Mahler measures and L-values of elliptic curves over real quadratic fields

2022/09/29 by Zhengyu Tao, Xuejun Guo, Tao, Zhengyu +3
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2209.14717

openalex publication_date 2022/09/29 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

A famous formula of Rodriguez Villegas shows that the Mahler measures m(k) of P_k(x,y)=x+1/x+y+1/y+k can be written as a Kronecker-Eisenstein series. We prove that the degree of k in Villegas' formula can be bounded by the class numbers of CM points. This fact allows us to systematically derive 28 new identities linking m(k) to L-values of cusp forms. Guided by Beilinson's conjecture, we also prove 5 formulas that express L-values of CM elliptic curves over real quadratic fields to some 2× 2 determinants of m(k). This extends a recent work of Guo (the second author of this paper), Ji, Liu, and Qin, in which they dealt with the cases when k=4± 4√(2).

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