I share some of my adventures in mathematical research and homeschooling in the time of COVID-19

Lalín, Matilde

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Journal of Humanistic Mathematics Volume 11 | Issue 2 July 2021 Arithmetics, Interrupted Matilde Lalín Université de Montréal Follow this and additional works at: https://scholarship.claremont.edu/jhm  Part of the Number Theory Commons, and the Online and Distance Education Commons Recommended Citation Lalín, M. "Arithmetics, Interrupted," Journal of Humanistic Mathematics, Volume 11 Issue 2 (July 2021), pages 265-270. . Available at: https://scholarship.claremont.edu/jhm/vol11/iss2/11 ©2021 by the authors. This work is licensed under a Creative Commons License. JHM is an open access bi-annual journal sponsored by the Claremont Center for the Mathematical Sciences and published by the Claremont Colleges Library | ISSN 2159-8118 | http://scholarship.claremont.edu/jhm/ The editorial staff of JHM works hard to make sure the scholarship disseminated in JHM is accurate and upholds professional ethical guidelines. However the views and opinions expressed in each published manuscript belong exclusively to the individual contributor(s). The publisher and the editors do not endorse or accept responsibility for them. See https://scholarship.claremont.edu/jhm/policies.html for more information. Arithmetics, Interrupted Cover Page Footnote The author is grateful to Malena Español and Alexandra Florea for helpful suggestions. This work is available in Journal of Humanistic Mathematics: https://scholarship.claremont.edu/jhm/vol11/iss2/11 Arithmetics, InterruptedMatilde LalínDépartement de mathématiques et de statistique, Université de Montréal, CANADAmatilde.lalin@umontreal.caSynopsisI share some of my adventures in mathematical research and homeschooling inthe time of COVID-19.Keywords: elliptic curves, hyperbolic triangles, Mahler measure, function fieldarithmetic, L-functions, homeschooling.A positive integer n is said to be congruent if it is equal to the area of a righttriangle with rational sides. It is known that n is congruent if and only if theequationEn : y2 = x3 − n2xhas a solution (x0, y0) ∈ Q2 with y0 6= 0. The equation En correspondsto an elliptic curve, and having a solution with y0 6= 0 is equivalent to saythat there are infinitely many rational solutions or, more precisely, that theelliptic curve has positive rank. The answer to this question is ultimatelyrelated to the Birch and Swinnerton-Dyer conjecture. These ideas have beenextended to the study of various geometric objects such as Heron triangles(see for example Goins and Maddox [6]).A hyperbolic triangle with area A, angles α, β, γ and sides a, b, c is a hyper-bolic Heron triangle ifea, eb, ec ∈ Q and eiα, eiβ, eiγ, eiA ∈ Q(i).In collaboration with Mila [10] we show that hyperbolic Heron triangles withfixed area and a fixed angle are parametrized by an elliptic curve family ofthe formEn,u : y2 = x(x− n)(x− n(u2 + 1)),Journal of Humanistic Mathematics Volume 11 Number 2 (July 2021)266 Arithmetics, Interruptedwhere u depends on the angle and n depends on both the area and the angle.This family has always a point of infinite order, which implies that we canfind infinitely many hyperbolic Heron triangles for most choices of the areaand angle. In particular, the congruent number problem has always infinitelymany solutions in this setting!———o———0———o———Oldest son wants to discuss the following problem: A “follower year” is a yearthat is written as the juxtaposition of two consecutive numbers. For example,78 and 2021 are follower years. Noélie sums the numbers that represent thefollower years between 12 and 2021 included. What is her result? [1]———o———0———o———The (logarithmic) Mahler measure of a non-zero polynomial P ∈ C[x1, . . . , xn]is defined bym(P ) =1(2π)n∫ 2π0· · ·∫ 2π0log |P (eiθ1 , . . . , eiθn)|dθ1 . . . dθn.For the one-variable case, this integral can be expressed in terms of theroots of the polynomial that lie outside the unit circle. In favorable cases,the Mahler measure of several-variable polynomials yields formulas involvingspecial values of arithmetically interesting formulas, such as the Riemannzeta function or L-functions of elliptic curves.Jointly with Gu [5] we prove a formula that expresses the Mahler measureof xa+b + 1 + (xa + 1)y + (xb − 1)z ∈ C[x, y, z] in terms of the Riemannzeta function value ζ(3) and a combination of Bloch–Wigner dilogarithms.The proof involves some new ideas regarding the integration of the regulator.This formula is a first on its kind for a three-variable family of polynomials,and it leads to an interesting application of the Boyd–Lawton formula! [2, 9]———o———0———o———Youngest son wants to discuss the following problem: “Pablo needs a stick toplay his favorite sport. Karen needs a ball to play her favorite sport. Imadneeds a basket to practise his throws. Binesi is tall, it is easy for them tosend the ball to the other side. Lakeisha hates golf and gymnastics. Xinchundoesn’t practise a team sports. Complete the following table with the favoritesport of each kid.” [11].Matilde Lalín 267soccer wheelchair basketball golf baseball gymnastics volleyballBinesiImadKarenLakeishaPabloXinchun———o———0———o———A Dirichlet character modulo k is a multiplicative function χ : Z → C suchthat χ(n) = χ(n+k) for all n, χ(n) = 0 iff (n, k) > 0, and χ(mn) = χ(m)χ(n)for all integers m,n. The Dirichlet L-series corresponding to χ is given byL(s, χ) =∞∑n=1χ(n)ns,where s is a complex variable. It converges for Re(s) > 1, and it has a mero-morphic continuation to the whole complex plane, the Dirichlet L-function,which satisfies a functional equation relating the value at s with the value at1 − s. A famous conjecture of Chowla predicts that L(12, χ) 6= 0. The values = 12is special (and difficult to estimate) because is the central value ofthe functional equation. The most studied case is the quadratic case, whenχ2(n) = 1 or 0 for all n. An important breakthrough came when Soundarara-jan [12] proved that at least 87.5% of the quadratic Dirichlet L-functions donot vanish at s = 12. For cubic characters (when χ3(n) = 1 or 0 for all n) thisis harder to analyze. A variant of this problem consists of studying Dirich-let L-functions over function fields, namely, constructed from the series thatsums over monic polynomials with coefficients in a finite field of q elementsFq:L(s, χ) =∑f∈Fq [T ]χ(f)qsdeg(f).In this case Donepudi and Li [4] proved that there is some vanishing for cubicL-functions, but the proportion that they found still goes to zero when thenumber of characters goes to infinity. In [3], we prove jointly with David andFlorea that the proportion of nonvanishing is of at least 0.4718e−e182 . Now,this is a very small number, but positive!268 Arithmetics, Interrupted———o———0———o———Oldest son needs help making a model of a paper plane for a big school projectdue soon. He also needs to make a paper model of a turbine. And a model ofa wing. And he has to interview a pilot. I also got a contact of an aircraftengineer whose son goes to the same school. Am I supposed to arrange thezoom call? Will the teacher do it?———o———0———o———Over number fields, the k-th divisor function dk(n) gives the number of waysof writing a positive integer as a product of k positive integers. It arises ascoefficients of the k-power of the Riemann zeta functionζ(s)k =∞∑n=1dk(n)ns, Re(s) > 1.The sum of the divisor function up to a positive real x is asymptotic to apolynomial in log x: ∑n≤xdk(n) ∼ xPk−1(log x),where Pk−1(T ) is certain polynomial of degree k − 1 (see [13, Chapter XII]).There is interest in studying the distribution of the error term ∆k(x) =∑n≤x dk(n) − xPk−1(log x) and in particular its variance. In [8] Keating,Rodgers, Rodity-Gershon, and Rudnick make conjectures on these distri-butions by studying the analogue problems over the function field (again,polynomials over Fq[T ]). When q → ∞, the distributions considered in [8]can be expressed in terms of integrals in random matrix theory, in particularover the set of unitary matrices. With Kuperberg [7] we are considering simi-lar problems, where the distribution takes places over the symplectic unitarymatrices. It is exciting to compare these results with the already knownunitary case!———o———0———o———Matilde Lalín 269Youngest son needs help with dictation, conjugating the French verb “aimer”:J’aime, tu aimes, il/elle aime, nous aimons, vous aimez, ils/elles aiment...je t’aime, je vous aime, même en pandémie! 1Personal bio: I am an Argentinian-Canadian professor of mathematics anda mother, who is passionate about research in number theory and related ar-eas. I became a mathematician by participating in Mathematical Olympiads,and I love solving problems and sharing my excitement with students andwith my children. COVID-19 has disrupted my life in many ways, and I amlooking forward to regaining my usual nerdy normalcy.References[1] Association Québécoise des Jeux Mathématiques (AQJM), 35e Cham-pionnat International des Jeux Mathématiques et Logiques, Problème 7,demi-finale du Québec, 2020-2021.[2] David W. Boyd, “Speculations concerning the range of Mahler’s mea-sure”, Canadian Mathematical Bulletin. Bulletin Canadien de Mathéma-tiques, Volume 24 Number 4 (1981), pages 453–469. MR644535[3] Chantal David, Alexandra Florea, and Matilde Lalin, “Non-vanishingfor cubic L–functions”, arXiv:2006.15661 (2020).[4] Ravi Donepudi and Wanlin Li, “Vanishing of Dirichlet L-functions at thecentral point over function fields”, to appear in Rocky Mountain Journalof Mathematics. arXiv:2012.15319 (2020).[5] Jarry Gu and Matilde Lalín, “The Mahler measure of a three-variablefamily and an application to the Boyd-Lawton formula”, Research inNumber Theory, Volume 7 Number 1 (2021), Paper No. 13, 23 pages.MR42159901 “I love, you love, he/she/it loves, we love, you love, they love... I love you, I love youall, even in pandemic times!”Answers to kids’ questions: Pablo:golf,Xinchun:gymnastics17990.Binesi:volleyball,Imad:wheelchairbasketball,Karen:soccer,Lakeisha:baseball,270 Arithmetics, Interrupted[6] Edray Herber Goins and Davin Maddox, “Heron triangles via ellipticcurves”, The Rocky Mountain Journal of Mathematics, Volume 36 Num-ber 5 (2006), pages 1511–1526. MR2285297[7] Vivian Kuperberg and Matilde Lalín, “Sums of divisor functions andvon Mangoldt convolutions in Fq[t] leading to symplectic distributions”,arXiv:2107.01437 (2021).[8] Jonathan P. Keating, Bradley W. Rodgers, Edva Roditty-Gershon, andZeév Rudnick, “Sums of divisor functions in Fq[t] and matrix integrals”,Mathematische Zeitschrift, Volume 288 Number 1-2 (2018), pages 167–198. MR3774409[9] Wayne M. Lawton, “A problem of Boyd concerning geometric means ofpolynomials”, Journal of Number Theory, Volume 16 Number 3 (1983),pages 356–362. MR707608[10] Matilde Lalín and Olivier Mila, “Hyperbolic Heron triangles and ellipticcurves”, arXiv:2102.05158 (2021).[11] Matériel Pédagogique Raphaëlle, Adapted from “Jeu d’inférence".[12] Kannan Soundararajan, “Nonvanishing of quadratic Dirichlet L-functions at s = 12“, Annals of Mathematics (2), Volume 152 Number 2(2000), pages 447–488. MR1804529[13] Edward C. Titchmarsh, The theory of the Riemann zeta-function, sec-ond ed., (edited and with a preface by D. R. Heath-Brown), The Claren-don Press, Oxford University Press, New York, 1986. MR882550

Arithmetics, Interrupted

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