p-adic families of modular forms and p-adic Abel-Jacobi maps

We show that p-adic families of modular forms give rise to certain p-adic Abel-Jacobi maps at their p-new specializations. We introduce the concept of differentiation of distributions, using it to give a new description of the Coleman-Teitelbaum cocycle that arises in the context of the LL -invariant.Nous associons certaines applications p-adiques d\u2019Abel-Jacobi aux familles analytiques de formes modulaires \ue0 ses poids nouveaux en p. Nous introduisons le concept de la d\ue9riv\ue9e d\u2019une distribution. Utilisant ce concept, nous donnons une nouvelle perspective sur le cocycle de Coleman-Teitelbaum dans le contexte de l\u2019invariant LL

Triple product p-adic L-functions for balanced weights

We construct p-adic triple product L-functions that interpolate (square roots of) central critical L-values in the balanced region. Thus, our construction complements that of Harris and Tilouine. There are four central critical regions for the triple product L-functions and two opposite settings, according to the sign of the functional equation. In the first case, three of these regions are of interpolation, having positive sign; they are called the unbalanced regions and one gets three p-adic L-functions, one for each region of interpolation (this is the Harris-Tilouine setting). In the other setting there is only one region of interpolation, called the balanced region: we produce the corresponding p-adic L-function. Our triple product p-adic L-function arises as p-adic period integrals interpolating normalizations of the local archimedean period integrals. The latter encode information about classical representation theoretic branching laws. The main step in our construction of p-adic period integrals is showing that these branching laws vary in a p-adic analytic fashion. This relies crucially on the Ash-Stevens theory of highest weight representations over affinoid algebras

Modular p-adic L-functions attached to real quadratic fields and arithmetic applications

Let f 08 Sk0+2(\u3930(Np)) be a normalized N-new eigenform with p 24 N and such that ap2 60 pk0+1 and ordp(ap) < k0 + 1. By Coleman's theory, there is a p-adic family of eigenforms whose weight k0 + 2 specialization is f. Let K be a real quadratic field and let \u3c8 be an unramified character of Gal(K\u305 /K). Under mild hypotheses on the discriminant of K and the factorization of N, we construct a p-adic L-function \u2112/K,\u3c8 interpolating the central critical values of the Rankin L-functions associated to the base change to K of the specializations of in classical weight, twisted by \u3c8. When the character \u3c8 is quadratic, \u2112/K,\u3c8 factors into a product of two Mazur-Kitagawa p-adic L-functions. If, in addition, has p-new specialization in weight k0 + 2, then under natural parity hypotheses we may relate derivatives of each of the Mazur-Kitagawa factors of \u2112/K,\u3c8 at k0 to Bloch\u2013Kato logarithms of Heegner cycles. On the other hand the derivatives of our p-adic L-functions encodes the position of the so called Darmon cycles

The Teitelbaum conjecture in the indefinite setting

Let f be a new cusp form on Gamma_0(N) of even weight k+2>=2. Suppose that there is a prime p dividing N and that we may write N=pN^{+}N^{-}, where N^{-} is the squarefree product of an even number of primes. There is a Darmon style L-invariant L_N^{-}(f) attached to this factorization, which is the Orton L-invariant when N^{-}=1. We prove that L_N^{-}(f) does not depend on the chosen factorization of N and it is equal to the other known L-invariants. We also give a formula for the computation of the logarithmic p-adic Abel-Jacobi image of the Darmon cycles. This formula is crucial for the computations of the derivatives of the p-adic L-functions of the weight variable attached to a real quadratic field K/Q such that the primes dividing N^{+} are split and the primes dividing pN^{-} are inert

Congruences and rationality of Stark-Heegner points

Let A/Q be a modular abelian variety attached to a weight 2 new modular form of level N=pM, where p is a prime and M is an integer prime to p. When K/Q is an imaginary quadratic extension the Heegner points, that are defined over the ring class fields H/K, can contribute to the growth of the rank of the Selmer groups over H. When K/Q is a real quadratic field the theory of Stark-Heegner points provides a conjectural explanation of the growth of these ranks under suitable sign conditions on the L-function of f/K. The main result of the paper relates the growth of the Selmer groups to the conjectured rationality of the Stark-Heegner points over the expected field of definition

Heegner cycles and derivatives of p-adic L-functions

Let f be an even weight k>=2 modular form on a p-adically uniformizable Shimura curve for a suitable gamma 0-type level structure. Let K=Q be an imaginary quadratic field, satisfying Heegner conditions assuring that the sign appearing in the functional equation of the complex L-function of f/K is negative. We may attach to f, or rather a deformation of it, a p-adic L-function of the weight variable , also depending on K. Our main result is a formula relating the derivative of this p-adic L-function at k to the Abel-Jacobi images of so called Heegner cycles

p-adic L-functions and the rationality of Darmon cycles

Darmon cycles are a higher weight analogue of Stark-Heegner points. They yield local cohomology classes in the Deligne representation associated with a cuspidal form on \u393_0(N) of even weight k 652 . They are conjectured to be the restriction of global cohomology classes in the Bloch-Kato Selmer group defined over narrow ring class fields attached to a real quadratic field. We show that suitable linear combinations of them obtained by genus characters satisfy these conjectures. We also prove p-adic Gross-Zagier type formulas, relating the derivatives of p-adic L-functions of the weight variable attached to imaginary (resp. real) quadratic fields to Heegner cycles (resp. Darmon cycles). Finally we express the second derivative of the Mazur-Kitagawa p-adic L-function of the weight variable in terms of a global cycle defined over a quadratic extension of Q

Stark-Heegner points and Selmer groups of abelian varieties

This thesis studies the image of Stark-Heegner points in the Selmer group of abelian varietie

Poincaré duality isomorphisms in tensor categories

If for a vector space V of dimension g over a characteristic zero field we denote by Lambda V-i its alternating powers, and by V-v its linear dual, then there are natural Poincare isomorphisms: Lambda V-i(v )congruent to Lambda Vg-i. We describe an analogous result for objects in rigid pseudo-abelian Q-linear ACU tensor categorie

L\mathcal L-invariants and Darmon cycles attached to modular forms

Let f be a modular eigenform of even weight k 652 and new at a prime p dividing exactly the level with respect to an indefinite quaternion algebra. The theory of Fontaine-Mazur allows to attach to f a monodromy module D_FM(f) and an L-invariant L_FM(f). The first goal of this paper is building a suitable p-adic integration theory that allows us to construct a new monodromy module D(f) and L-invariant L(f), in the spirit of Darmon. The two monodromy modules are isomorphic, and in particular the two L-invariants are equal. Let K be a real quadratic field and assume the sign of the functional equation of the L-series of f over K is 121 . The Bloch-Beilinson conjectures suggest that there should be a supply of elements in the Selmer group of the motive attached to f over the tower of narrow ring class fields of K. Generalizing work of Darmon for k=2, we give a construction of local cohomology classes which we expect to arise from global classes and satisfy an explicit reciprocity law, accounting for the above prediction