Power series expansions of modular forms and p-adic interpolation of\n the square roots of Rankin-Selberg special values

2019/10/22 by Andrea Mori, Mori, Andrea · 1 citation
Mathematics · #11F67 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1910.09992

openalex publication_date 2019/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let f be a newform of even weight 2\κ for D^\×, where D is a\npossibly split indefinite quaternion algebra over \ℚ. Let K be a\nquadratic imaginary field splitting D and p an odd prime split in K. We\nextend our theory of p-adic measures attached to the power series expansions\nof f around the Galois orbit of the CM point corresponding to an embedding\nK hookrightarrow D to forms with any nebentypus and to p dividing the level\nof f. For the latter we restrict our considerations to CM points\ncorresponding to test objects endowed with an arithmetic p-level structure.\nAlso, we restrict these p-adic measures to \ℤ_p^\× and compute\nthe corresponding Euler factor in the formula for the p-adic interpolation of\nthe "square roots" of the Rankin-Selberg special values\nL(\π_K\⊗\ξ_r, frac12) where \π_K is the base change to K of the\nautomorphic representation of \GL₂ associated, up to\nJacquet-Langland correspondence, to f and \ξ_r is a compatible family of\ngr "ossencharacters of K with infinite type \ξ_r,\∞(z)=(z/ nz)^\κ+r.\n

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• On quaternionic ordinary families of modular forms and p-adic L-functions

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