32 research outputs found
Gauss-Manin connections for p-adic families of nearly overconvergent modular forms
We interpolate the Gauss-Manin connection in p-adic families of nearly
overconvergent modular forms. This gives a family of Maass-Shimura type
differential operators from the space of nearly overconvergent modular forms of
type r to the space of nearly overconvergent modular forms of type r + 1 with
p-adic weight shifted by 2. Our construction is purely geometric, using
Andreatta-Iovita-Stevens and Pilloni's geometric construction of eigencurves,
and should thus generalize to higher rank groups.Comment: Final version accepted for publication in the Annales de l'Institut
Fourier. Minor revisions. 11 page
Iwasawa theory for symmetric powers of CM modular forms at non-ordinary primes
Let f be a cuspidal newform with complex multiplication (CM) and let p be an
odd prime at which f is non-ordinary. We construct admissible p-adic
L-functions for the symmetric powers of f, thus verifying general conjectures
of Dabrowski and Panchishkin in this special case. We also construct their
"mixed" plus and minus counterparts and prove an analogue of Pollack's
decomposition of the admissible p-adic L-functions into mixed plus and minus
p-adic L-functions. On the arithmetic side, we define corresponding mixed plus
and minus Selmer groups. We unite the arithmetic with the analytic by first
formulating the Main Conjecture of Iwasawa Theory relating the plus and minus
Selmer groups with the plus and minus p-adic L-functions, and then proving the
exceptional zero conjecture for the admissible p-adic L-functions. The latter
result takes advantage of recent work of Benois, while the former uses recent
work of the second author, as well as the main conjecture of Mazur-Wiles.Comment: Submitted. 24 pages. Comments welcome
Explicit computations of Hida families via overconvergent modular symbols
In [Pollack-Stevens 2011], efficient algorithms are given to compute with
overconvergent modular symbols. These algorithms then allow for the fast
computation of -adic -functions and have further been applied to compute
rational points on elliptic curves (e.g. [Darmon-Pollack 2006, Trifkovi\'c
2006]). In this paper, we generalize these algorithms to the case of families
of overconvergent modular symbols. As a consequence, we can compute -adic
families of Hecke-eigenvalues, two-variable -adic -functions,
-invariants, as well as the shape and structure of ordinary Hida-Hecke
algebras.Comment: 51 pages. To appear in Research in Number Theory. This version has
added some comments and clarifications, a new example, and further
explanations of the previous example
On Greenberg's -invariant of the symmetric sixth power of an ordinary cusp form
We derive a formula for Greenberg's -invariant of Tate twists of the
symmetric sixth power of an ordinary non-CM cuspidal newform of weight ,
under some technical assumptions. This requires a "sufficiently rich" Galois
deformation of the symmetric cube which we obtain from the symmetric cube lift
to \GSp(4)_{/\QQ} of Ramakrishnan--Shahidi and the Hida theory of this group
developed by Tilouine--Urban. The -invariant is expressed in terms of
derivatives of Frobenius eigenvalues varying in the Hida family. Our result
suggests that one could compute Greenberg's -invariant of all symmetric
powers by using appropriate functorial transfers and Hida theory on higher rank
groups.Comment: 20 pages, submitte