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Mahler Measures as Linear Combinations of L–values of Multiple Modular Forms
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- Journal:
- Canadian Journal of Mathematics / Volume 67 / Issue 2 / April 2015
- Published online by Cambridge University Press:
- 20 November 2018, pp. 424-449
- Print publication:
- April 2015
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- Article
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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.
