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Congruences and rationality of Stark-Heegner points
No full textLet 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
The Teitelbaum conjecture in the indefinite setting
No full textLet 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
Heegner cycles and derivatives of p-adic L-functions
No full textLet 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 families of modular forms and p-adic Abel-Jacobi maps
Full text linkWe 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’Abel-Jacobi aux familles analytiques de formes modulaires à ses poids nouveaux en p. Nous introduisons le concept de la dérivée d’une distribution. Utilisant ce concept, nous donnons une nouvelle perspective sur le cocycle de Coleman-Teitelbaum dans le contexte de l’invariant LL
p-adic L-functions and the rationality of Darmon cycles
No full textDarmon 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 Γ_0(N) of even weight k≥2 . 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
The Arithmetic Theory of Local Constants for Abelian Varieties
No full textWe present a generalization of the theory of local constant developed by B. Mazur and K. Rubin in order to cover the case of abelian varieties, with emphasis to abelian varieties with real multiplication. Let l be an odd rational prime and let L/K be an abelian l -power extension. Assume that we are given a quadratic extension K/k such that L/k is a dihedral extension and the abelian variety A/k is defined over k and polarizable. This theory can be used to relate the rank of the l-Selmer group of A over K to the rank of the l-Selmer group of A over L
p-adic families of cohomological modular forms for indefinite quaternion algebras and the Jacquet-Langlands correspondence
No full textWe use the method of Ash and Stevens to prove the existence of small slope p-adic families of cohomological modular forms for an indefinite quaternion algebra B. We prove that the Jacquet-Langlands correspondence relating modular forms on GL2/Q and cohomomological modular forms for B is compatible with the formation of p-adic families. This result is an analogue of a theorem of Chenevier concerning definite quaternion algebras
Formal period integrals and special value formulas
No full textMotivated by the conjectures of Gan-Gross-Prasad, we develop a p-adic formalism for placing these conjectures in a p-adic setting which is suited for p-adic interpolation
Stark-Heegner points and Selmer groups of abelian varieties
No full textThis thesis studies the image of Stark-Heegner points in the Selmer group of abelian varietie
Modular p-adic L-functions attached to real quadratic fields and arithmetic applications
Full text linkLet f ∈ Sk0+2(Γ0(Np)) be a normalized N-new eigenform with p ∤ N and such that ap2 ≠ 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 ψ be an unramified character of Gal(K̅ /K). Under mild hypotheses on the discriminant of K and the factorization of N, we construct a p-adic L-function L/K,ψ 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 ψ. When the character ψ is quadratic, L/K,ψ 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 L/K,ψ at k0 to Bloch–Kato 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
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