Mahler measures of elliptic modular surfaces

Brunault, François; Neururer, Michael

doi:10.1090/tran/7524

Mahler measures of elliptic modular surfaces
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by François Brunault and Michael Neururer PDF

Trans. Amer. Math. Soc. 372 (2019), 119-152 Request permission

Abstract:

We develop a new method for relating Mahler measures of three-variable polynomials that define elliptic modular surfaces to $L$-values of modular forms. Using an idea of Deninger’s, we express the Mahler measure as a Deligne period of the surface and then apply the first author’s extension of the Rogers–Zudilin method for Kuga–Sato varieties to arrive at an $L$-value.

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