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Gauss–Manin connections for p -adic families of nearly overconvergent modular forms

Robert Harron;  Liang Xiao — 2014

Annales de l’institut Fourier

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.

Iwasawa theory for symmetric powers of CM modular forms at non-ordinary primes

Robert Harron;  Antonio Lei — 2014

Journal de Théorie des Nombres de Bordeaux

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 conjectures of Dabrowski and Panchishkin in this special case. We combine this with recent work of Benois to prove the trivial zero conjecture in this setting. We also construct “mixed” plus and minus p -adic L -functions and prove an analogue of Pollack’s decomposition of the admissible p -adic L -functions....

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