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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.
Research in the Mathematical Sciences – Springer Journals
Published: Jul 1, 2016
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