Mahler Measures as Linear Combinations of L–values of Multiple Modular Forms. (20th April 2015)

• Record Type: Journal Article
• Authors: Samart, Detchat
• Is Part Of: Canadian journal of mathematics. Volume 67:Number 2(2015)
• Page Start: 424
• Page End: 449
• Publication Date: 2015-04-20
• Subjects: 11F67 -- 33C20; Mahler measures -- Eisenstein–Kronecker series -- L–functions -- hypergeometric series; Mathematics -- Periodicals; Mathematics; Electronic journals; Periodicals; 510
• DOI: 10.4153/CJM-2014-012-8
• Languages: English
• ISSNs: 0008-414X

Abstract

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.