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# Lalin, Matilde

Full Professor

Faculty of Arts and Science - Department of Mathematics and Statistics

514 343-6689

### Affiliations

• Membre Centre de recherches mathématiques
• Membre CRM  Centre de recherches mathématiques

### Student supervision Expand allCollapse all

Polylogarithmes et mesure de Mahler Theses and supervised dissertations / 2020-09
Gu, Jarry
Abstract
The main purpose of this thesis is to compute the logarithmic Mahler measure of the three variable polynomial family xn + 1 + (xn−1 + 1)y + (x − 1)z. In order to accomplish this, we integrate regulators defined on polylogarithmic motivic complexes. To understand these regulators, we explore the properties of polylogarithms and show some polylogarithmic identities. The regulators are then applied to simplify the integrand. Our result is a formula relating the Mahler measure of the family of polynomials to the Bloch–Wigner Dilogarithm and the Riemann zeta function.

Generalizations of monsky matrices for elliptic curves in legendre form Theses and supervised dissertations / 2020-04
Mokrani, Youcef
Abstract
A positive integer n is said to be congruent if it is the area of a right triangle whose sides are all of rational length. The task of finding which integers are congruent is an old and famous yet still open question in arithmetic geometry called the congruent number problem. It is linked to the theory of elliptic curves as the integer n is congruent if and only if the elliptic curve y²=x³-n²x has a rational point of infinite order. The link between congruent numbers and elliptic curves enables the application of techniques from algebraic geometry to study the problem. One of these methods is the concept of Monsky matrices that can be used to calculate the size of the 2-Selmer group of the elliptic curve y²=x³-n²x. One can use these matrices in order to find new infinite families of non-congruent numbers. The connection to elliptic curves also introduces generalizations to the congruent number problem. For example, one may consider the θ-congruent number problem which studies triangles with a fixed angle of θ instead of only right triangles. This problem is also related to elliptic curves and the concept of Monsky matrices can be generalized to it. In fact, Monsky matrices can be generalized to any elliptic curve that has a Legendre form over the rationals. The goal of this thesis is to construct such a generalization and then to apply it to relevant problems in arithmetic geometry to efficiently reprove old results and find new ones.

Generalized Mahler measure of a family of polynomials Theses and supervised dissertations / 2019-12
Roy, Subham
Abstract
In this thesis we consider a variation of the Mahler measure where the defining integral is performed over a more general torus. Our work is based on a tempered family of polynomials originally studied by Boyd, Boyd P_k (x, y) = x + 1/x + y + 1/y + k with k ∈ R_{>4}. For the k = 4 case we use special values of the Bloch-Wigner dilogarithm to obtain the Mahler measure of P_4 over an arbitrary torus (T_ {a, b})^2 = {(x, y) ∈ C* X C* : | x | = a, | y | = b } with a, b ∈ R_{> 0}. Next we establish a relation between the Mahler measure of P_8 over a torus(T_ {a, √a} )^2 and its standard Mahler measure. The combination of this relation with results due to Lalin, Rogers, and Zudilin leads to a formula involving the generalized Mahler measure of this polynomial given in terms of L'(E, 0). In the end, we propose a strategy to prove some similar results for the general case k > 4 over (T_ {a, b})^2 with some restrictions on a, b.

La mesure de Mahler d'une forme de Weierstrass Theses and supervised dissertations / 2019-05
Giard, Antoine
Abstract
This master’s thesis goal is to give a simple and brief introduction about Mahler measure and its connections with elliptic curves L-functions. This theory culminates with the Bloch-Beı̆linson conjectures which we try to explain at the end of chapter 2, the first two chapters serving to introduce the necessary requirements to understand them and as a stepping stone to this master’s main question which is to find a link between the Mahler measure of y 2 + 4xy + 2y − x 3 and the L-function associated to it. For this purpose, we see that the conjectured relation given by D. Boyd [1] is false but we still study the homology cycles and integration paths associated to the elliptic curves y 2 + 4xy + 2y − x 3 = 0 and (1 + x)(1 + y)(x + y) + 2xy = 0.

Étude du nombre de polynômes irréductibles dans les corps finis avec certaines contraintes imposées aux coefficients Theses and supervised dissertations / 2016-08
Beauchamp Houde, Gabriel
Abstract
The objective of this thesis is to count monic irreducible polnomials over a finite field under some conditions on the coefficients of the polynomial. These conditions will be simply to fix some coefficients, or to fix their sign, cubicity or quarticity.

Dénombrement des polynômes irréductibles unitaires dans les corps finis avec différentes contraintes sur les coefficients Theses and supervised dissertations / 2014-09
Larocque, Olivier
Abstract
The objective of this thesis is to count the monic irreducible polynomial over finite field with some conditions related to coefficient. First, we will determine the trace of the polynomial. Thereafter, we will elect the cotrace when the trace is already fixed to zero. Finally, we will discuss the case where the trace and the constant term are fixed at the same time

Évaluation du régulateur sur une courbe modulaire et valeurs particulières Theses and supervised dissertations / 2014-09
Bouchard, Nicolas
Abstract
Bloch and Beilinson conjectured many relations regarding the regulator of a modular curve. This function from the algebraic K-theory of the modular curve is supposed to be related to special values of L functions. Let N be a positive integer et consider the congruence subgroup $\Gamma_0(N)$. This thesis relates explicitly the regulator of the modular curve $X_0(N)$ applied to some newform with a special value of the newform's L function.

A Generalization of a Theorem of Boyd and Lawton Theses and supervised dissertations / 2012-08
Issa, Zahraa
Abstract
This thesis applies to study first, in part 1, the Mahler measure of polynomials in one variable. It starts by giving some definitions and results that are important for calculating this height. It also addresses the topic of Lehmer’s question, an interesting conjecture in the field, and it gives some examples and results aimed at resolving the issue. The extension of the Mahler measure to several variable polynomials is then considered including the subject of limit points with some examples. In the second part, we first give definitions of a higher order for the Mahler measure, and generalize from single variable polynomials to multivariable polynomials. Lehmer’s question has a counterpart in the area of the higher Mahler measure, but with totally different answers. At the end, we reach our goal, where we will demonstrate the generalization of a theorem of Boyd-Lawton. This theorem shows a relation between the limit of Mahler measure of multivariable polynomials with Mahler measure of polynomials in one variable. This result has implications in terms of Lehmer's conjecture and serves to clarify the relationship between the Mahler measure of one variable polynomials, and the Mahler measure of multivariable polynomials, which are very different.

Mesure de Mahler supérieure de certaines fonctions rationelles Theses and supervised dissertations / 2012-08
Lechasseur, Jean-Sébastien
Abstract
The 2-higher and 3-higher Mahler measure of some rational functions are given in terms of special values of the Riemann zeta function, a Dirichlet L-function and multiple polylogarithms. Our results generalize those obtained in [10] for the classical Mahler measure. We improve one of our results by providing a reduction for a certain linear combination of multiple polylogarithms in terms of Dirichlet L-functions. We conclude by giving a complete reduction of a special case.

### Selected publications Expand allCollapse all

#### Higher Mahler measure of an $n$-variable family

Lal\'in, Matilde et Lechasseur, Jean-S\'ebastien, Higher Mahler measure of an $n$-variable family , (2015), , Acta Arith.

#### Secant zeta functions

Lalín, Matilde, Rodrigue, Francis et Rogers, Mathew, Secant zeta functions 409, 197--204 (2014), , J. Math. Anal. Appl.

#### Statistics for ordinary Artin-Schreier covers and other $p$-rank strata

Bucur, Alina, David, Chantal, Feigon, Brooke, et Lalín, Matilde, Statistics for ordinary Artin-Schreier covers and other $p$-rank strata , (2014), , Trans. Amer. Math. Soc.

#### The number of irreducible polynomials with first two prescribed coefficients over a finite field

Lal\'in, Matilde et Larocque, Olivier, The number of irreducible polynomials with first two prescribed coefficients over a finite field , (2014), , Rocky Mountain J. Math.

#### The distribution of $\mathbb{F}_q$-points on cyclic $\ell$-covers of genus $g$

Bucur, Alina, David, Chantal, Feigon, Brooke, Kaplan, Nathan, Lalín, Matilde, Ozman, Ekin, et Wood, Melanie Matchett, The distribution of $\mathbb{F}_q$-points on cyclic $\ell$-covers of genus $g$ , (2014), , Int. Math. Res. Not. IMRN

#### Equations for Mahler measure and isogenies

Lal\'in, Matilde, Equations for Mahler measure and isogenies 25, 387--399 (2013), , Proceedings of the Cuartas Jornadas de Teoria de Numeros.'' J. Th\'eor. Nombres Bordeaux

#### Mahler measure of some singular $K3$-surfaces

Bertin, Marie-Jos\'e, Feaver, Amy, Fuselier, Jenny, Lal\'in, Matilde et Manes, Michelle, Mahler measure of some singular $K3$-surfaces 606, 149--169 (2013), , WIN2 - Women in Numbers 2, Contemp. Math., Amer. Math. Soc., Providence, RI

#### Mahler measure of multivariable polynomials

Bertin, Marie-Jos\'e et Lal\'in, Matilde N. , Mahler measure of multivariable polynomials 606, 125--147 (2013), , WIN2 - Women in Numbers 2, Contemp. Math., Amer. Math. Soc., Providence, RI

#### WIN2 - Women in Numbers 2

David, Chantal, Lal\'in, Matilde, Manes, Michelle, \'Editrices, WIN2 - Women in Numbers 2 606, x+207 (2013), , Comtemp. Math., Amer. Math. Soc. Providence, RI

#### Unimodularity of zeros of self-inversive polynomials

Lalín, Matilde N. et Smyth, Chris J., Unimodularity of zeros of self-inversive polynomials 138, 85--101 (2013), , Acta Math. Hungar.

#### Variations of the Ramanujan polynomials and remarks on $\zeta(2j+1)/\pi^{2j+1}$

Lalín, Matilde N. et Rogers, Mathew D., Variations of the Ramanujan polynomials and remarks on $\zeta(2j+1)/\pi^{2j+1}$ 48, 91--111 (2013), , Funct. Approx. Comment. Math.

#### A generalization of a theorem of Boyd and Lawton

Issa, Zahraa et Lalín, Matilde, A generalization of a theorem of Boyd and Lawton 56, 759--768 (2013), , Canad. Math. Bull.

#### Mahler measure and elliptic curve $L$-functions at $s=3$

Lal\'in, Matilde N. , Mahler measure and elliptic curve $L$-functions at $s=3$ , (2013), , J. Reine Angew. Math.

#### Distribution of zeta zeroes of Artin-Schreier covers

Bucur, Alina, David, Chantal, Feigon, Brooke, Lalín, Matilde et Sinha, Kaneenika, Distribution of zeta zeroes of Artin-Schreier covers 19, 1329--1356 (2012), , Math. Res. Lett.

#### Higher Mahler measure for cyclotomic polynomials and Lehmer's question

Lalín, Matilde et Sinha, Kaneenika, Higher Mahler measure for cyclotomic polynomials and Lehmer's question 26, 257--294 (2011), , Ramanujan J.

#### Biased statistics for traces of cyclic p-fold covers over finite fields

Bucur, Alina, David, Chantal, Feigon, Brooke et Lal\'in, Matilde, Biased statistics for traces of cyclic p-fold covers over finite fields 60, 121--143 (2011), , WIN - Women in Numbers, Fields Institute Communications, Amer. Math. Soc., Providence, RI

#### Higher Mahler measure as a Massey product in Deligne Cohomology, Low-dimensional Topology and Number Theory

LALIN Maltide, Higher Mahler measure as a Massey product in Deligne Cohomology, Low-dimensional Topology and Number Theory 7 no.3, (2010), , Oberwolfach Reports

#### Fluctuations in the number of points on smooth plane curves over finite fields

Bucur, Alina, David, Chantal, Feigon, Brooke et Lalín, Matilde, Fluctuations in the number of points on smooth plane curves over finite fields 130, 2528--2541 (2010), , J. Number Theory

#### On a conjecture by Boyd

Lalín, Matilde N., On a conjecture by Boyd 6, 705--711 (2010), , Int. J. Number Theory

#### Higher Mahler measure as a Massey product in Deligne Cohomology

Lal\'in, Matilde N., Higher Mahler measure as a Massey product in Deligne Cohomology 7, 2101--2163 (2010), , Low-dimensional Topology and Number Theory. Abstracts from the workshop held August 15-August 21, 2010. Organized by Paul E. Gunnells, Walter Neumann, Adam S. Sikora and Don Zagier. Oberwolfach Reports.

#### Statistics for traces of cyclic trigonal curves over finite fields

Bucur, Alina, David, Chantal, Feigon Brooke et Lal\'in Matilde, Statistics for traces of cyclic trigonal curves over finite fields no. 5, , 932--967 (2010), , Int. Math. Res. Not. IMRN

#### Mahler measure under variations of the base group

Dasbach, Oliver T. et Lal\'in, Matilde N., Mahler measure under variations of the base group 21, 621--637 (2009), , Forum Math.

#### On the recurrence of coefficients in the L\"uck-Fuglede-Kadison determinant

Dasbach, Oliver T. et Lal\'in, Matilde N., On the recurrence of coefficients in the L\"uck-Fuglede-Kadison determinant , 119--134 (2008), , Proceedings of the Segundas Jornadas de Teoria de Numeros.'' Bib. Rev. Mat. Iberoam. Rev. Mat. Iberoamericana, Madrid

#### Higher Mahler measures and zeta functions

Kurokawa, Nobushige , Lalín, Matilde N. et Ochiai, Hiroyuki., Higher Mahler measures and zeta functions 135, 269--297 (2008), , Acta Arith.

#### Mahler measures and computations with regulators

Lalín, Matilde N., Mahler measures and computations with regulators 128, 1231--1271 (2008), , J. Number Theory

#### An algebraic integration for Mahler measure

Lal\'in, Matilde N., An algebraic integration for Mahler measure 138, 391--422 (2007), , Duke Math. J.

#### On the Mahler measure of resultants in small dimensions

D'Andrea, Carlos A. et Lalín, Matilde N., On the Mahler measure of resultants in small dimensions 209, 393--410 (2007), , J. Pure Appl. Algebra

#### Functional equations for Mahler measures of genus-one curves

Lal\'in, Matilde et Rogers, Mathew D., Functional equations for Mahler measures of genus-one curves 1, 87--117 (2007), , Algebra Number Theory

#### On certain combination of colored multizeta values

Lalín, Matilde N., On certain combination of colored multizeta values 21, 115--127 (2006), , J. Ramanujan Math. Soc.

#### Mahler measure of some $n$-variable polynomial families

Lalín, Matilde N., Mahler measure of some $n$-variable polynomial families 116, 102--139 (2006), , J. Number Theory

#### Mahler measure and volumes in hyperbolic space

Lalín, Matilde N., Mahler measure and volumes in hyperbolic space 107, 211--234 (2004), , Geom. Dedicata

#### Some examples of Mahler measures as multiple polylogarithms

Lalín, Matilde N., Some examples of Mahler measures as multiple polylogarithms 103, 85--108 (2003), , J. Number Theory