Generalization of a theorem of Greenberg and Stevens to the case of the symmetric square of a modular form and an application to the Selmer group

This thesis is devoted to the study of certain cases of a conjecture of Greenberg and Benois on derivative of p-adic L-functions using the method of Greenberg and Stevens. We first prove this conjecture in the case of the symmetric square of a parallel weight 2 Hilbertmodular form over a totally real field where p is inert and whose associated automorphic representation is Steinberg in p, assuming certain hypotheses on the conductor. This is a direct generalization of (unpublished) results of Greenberg and Tilouine. Subsequently, we deal with the symmetric square of a finite slope, elliptic, modular form which is Steinberg at p. To construct the two-variable p-adic L-function, necessary to apply the method of Greenberg and Stevens, we have to appeal to the recently developed theory of nearly overconvergent forms of Urban. We furtherstrengthen the above result, removing the assumption that the conductorof the form is even, using the construction of the p-adic L-function byBocherer and Schmidt. In the final chapter we recall the definition and the calculation of the algebraic L-invariant a la Greenberg-Benois, and explain how some of the above-mentioned results could be generalized to higher genus Siegel modular forms.1 Introduction 2 Dérivée en s = 1 de la fonction L p-adique du carré symétrique d’une courbe elliptique sur un corps totalement réel 3 Derivative at s = 1 of the p-adic L-function of the symmetric square of a Hilbert modular form 4 A formula for the derivative of the p-adic L-function of the symmetric square of a finite slope modular form 5 Derivative of symmetric square p-adic L-functions via pull-back formula 6 The arithmetic L-invariant 7 Conclusion Bibliography Curriculum List of publicationsstatus: publishe