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Kurokawa–Mizumoto congruences and degree-8 L-values

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Abstract

Let f be a Hecke eigenform of weight k, level 1, genus 1. Let \(E^k_{2,1}(f)\) be its genus-2 Klingen–Eisenstein series. Let F be a genus-2 cusp form whose Hecke eigenvalues are congruent modulo \({\mathfrak {q}}\) to those of \(E^k_{2,1}(f)\), where \({\mathfrak {q}}\) is a “large” prime divisor of the algebraic part of the rightmost critical value of the symmetric square L-function of f. We explain how the Bloch–Kato conjecture leads one to believe that \({\mathfrak {q}}\) should also appear in the denominator of the “algebraic part” of the rightmost critical value of the tensor product L-function \(L(s,f\otimes F)\), i.e. in an algebraic ratio obtained from the quotient of this with another critical value. Using pullback of a genus-5 Siegel–Eisenstein series, we prove this, under weak conditions.

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Authors and Affiliations

  1. School of Mathematics and Statistics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield, S3 7RH, UK

    Neil Dummigan & Angelo Rendina

  2. Department of Applied Information Technology, German University of Technology in Oman (GUtech), Corner of Beach Road and Wadi Athaibah Way, PO Box 1816, 130, Athaibah, Sultanate of Oman

    Bernhard Heim

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  1. Neil Dummigan
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  2. Bernhard Heim
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  3. Angelo Rendina
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Correspondence to Neil Dummigan.

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Dummigan, N., Heim, B. & Rendina, A. Kurokawa–Mizumoto congruences and degree-8 L-values. manuscripta math. 160, 217–237 (2019). https://doi.org/10.1007/s00229-018-1061-9

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  • DOI: https://doi.org/10.1007/s00229-018-1061-9

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