1,721,030 research outputs found
On Shafarevich-Tate groups and analytic ranks in families of modular forms, I. Hida families
Full text linkLet be a newform of weight , square-free level and trivial character,
let be the abelian variety attached to and for every good ordinary
prime for let be the -adic Hida family through
. We prove that, for all but finitely many primes as above, if is
an elliptic curve such that has rank and the -primary
part of the Shafarevich-Tate group of over is finite then all
specializations of of weight congruent to modulo
and trivial character have finite (-primary) Shafarevich-Tate group
and -dimensional image of the relevant -adic \'etale Abel-Jacobi map.
Analogous results are obtained also in the rank case. As a second
contribution, with no restriction on the dimension of but assuming the
non-degeneracy of certain height pairings \`a la Gillet-Soul\'e between Heegner
cycles, we show that if has analytic rank then, for all but finitely
many , all specializations of of weight congruent to
modulo and trivial character have analytic rank . This result
provides some evidence in rank and weight larger than for a conjecture
of Greenberg predicting that the analytic ranks of even weight modular forms in
a Hida family should be as small as allowed by the functional equation, with at
most finitely many exceptions.Comment: Slight revision following the referee's report; 37 pages. Final
version, to appear in Annali della Scuola Normale Superiore di Pisa, Classe
di Scienz
On Bloch-Kato Selmer groups and Iwasawa theory of p-adic Galois representations
No full textAresultduetoR.Greenberggivesarelationbetweenthe cardinalityof Selmergroupsof ellipticcurvesovernumberfieldsand thecharacteristicpowerseriesofPontryagindualsofSelmergroupsover cyclotomicZp-extensionsatgoodordinaryprimesp.WeextendGreenberg’sresulttomoregeneralp-adicGaloisrepresentations, includinga largesubclassof thoseattachedtop-ordinarymodular formsofweight at least4andlevelΓ0(N)withp N
Quaternionic Darmon points on abelian varieties
No full text[email protected] In the first part ofthe paper we prove formulasfor the p-Adic logarithm of quaternionic Darmon points on modular abelian varieties over Q with toric reduction at p. These formulas are amenable to explicit computations and are the first to treat Stark-Heegner type points on higher-dimensional abelian varieties. In the second part of the paper we explain how these formulas, together with a mild generalization of results of Bertolini and Darmon on Hida families of modular forms and rational points, can be used to obtain rationality results over genus fields of real quadratic fields for Darmon points on abelian varieties
Quaternionic darmon points on abelian varieties
Full text link[email protected] In the first part ofthe paper we prove formulasfor the p-Adic logarithm of quaternionic Darmon points on modular abelian varieties over Q with toric reduction at p. These formulas are amenable to explicit computations and are the first to treat Stark-Heegner type points on higher-dimensional abelian varieties. In the second part of the paper we explain how these formulas, together with a mild generalization of results of Bertolini and Darmon on Hida families of modular forms and rational points, can be used to obtain rationality results over genus fields of real quadratic fields for Darmon points on abelian varieties
On ring class eigenspaces of Mordell-Weil groups of elliptic curves over global function fields
No full textIf E is a non-isotrivial elliptic curve over a global function field F of odd characteristic we show that certain Mordell-Weil groups of E have 1-dimensional eigenspace relative to a fixed complex ring class character provided that the projection onto this eigenspace of a suitable Drinfeld-Heegner point is nonzero. This represents the analogue in the function field setting of a theorem for rational elliptic curves due to Bertolini and Darmon, and at the same time is a generalization of the main result proved by Brown in his monograph on Heegner modules. As in the number field case, our proof employs Kolyvagin-type arguments, and the cohomological machinery is started up by the control on the Galois structure of the torsion of E provided by classical results of Igusa in positive characteristic
The Rationality of Quaternionic Darmon Points Over Genus Fields of Real Quadratic Fields
Full text linkDarmon points on p-adic tori and Jacobians of Shimura curves over Q were introduced in joint articles with Rotger as generalizations of Darmon's Stark-Heegner points. In this article, we study the algebraicity over extensions of a real quadratic field K of the projections of Darmon points to elliptic curves, which coincide with the points on elliptic curves previously defined by M. Greenberg. More precisely, we prove that linear combinations of Darmon points on elliptic curves weighted by certain genus characters of K are rational over the predicted genus fields of K. This extends to an arbitrary quaternionic setting the main theorem on the rationality of Stark-Heegner points obtained by Bertolini and Darmon, and at the same time gives evidence for the rationality conjectures formulated in a joint paper with Rotger and by Greenberg in his article on Stark-Heegner points. In light of this result, quaternionic Darmon points represent the first instance of a systematic supply of points of Stark-Heegner type other than Darmon's original ones for which explicit rationality results are known. © 2013 The Author(s) 2013. Published by Oxford University Press. All rights reserved
Vanishing of special values and central derivatives in Hida families
Full text linkThe theme of this work is the study of the Nekovář-Selmer group H1f(K, †) attached to a twisted Hida family † of Galois representations and a quadratic number field K. The results that we obtain have the following shape: if a twisted L-function of a suitable modular form in the Hida family has order of vanishing r ≤ 1 at the central critical point then the rank of H1f(K, †) as a module over a certain local Hida-Hecke algebra is equal to r. Under the above assumption, we also show that infinitely many twisted L-functions of modular forms in the Hida family have the same order of vanishing at the central critical point. Our theorems extend to more general arithmetic situations results obtained by Howard when K is an imaginary quadratic field and all the primes dividing the tame level of the Hida family split in K
A note on Control Theorems for quaternionic Hida families of modular forms
Full text linkWe extend a result of Greenberg and Stevens on the interpolation of
modular symbols in Hida families to the context of non-split rational quaternion algebras.
Both the denite case and the indenite case are considered
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