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Mahler Measures as Linear Combinations of $L$-values of Multiple Modular Forms

  • Detchat Samart,
    Department of Mathematics, Texas A&M University, College Station, TX 77843, USA
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Abstract

We study the Mahler measures of certain families of Laurent polynomials in two and three variables. Each of the known Mahler measure formulas for these families involves $L$-values of at most one newform and/or at most one quadratic character. In this paper, we show, either rigorously or numerically, that the Mahler measures of some polynomials are related to $L$-values of multiple newforms and quadratic characters simultaneously. The results suggest that the number of modular $L$-values appearing in the formulas significantly depends on the shape of the algebraic value of the parameter chosen for each polynomial. As a consequence, we also obtain new formulas relating special values of hypergeometric series evaluated at algebraic numbers to special values of $L$-functions.
Keywords: Mahler measures, Eisenstein-Kronecker series, $L$-functions, hypergeometric series Mahler measures, Eisenstein-Kronecker series, $L$-functions, hypergeometric series
MSC Classifications: 11F67, 33C20 show english descriptions Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols
Generalized hypergeometric series, ${}_pF_q$
11F67 - Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols
33C20 - Generalized hypergeometric series, ${}_pF_q$
 

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