Special values of the triple product p-adic l-function

• Autores: Francesca Gatti
• Directores de la Tesis: Victor Rotger Cerdà (dir. tes.)
• Lectura: En la Universitat Politècnica de Catalunya (UPC) ( España ) en 2021
• Idioma: español
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• Resumen

  • Let E be an elliptic curve defined over the field of rational numbers and let f be the corresponding newform of weight 2. Let g,h be two weight one newforms whose nebentype characters are mutually inverse. Let V be the tensor product of the Artin representations attached to g and h, let L be the number field generated by the Fourier coefficients of g and h and let H be the number field cut by V.

    The points of E having coefficients in H form a finitely generated abelian group E(H), and one can attach to the pair (E,V) the finite-dimensional L-vector space MW(E,V) consisting of the homomorphisms from V to E(H) which are equivariant with respect to the action of the absolute Galois group of the rational numbers. We call algebraic rank of (E,V) the L-dimension of MW(E,V).

    On the other hand, a complex L-function L(E,V;s), which satisfies a functional equation whose center of symmetry is s=1, can be attached to (E,V). The Galois-equivariant version of the Birch and Swinnerton–Dyer conjecture predicts that the algebraic rank of (E,V) should equal the order of vanishing of L(E,V;s) at s=1, that we call analytic rank of (E,V).

    An interesting approach to these sort of problems (which are in general still unsolved) is via p-adic analysis. In our setting, we will consider the triple product p-adic L-function Lp(F,G,H)(k,l,m) attached by Darmon and Rotger to a triple of Hida families F,G,H passing through f,g,h respectively at weights 2,1,1. The function Lp(F,G,H)(k,l,m) interpolates p-adically the central values of the complex L-functions attached to the specialisations of the families F,G and H at classical weights k,l,m for which the weight l is greater or equal to the sum of the other two (in this case we say that l is ’dominant’). In particular, the triple (2,1,1), which corresponds to the pair (E,V) lies outside the region of classical interpolation of the triple product L-function. In this thesis we study the value Lp(F,G,H)(2,1,1), and more in general the values of Lp(F,G,H) at weights (k,l,m) with k dominant, in different settings.

    In the case in which the ranks of (E,V) are 2, with the elliptic Stark conjecture Darmon, Lauder and Rotger predict (and prove, in some special cases) that Lp(F,G,H)(2,1,1) can be expressed as a p-adic regulator involving the p-adic logarithm of points on E(H), divided by the logarithm of a so- called Stark unit. We generalise this conjecture to general weights (k,l,m) with k dominant and we prove some cases of this conjecture. Here p is an odd prime which does not divide the conductors of g and h and divides at most once the conductor of E.

    If p does not divide any of the conductors and if the analytic and the alge- braic ranks of (E,V) are 0 we prove, under the additional hypothesis that the Selmer group of (E,V) is trivial, a formula for the value Lp(F,G,H)(2,1,1) in terms of the Bloch–Kato logarithm of a canonical non-cristalline class which belongs to the p-relaxed Selmer group of (E,V), along a certain cristalline direction. As a corollary we prove that, in the special case in which g and h are theta series of Hecke characters of the same imaginary quadratic field in which p splits, the triple product L-function vanishes at (2,1,1).

    Finally we study the case in which E has multiplicative reduction at p, g and h are theta series of the same imaginary quadratic field K in which p is inert, p does not divide the conductors of g and h and the analytic rank of (E,V) is 0. In this setting, Bertolini and Darmon constructed a so–called Kolyvagin class with interesting arithmetic properties, which lies in the relaxed Selmer group of E over the field H cut out by V. We prove a formula which relates the value Lp(F,G,H)(2,1,1) to the p-adic logarithm of the projection along a cristalline direction of the Kolyvagin class.