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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
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References:

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