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Mahler measures, special values of L-functions and complex multiplication

Abstract : This thesis studies the relations between special values of $L$-functions of arithmetic objects and heights, as well as the arithmetic of torsion points on elliptic curves with complex multiplication. The first of the main results of this thesis, exposed in its last chapter, shows that the special value $L^\ast(E,0)$ of the $L$-function associated to an elliptic curve $E$ defined over $\mathbb{Q}$ which has complex multiplication can be expressed as an explicit rational linear combination of a logarithm of an algebraic number and the Mahler measure of a polynomial. The other main result of this thesis, exposed in its penultimate chapter and obtained in collaboration with Francesco Campagna, shows that the family of $p^\infty$-division fields associated to an elliptic curve $E$ defined over a number field $F$ containing the CM field $K$ becomes linearly disjoint after removing a finite and explicit subfamily of fields, which we expect to be never linearly disjoint over $F$ as soon as it contains more than one element and $F(E_\text{tors}) \subseteq K^\text{ab}$. We prove this expectation if $F = K$ and $E$ is the base-change of an elliptic curve defined over $\mathbb{Q}$. The content of this thesis is articulated in the following chapters: the first chapter contains background material on the notion of height, and on Diophantine properties of heights; the second chapter contains background material on motives, motivic cohomology and regulators; the third chapter contains background material on $L$-functions, together with some results concerning the finiteness of the family of $L$-functions having bounded special values, which is based on joint work in progress with Fabien Pazuki; the fourth chapter contains background material on the Mahler measure, as well as some computations concerning explicit families of polynomials; the fifth chapter contains the outline of an ongoing project joint with François Brunault, whose aim is to give a geometric interpretation of results by Lalín, inspired by an insight from Maillot, concerning the Mahler measures associated to polynomials satisfying a suitable exactness condition; the sixth chapter, which is based on joint work in progress with Francesco Campagna, introduces the notion of ray class fields associated to orders in algebraic number fields. This is probably well known to the experts but not so well documented in the literature; the seventh chapter contains background material on elliptic curves and abelian varieties with complex multiplication, together with the proof of an optimal upper bound for the index of the image of the Galois representation attached to the torsion points of an elliptic curve with complex multiplication, which is based on joint work in progress with Francesco Campagna; the eight and ninth chapter contain the expositions of the main results of this thesis, which were described in the previous paragraph; the appendix contains the tables mentioned in the main body of the thesis.
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https://hal-ens-lyon.archives-ouvertes.fr/tel-03037598
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Submitted on : Tuesday, October 5, 2021 - 5:21:11 PM
Last modification on : Wednesday, October 27, 2021 - 7:38:00 AM

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Riccardo Pengo. Mahler measures, special values of L-functions and complex multiplication. Number Theory [math.NT]. København Universiteit, 2020. English. ⟨tel-03037598v2⟩

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