x^n+a_1x^{n-1}+\cdots+a_n$ with $\max\{|a_1|,\ldots,|a_n|\}\leq H$, how many have associated Galois group that is not the full symmetric group $S_n$? There are clearly $\gg H^{n-1}$ such polynomials, as can be seen by setting $a_n=0$. In 1936, van der Waerden conjectured that $O(H^{n-1})$ should in fact also be the correct upper bound for the count of such polynomials. The conjecture has been known for $n\leq 4$ due to work of van der Waerden and Chow and Dietmann. In this talk, we describe how to prove van der Waerden's Conjecture for all degrees $n$.

Thursday, July 15, 12:00 ~ 12:30 UTC-3

## Expansion, divisibility and parity

### Harald Helfgott

#### U. Gottingen, Germay - This email address is being protected from spambots. You need JavaScript enabled to view it.

We will discuss a graph that encodes the divisibility properties of integers by primes. We show that this graph is shown to have a strong local expander property almost everywhere. We then obtain several consequences in number theory, beyond the traditional parity barrier. For instance: for $\lambda$ the Liouville function (that is, the completely multiplicative function with $\lambda(p) = -1$ for every prime), $$ \frac{1}{\log x} \sum_{n\leq x} \frac{\lambda(n) \lambda(n+1)}{n} = O\left(\frac{1}{{\sqrt{\log \log x}}}\right), $$ which is stronger than a well-known result by Tao. We also manage to prove, for example, that $\lambda(n+1)$ averages to 0 at almost all scales when $n$ is restricted to have a specific number of prime divisors $\Omega(n)=k$, for any "popular" value of $k$ (that is, $k = \log \log N + O({\sqrt{\log \log N}})$ for $n\le N$).

Joint work with M. Radziwill (Caltech, USA).

Thursday, July 15, 12:30 ~ 13:00 UTC-3

## Sums of certain arithmetic functions over $\mathbb{F}_q[T$] and symplectic distributions

### Matilde Lalin

#### U. Montreal, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

In 2018 Keating, Rodgers, Roditty-Gershon and Rudnick established relationships of the mean-square of sums of the divisor function $d_k(f)$ over short intervals and over arithmetic progressions for the function field $\mathbb{F}_q[T]$ to certain integrals over the ensemble of unitary matrices. We consider similar problems leading to distributions over the ensemble of symplectic matrices. We also consider analogous questions involving convolutions of the von Mangoldt function.

Joint work with Vivian Kuperberg (Stanford U., USA).

Thursday, July 15, 13:00 ~ 13:30 UTC-3

## Irreducibility of random polynomials of large degree

### Dimitris Koukoulopoulos

#### U. Montreal, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $f(x)=a_0+a_1x+\cdots+a_{n-1}x^{n-1}+x^n$ be a random monic polynomial, where $a_j$ is chosen uniformly at random from $\{0,1\}$ and independently of the other coefficients. In 1993, Odlyzko and Poonen conjectured that $f(x)$ is irreducible with probability $\sim1/2$ when $n\to\infty$. Breuillard and Varj\'u proved that this expectation is indeed true under the Generalized Riemann Hypothesis. In this talk, I will present recent joint work with Bary-Soroker and Kozma that proves that $f(x)$ is irreducible with probability $\ge1/1000$ for all large enough $n$. In addition, if we condition on the event that $f(x)$ is irreducible, then we prove that the Galois group of $f(x)$ contains the alternating group $A_n$ with conditional probability $\sim1$. The proofs use a fun mixture of ideas from sieve methods, the arithmetic of polynomials over finite fields, $p$-adic Fourier analysis, primes with restricted digits, Galois theory and group theory.

Thursday, July 15, 13:30 ~ 14:00 UTC-3

## Trace formulas for locally supercuspidal cusp forms

### Andrew Knightly

#### U. Maine, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

I will report on work in progress to compute trace formulas on spaces of GL(2) cusp forms with prescribed supercuspidal local behavior.

Thursday, July 15, 14:00 ~ 14:30 UTC-3

## Trinomials, singular moduli and Riffaut's conjecture

### Amalia Pizarro

#### U. Valparaiso, Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.

Riffaut (2019) conjectured that a singular modulus of degree $h > 2$ cannot be a root of a trinomial with rational coefficients. In this talk, we will show that this conjecture follows from the GRH, and obtain partial unconditional results.

Joint work with Yu. F. Bilu (U. Bordeaux, France) and F. Luca (Wits, South Africa and UNAM, Mexico).

Thursday, July 15, 14:30 ~ 15:00 UTC-3

## Some recent development on the Iwasawa theory of fine Selmer groups

### Antonio Lei

#### U. Laval, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

In Iwasawa theory, we are interested in the asymptotic behaviour of arithmetic objects over a tower of extensions. In the early 2000's, Coates and Sujatha initiated the studies of the Iwasawa theory of the fine Selmer group of an elliptic curve. Fine Selmer groups have the advantage that many arithmetic properties at a fixed prime are independent of the reduction type of the elliptic curve, giving rise to uniform results that resemble those in classical Iwasawa Theory. In this talk, we will discuss some of these properties and recent results on certain relations between fine Selmer groups and the growth of Mordell-Weil ranks of an elliptic curve.

Thursday, July 15, 15:00 ~ 15:30 UTC-3

## Zero-sum squares in bounded discrepancy $\mathbf{\{-1,1\}}$-matrices

### Amanda Montejano

#### UNAM, Juriquilla, Mexico - This email address is being protected from spambots. You need JavaScript enabled to view it.

A \emph{square} in a matrix ${\mathcal M}=(a_{ij})$ is a $2\times 2$ sub-matrix of ${\mathcal M}$ with entries $a_{i,j}, a_{i+s,j}, a_{i,j+s}, a_{i+s,j+s}$ for some $s\ge 1$. An \emph{Erickson matrix} is a square binary matrix that contains no squares with constant entries. In [4], Erickson asked for the maximum value of $n$ for which there exists an $n\times n$ Erickson matrix. In [2] Axenovich and Manske gave an upper bound of around $2^{2^{40}}$. This gargantuan bound was later improved by Bacher and Eliahou in [3] using computational means to the optimal value of $15$.

In this talk we present the study of a zero-sum analogue of the Erickson matrices problem where we consider binary matrices with entries in $\{-1,1\}$. For this purpose, of course, we need to take into account the discrepancy or deviation of the matrix, defined as the sum of all its entries, that is \[ {\rm disc}({\mathcal M})=\sum_{\substack{1\le i\le n \\ 1\le j\le m}} a_{i,j}. \]

A \emph{zero-sum square} is a square $S$ with ${\rm disc}(S)=0$. A natural question is, for example, the following: is it true that for sufficiently large $n$ every $n\times n$ $\{-1,1\}$-matrix ${\mathcal M}$ with ${\rm disc}({\mathcal M})=0$ contains a zero-sum square? We answered positive to this question. Since, our proof uses an induction argument, in order for the induction to work we prove the following stronger statement: For $n\ge 5$ and $m\in \{n,n+1\}$, every $n\times m$ $\{-1,1\}$-matrix $M$ with ${\rm abs}{\rm disc}(M)\le n$ contains a zero-sum square except for the split matrix (up to symmetries), where a split matrix is a matrix with all entries above the diagonal equal to $-1$ and all remaining entries equal to $1$.

\medskip

\par\noindent [1] \quad A. R. Ar\'evalo, A. Montejano and E. Rold\'an-Pensado, \emph{Zero-sum squares in bounded discrepancy $\{-1,1\}$-matrices}, arXiv:2005.07813 (2020). \par\noindent [2] M.~Axenovich and J.~Manske, \emph{On monochromatic subsets of a rectangular grid}, Integers \textbf{8} (2008), A21, 14. \par\noindent [3] R.~Bacher and S.~Eliahou, \emph{Extremal binary matrices without constant 2-squares}, J. Comb. \textbf{1} (2010), no.~1, 77--100. \par\noindent [4] M.~J. Erickson, \emph{Introduction to combinatorics}, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley \& Sons, Inc., New York, 1996, A Wiley-Interscience Publication.

Joint work with Alma R. Ar\'evalo (UNAM, Mexico) and Edgardo Rold\'an Pensado (UNAM, Mexico).

Thursday, July 15, 15:30 ~ 16:00 UTC-3

## Rational points of degree D on a planar curve

### Andrew Granville

#### U. Montreal, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

Given a planar curve C we ask for which integers d are there rational points on the curve in a field of degree d. There is a lot of structure in the set of values d, some of which we will explain in this talk.

Joint work with Lea Beneish (McGill University, Canada).

Thursday, July 22, 16:00 ~ 16:30 UTC-3

## Unconditional discriminant lower bounds exploiting violations of the Generalized Riemann Hypothesis

### Eduardo Friedman

#### U. Chile, Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.

In the 1970's Andrew Odlyzko proved good lower bounds for the discriminant of a number field. He also showed that his results could be sharpened by assuming the Generalized Riemann Hypothesis. Some years later Odlyzko suggested that it might be possible to do without GRH. I shall explain Odlyzko's ideas and sketch how for number fields of reasonably small degree (say up to degree 11 or 12) one can indeed improve the lower known bounds by exploiting hypothetical violations of GRH. Our method extends unpublished results of Matias Atria.

Joint work with Karim Belabas (U. Bordeaux, France) and Francisco Diaz y Diaz (U. Bordeaux, France).

Thursday, July 22, 16:30 ~ 17:00 UTC-3

## Hecke characters and some diophantine equations

### Ariel Pacetti

#### U. N. Cordova, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk we will study solutions to the equation $x^2 + dy^6 = z^p$. We will explain how to attach a ${\mathbb Q}$-curve over a quadratic field to a putative solution, and how to extend the representation to a rational one of a concrete level and Nebentypus (corresponding to a classical modular form via Serre's conjectures). The way to explicitly obtain the level and nebentypus is via the construction of a Hecke character with some desired properties. After computing the respective space of classical modular forms, some classical elimination techniques allow us to prove that no non-trivial solution exists for some values of $d$ and all $p$ sufficiently large.

Thursday, July 22, 17:00 ~ 17:30 UTC-3

## Genus fields of Kummer extensions of rational function fields

### Martha Rzedowski

#### CINVESTAV, IPN, Mexico - This email address is being protected from spambots. You need JavaScript enabled to view it.

We obtain the genus field of a general Kummer extension of a global rational function field. First we study the case of a Kummer extension of degree a power of a prime. Then we prove that the genus field of the composite of two abelian extensions of a global rational function field of relatively prime degrees is equal to the composite of the genus fields of the extensions. As a consequence is obtained the genus field of a general Kummer extension.

Joint work with Gabriel Villa (CINVESTAV, IPN, Mexico).

Thursday, July 22, 17:30 ~ 18:00 UTC-3

## Ramification in tamely ramified Galois towers of function fields

### Ricardo Toledano

#### U. N. Litoral, Santa Fe, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

We study the set of all ramified places in a finite Galois extension E/F of function fields. We give an upper bound for the size of this set in terms of the index of the conorm of Div(F), the divisor group of F, in the subgroup of Div(E) defined for the fixed divisors of E under the action of the Galois group of E/F. We use this bound to give necessary and sufficient conditions to have finite ramification locus in a tamely ramified Galois tower of function fields over a finite field.

Thursday, July 22, 18:00 ~ 18:30 UTC-3

## Hasse-Witt matrices and mirror toric pencils

### Adriana Salerno

#### Bates College, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

Mirror symmetry predicts unexpected relationships between arithmetic properties of distinct families of algebraic varieties. For example, Wan and others have shown that for some mirror pairs, the number of rational points over a finite field matches modulo the order of the field. In this talk, we obtain a similar result for certain mirror pairs of toric varieties. We use recent results by Huang, Lian, Yau and Yu describing the relationship between the Picard-Fuchs equations and the Hasse-Witt matrix of these varieties, which encapsulates information about the number of points. The result allows us to compute the number of points modulo the order of the field explicitly, and we illustrate this by computing K3 surface examples related to hypergeometric functions.

Joint work with Ursula Whitcher (U. Michigan, USA).

Thursday, July 22, 18:30 ~ 19:00 UTC-3

## Fractal geometry of the Markov and Lagrange spectra and their set difference

### Carlos Gustavo Moreira

#### IMPA, Brasil - This email address is being protected from spambots. You need JavaScript enabled to view it.

We will discuss some recent results on the fractal geometry of the Markov and Lagrange spectra from Diophantine approximations, and their set difference - in particular we show, in collaboration with Carlos Matheus, Mark Pollicott and Polina Vytnova that the smallest element $t_1$ of the Lagrange spectrum $L$ such that $L\cap (-\infty,t_1)$ has Hausdorff dimension $1$ has decimal representation starting by $0.334384$, proving rigorously and improving previous heuristic estimates by R. Bumby, which suggest that $0.33437

Thursday, July 22, 19:00 ~ 19:30 UTC-3

## Generalized Fibonacci and Pell numbers

### Jhon Bravo

#### U. Cauca, Colombia - This email address is being protected from spambots. You need JavaScript enabled to view it.

The Pell sequence $(P_n)_{n\geq 0}$ is the second order linear recurrence defined by $P_n=2P_{n-1}+P_{n-2}$ with initial conditions $P_0=0$ and $P_1=1$. In this talk, we present some recent work on a generalization of the Pell sequence called the $k$-Pell sequence $(P_n^{(k)})_{n}$ which is generated by a recurrence relation of a higher order. We report about some arithmetic properties of $(P_n^{(k)})_{n}$ and study some Diophantine equations involving Fibonacci and $k$-Pell numbers.

Joint work with J. L. Herrera (U. Cauca, Colombia) and F. Luca (Wits, South Africa and UNAM, Mexico).

Thursday, July 22, 19:30 ~ 20:00 UTC-3

## Joint Shapes of Quartic Fields and Their Cubic Resolvents

### Piper H

#### University of Toronto , Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

In studying the (equi)distribution of shapes of quartic number fields, one relies heavily on Bhargava's parametrizations which brings with it a notion of resolvent ring. Maximal rings have unique resolvent rings so it is possible to live a long and healthy life without understanding what they are. The authors have decided, however, to forsake such bliss and look into what ever are these rings and what happens if we consider their shapes along with our initial number fields.

Joint work with Christelle Vincent (University of Vermont, USA).

Thursday, July 22, 20:30 ~ 21:00 UTC-3

## Integral theta correspondence between two $\lambda$-resolvent Green functions

### Hugo Chapdelaine

#### U. Laval, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $F$ be a real quadratic field and let $\{\infty_1,\infty_2\}$ be its two real places. Let $B_1/F$ and $B_2/F$ be two quaternion algebras defined over $F$. We shall assume that $B_1$ is everywhere unramified (so that $B_1\simeq M_2(F)$) and that $B_2$ is ramified exactly in the two places $\{\infty_1,w\}$ where $w$ is a finite place of $F$. Let $O_i\subseteq B_i$ ($i=1,2$) be two orders which have been suitably chosen. One may associate to $O_i$ a couple $(V_i,\Delta_i)$ where $V_i$ is a Hilbert vector space of automorphic functions and where $\Delta_i$ is a Laplacian-like linear operator. The resolvent of $\Delta_i$, namely $(\Delta_i-\lambda)^{-1}$, can be written as an integral against a kernel which is given by some explicit automorphic Green function $G_{\lambda}^{i}$ ($i=1,2$). In this talk which shall present an equality between two integrals where $G_{\lambda}^{1}$ appears on the left hand side while $G_{\lambda}^{2}$ appears on the right hand side. Afterwards, we shall sketch how one can develop the integrals which appear on both side of this equality in order to obtain, a priori, non-trivial \,\lq\lq automorphic identities\rq\rq. A main motivation for this project was the famous Jacquet-Langlands correspondence which was published in 1970.

Thursday, July 22, 20:30 ~ 21:00 UTC-3

## Values of the Ramanujan-tau function

### Mike Bennett

#### UBC, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

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.

Joint work with Adela Gherga (U. Warwick, UK), Vandita Patel (U. Manchester, UK) and Samir Siksek (U. Warwick, UK).