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An $$l \ne p$$ l ≠ p -interpolation of genuine p-adic L-functions

An $$l \ne p$$ l ≠ p -interpolation of genuine p-adic L-functions Let $$\mathcal {F}$$ F be a totally real field, l and p distinct odd prime unramified in $$\mathcal {F}$$ F and $${\mathfrak {l}}$$ l a prime above l. Let $$\mathcal {K}/\mathcal {F}$$ K / F be a p-ordinary CM quadratic extension and $$\lambda $$ λ an arithmetic Hecke character over $$\mathcal {K}$$ K . Hida constructed a measure on the $${\mathfrak {l}}$$ l -anticyclotomic class group of $$\mathcal {K}$$ K interpolating the normalised Hecke L-values $$L^{\mathrm{alg},{\mathfrak {l}}}(0,\lambda \nu )$$ L alg , l ( 0 , λ ν ) , as $$\nu $$ ν varies over the finite order $${\mathfrak {l}}$$ l -power conductor anticyclotomic characters. In this article, we interpolate the measures as $$\lambda $$ λ varies in a p-adic family. In particular, this gives p-adic deformation of the measures. An analogue holds in the case of self-dual Rankin–Selberg convolution of a Hilbert modular form and a theta series. In the case of root number $$-1$$ - 1 , we describe an upcoming analogous interpolation of the p-adic Abel–Jacobi image of generalised Heegner cycles associated with the convolution. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Research in the Mathematical Sciences Springer Journals

An $$l \ne p$$ l ≠ p -interpolation of genuine p-adic L-functions

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References (22)

Publisher
Springer Journals
Copyright
Copyright © 2016 by Burungale.
Subject
Mathematics; Mathematics, general; Applications of Mathematics; Computational Mathematics and Numerical Analysis
eISSN
2197-9847
DOI
10.1186/s40687-016-0060-2
Publisher site
See Article on Publisher Site

Abstract

Let $$\mathcal {F}$$ F be a totally real field, l and p distinct odd prime unramified in $$\mathcal {F}$$ F and $${\mathfrak {l}}$$ l a prime above l. Let $$\mathcal {K}/\mathcal {F}$$ K / F be a p-ordinary CM quadratic extension and $$\lambda $$ λ an arithmetic Hecke character over $$\mathcal {K}$$ K . Hida constructed a measure on the $${\mathfrak {l}}$$ l -anticyclotomic class group of $$\mathcal {K}$$ K interpolating the normalised Hecke L-values $$L^{\mathrm{alg},{\mathfrak {l}}}(0,\lambda \nu )$$ L alg , l ( 0 , λ ν ) , as $$\nu $$ ν varies over the finite order $${\mathfrak {l}}$$ l -power conductor anticyclotomic characters. In this article, we interpolate the measures as $$\lambda $$ λ varies in a p-adic family. In particular, this gives p-adic deformation of the measures. An analogue holds in the case of self-dual Rankin–Selberg convolution of a Hilbert modular form and a theta series. In the case of root number $$-1$$ - 1 , we describe an upcoming analogous interpolation of the p-adic Abel–Jacobi image of generalised Heegner cycles associated with the convolution.

Journal

Research in the Mathematical SciencesSpringer Journals

Published: Jul 1, 2016

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