Aspects explicites des fonctions L et applications

Abstract : This thesis focuses on L-functions, their explicit aspects and their applications.In the first chapter, we give a precise definition of L-functions and their main properties, especially about the invariants called local parameters. Then, we deal with Artin L-functions. For them, we have created a computer program in PARI/GP which gives the coefficients and the invariants for an Artin L-function above Q.In the second chapter, we make explicit a theorem of Henryk Iwaniec and Emmanuel Kowalski, which distinguishes between two L-functions by considering their local parameters for primes up to a theoretical bound.Actually, distinguishing between sums of local parameters of Artin L-functions is the same as separating the associated characters by the Frobenius automorphism. This is the subject of the third chapter, that can be related to Chebotarev Theorem. By applying the result to conjugate characters of the alternating group, we get a bound for a prime p giving the factorization modulo $p$ of a certain polynomial. This work has to be compared with a result from Joël Bellaïche (2013).Finally, we numerically illustrate our results by studying the evolution of the bound on polynomials X^n+uX+v, for n=5, 7 and 13.
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Charlotte Euvrard. Aspects explicites des fonctions L et applications. Théorie des nombres [math.NT]. Université de Franche-Comté, 2016. Français. ⟨NNT : 2016BESA2074⟩. ⟨tel-01661548⟩

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