## On the Mahler measure of a family of genus 2 curves.(English)Zbl 1347.11076

The (logarithmic) Mahler measure of a polynomial $$P(x,y)$$ is the average of $$\log|P(x,y)|$$ over the 2-torus $$\{|x| = 1, |y| = 1\}$$. The reviewer in [Exp. Math. 7, No. 1, 37–82 (1998; Zbl 0932.11069)] was led by extensive numerical experiments on families of polynomials such as $$P_k(x,y) = (x+1)y^2 + (x^2 +kx + 1)y + (x^2+x)$$ to conjecture results such as $$m(P_k) = r_k L'(E_k,0)$$, where $$r_k$$ are explicitly given rational numbers, $$E_k$$ is the curve defined by $$P_k = 0$$ which is generically an elliptic curve, and $$L(E_k,s)$$ is the $$L$$-function of $$E_k$$. (Here $$k$$ is taken to be an integer but for certain families, F. Rodriguez Villegas [in: Topics in number theory. In honor of B. Gordon and S. Chowla. Pennsylvania State University, University Park, USA,1997. Dordrecht: Kluwer Academic Publishers. 17–48 (1999; Zbl 0980.11026)] showed that it is more appropriate to allow $$k^2$$ to be an integer). Some of these results have now been established for this family and some other families but the conjectures are mostly still unproved. In the same paper, the reviewer considered some families of genus 2 curves whose Jacobians split into a product of 2 (generically) elliptic curves and for which similar formulas hold involving just one of the two factors of the Jacobian. For example, the family $$Q_k(x,y) = y^2 + (x^4 + kx^3 + 2kx^2 + k + 1)y + x^4$$ is one such family for which such formulas were conjectured. This family was considered in J. Bosman’s thesis [Boyd’s conjecture for a family of genus 2 curves. Utrecht: Universiteit Utrecht (Master Thesis) (2004)] and $$m(Q_k)$$ was evaluated for $$k = 2, -1$$ and $$8$$. It happens that one of the factors of the Jacobian of $$Q_k = 0$$ is the curve $$P_{2-k} = 0$$ and the conjectures of the reviewer would imply that $$m(Q_k) = 2m(P_{2-k})$$ for $$0 \leq k \leq 4$$ and $$m(Q_k) = m(P_{2-k})$$ for $$k \leq -1$$. This result is proved in the paper under review by a clever analysis which reduces the derivatives with respect to $$k$$ of the Mahler measures in question to (sometimes incomplete) elliptic integrals. The method seems quite robust and indeed the authors preview a forthcoming paper in which further results of this type will be proved.

### MSC:

 11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure 11G05 Elliptic curves over global fields 11G30 Curves of arbitrary genus or genus $$\ne 1$$ over global fields 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols

### Citations:

Zbl 0932.11069; Zbl 0980.11026
Full Text:

### References:

 [1] Berndt, B.C.: Ramanujan’s Notebooks, Part V. Springer, New York (1998) · Zbl 0886.11001 [2] Bertin, MJ, Une mesure de Mahler explicite, C. R. Acad. Sci. Paris Sér. I Math., 333, 1-3, (2001) · Zbl 1011.11069 [3] Bertin, M.J., Zudilin, W.: On the Mahler measure of hyperelliptic families, Preprint (2016); arXiv:1601.07583 [math.NT] · Zbl 1434.11211 [4] Borwein, JM; Straub, A; Wan, J; Zudilin, W, Densities of short uniform random walks. with an appendix by don Zagier, Can. J. Math., 64, 961-990, (2012) · Zbl 1296.33011 [5] Bosman, J.: Boyd’s conjecture for a family of genus $$2$$ curves, Thesis (2004). http://www.uni-due.de/ ada649b/papers/scriptie.pdf · Zbl 1386.11129 [6] Boyd, D, Mahler’s measure and special values of $$L$$-functions, Exp. Math., 7, 37-82, (1998) · Zbl 0932.11069 [7] Goursat, É, Sur la réduction des intégrales hyperelliptiques, Bull. Soc. Math. Fr., 13, 143-162, (1885) · JFM 17.0466.01 [8] Mellit, A.: Elliptic dilogarithms and parallel lines, Preprint (2009, 2011); arXiv:1207.4722 [math.NT] · Zbl 1464.11112 [9] Rodriguez Villegas, F.: Modular Mahler measures. I. In: Topics in Number Theory. University Park, PA, 1997, Math. Appl. 467 (Kluwer Acadamic Publication, Dordrecht, 1999), pp. 17-48 · Zbl 0980.11026 [10] Rogers, M; Zudilin, W, From $$L$$-series of elliptic curves to Mahler measures, Compos. Math., 148, 385-414, (2012) · Zbl 1260.11062 [11] Verrill, H.A.: Picard-Fuchs equations of some families of elliptic curves. In: Proceedings on Moonshine and Related Topics (Montréal, QC, 1999), CRM Proc. Lecture Notes 30 (Amer. Math. Soc., Providence, RI, 2001), pp. 253-268 · Zbl 1082.14503 [12] Zudilin, W, Regulator of modular units and Mahler measures, Math. Proc. Camb. Philos. Soc., 156, 313-326, (2014) · Zbl 1386.11129
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.