Abstract. Broadhurst [12] conjectured that the Feynman integral associated to the polynomial corresponding to t = 1 in the one-parameter family (1 +, where f is a cusp form of weight 3 and level 15. Bloch, Kerr and Vanhove [8] have recently proved that the conjecture holds up to a rational factor. In this paper, we prove that Broadhurst's conjecture is true. Similar identities involving Feynman integrals associated to other polynomials in the same family are also established.
Broadhurst [12] conjectured that the Feynman integral associated to the polynomial corresponding to t = 1 in the one-parameter family (1 +, where f is a cusp form of weight 3 and level 15. Bloch, Kerr and Vanhove [8] have recently proved that the conjecture holds up to a rational factor. In this paper, we prove that Broadhurst's conjecture is true. Similar identities involving Feynman integrals associated to other polynomials in the same family are also established.
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.
We prove that the (logarithmic) Mahler measure m(P ) of P (x, y) = x + 1/x + y + 1/y + 3 is equal to the L-value 2L ′ (E, 0) attached to the elliptic curve E : P (x, y) = 0 of conductor 21. In order to do this we investigate the measure of a more general Laurent polynomialand show that the wanted quantity m(P ) is related to a "half-Mahler" measure ofP (x, y) = P √ 7,1,3 (x, y). In the finale we use the modular parametrization of the elliptic curveP (x, y) = 0, again of conductor 21, due to Ramanujan and the Mellit-Brunault formula for the regulator of modular units.
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