On the Mahler measure associated to \(X_1(13)\). arXiv:1503.04631
Preprint, arXiv:1503.04631 [math.NT] (2015).
Summary: We show that the Mahler measure of a defining equation of the modular curve \(X_1(13)\) is equal to the derivative at \(s=0\) of the \(L\)-function of a cusp form of weight 2 and level 13 with integral Fourier coefficients. The proof combines Deninger’s method, an explicit version of Beilinson’s theorem together with an idea of Merel to express the regulator integral as a linear combination of periods. Finally, we present further examples related to the modular curves of level 16, 18 and 25.
MSC:
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
| 11R06 | PV-numbers and generalizations; other special algebraic numbers; Mahler measure |
| 11F11 | Holomorphic modular forms of integral weight |
| 11G16 | Elliptic and modular units |
| 19F27 | Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects) |
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