The Mahler measure of a genus 3 family. (English) Zbl 1468.11222
In the paper under review, the authors show that the two polynomials
\[
P_{k}(x,y):=y^{2}+(x^{6}+kx^{5}-x^{4}+2(1-k)x^{3}-x^{2}+kx+1)y+x^{6}
\]
and
\[
Q_{k}(x,y):=xy^{2}+(kx-1)y-x^{2}+x,
\]
where \(k\in \mathbb{N}\) and \(k\geq 2\), have the same Mahler measure.
The method of proof of this conjectural identity of [H. Liu and H. Qin, “Data for Mahler measure of polynomials defining genus 2 and 3 curves”, https://github.com/liuhangsnnu/mahler-measure-of-genus-2-and-3-curves] is similar to that employed in [M. Lalín and G. Wu, Int. J. Number Theory 15, No. 5, 945–967 (2019; Zbl 1448.11191)], establishing identities between the regulators, but new strategies are used to simplify the regulator of the genus \(3\) curve \(P_{k}(x,y)=0\), when \(k\geq 3\), before comparing it to the regulator of the curve \(Q_{k}(x,y)=0\) of genus \(1\).
The method of proof of this conjectural identity of [H. Liu and H. Qin, “Data for Mahler measure of polynomials defining genus 2 and 3 curves”, https://github.com/liuhangsnnu/mahler-measure-of-genus-2-and-3-curves] is similar to that employed in [M. Lalín and G. Wu, Int. J. Number Theory 15, No. 5, 945–967 (2019; Zbl 1448.11191)], establishing identities between the regulators, but new strategies are used to simplify the regulator of the genus \(3\) curve \(P_{k}(x,y)=0\), when \(k\geq 3\), before comparing it to the regulator of the curve \(Q_{k}(x,y)=0\) of genus \(1\).
Reviewer: Toufik Zaïmi (Riyadh)
MSC:
| 11R06 | PV-numbers and generalizations; other special algebraic numbers; Mahler measure |
| 11G05 | Elliptic curves over global fields |
| 11F66 | Langlands \(L\)-functions; one variable Dirichlet series and functional equations |
| 19F27 | Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects) |
| 33E05 | Elliptic functions and integrals |
Keywords:
Mahler measure; special values of \(L\)-functions; genus 3 curves; elliptic curves; elliptic regulatorCitations:
Zbl 1448.11191Software:
GitHubReferences:
| [1] | Beĭlinson, AA, Higher regulators and values of \(L\)-functions of curves, Funktsional. Anal. Prilozhen., 14, 2, 46-47 (1980) · Zbl 0475.14015 |
| [2] | Bloch, SJ, Higher Regulators, Algebraic \(K\)-Theory, and Zeta Functions of Elliptic Curves (2000), Providence: American Mathematical Society, Providence · Zbl 0958.19001 |
| [3] | Bosman, J.: Boyd’s conjecture for a family of genus 2 curves, Thesis-Universiteit Utrecht (2004) |
| [4] | Boyd, DW, Speculations concerning the range of Mahler’s measure, Can. Math. Bull., 24, 4, 453-469 (1981) · Zbl 0474.12005 · doi:10.4153/CMB-1981-069-5 |
| [5] | Boyd, DW, Mahler’s measure and special values of \(L\)-functions, Exp. Math., 7, 1, 37-82 (1998) · Zbl 0932.11069 · doi:10.1080/10586458.1998.10504357 |
| [6] | Bertin, MJ; Zudilin, W., On the Mahler measure of a family of genus 2 curves, Math. Z., 283, 3-4, 1185-1193 (2016) · Zbl 1347.11076 · doi:10.1007/s00209-016-1637-6 |
| [7] | Bertin, MJ; Zudilin, W., On the Mahler measure of hyperelliptic families, Ann. Math. Qué., 41, 1, 199-211 (2017) · Zbl 1434.11211 · doi:10.1007/s40316-016-0068-4 |
| [8] | Deninger, C., Deligne periods of mixed motives, \(K\)-theory and the entropy of certain \({ Z}^n\)-actions, J. Am. Math. Soc., 10, 2, 259-281 (1997) · Zbl 0913.11027 · doi:10.1090/S0894-0347-97-00228-2 |
| [9] | Lehmer, DH, Factorization of certain cyclotomic functions, Ann. Math., 34, 3, 461-479 (1933) · Zbl 0007.19904 · doi:10.2307/1968172 |
| [10] | Liu, H., Qin, H.: Data for Mahler measure of polynomials defining genus 2 and 3 curves. https://github.com/liuhangsnnu/mahler-measure-of-genus-2-and-3-curves |
| [11] | Liu, H., Qin, H.: Mahler measure of polynomials defining genus 2 and 3 curves. arXiv:1910.10884 [math.NT] (2019) |
| [12] | Lalín, M.; Gang, W., Regulator proofs for Boyd’s identities on genus 2 curves, Int. J. Number Theory, 15, 5, 945-967 (2019) · Zbl 1448.11191 · doi:10.1142/S1793042119500519 |
| [13] | Mellit, A., Elliptic dilogarithms and parallel lines, J. Number Theory, 204, 1-24 (2019) · Zbl 1464.11112 · doi:10.1016/j.jnt.2019.03.019 |
| [14] | Rodriguez-Villegas, F., Modular Mahler Measures. I, Topics in Number Theory (University Park, PA), 17-48 (1997), Dordrecht: Kluwer Academic Publisher, Dordrecht · Zbl 0980.11026 |
| [15] | Rodriguez-Villegas, F., Identities Between Mahler Measures, 223-229 (2002), Natick: A K Peters, Natick · Zbl 1029.11054 |
| [16] | Rogers, M.; Zudilin, W., From \(L\)-series of elliptic curves to Mahler measures, Compos. Math., 148, 2, 385-414 (2012) · Zbl 1260.11062 · doi:10.1112/S0010437X11007342 |
| [17] | Smyth, CJ, On measures of polynomials in several variables, Bull. Aust. Math. Soc., 23, 1, 49-63 (1981) · Zbl 0442.10034 · doi:10.1017/S0004972700006894 |
| [18] | Zagier, D.; Gangl, H., Classical and Elliptic Polylogarithms and Special Values of \(L\)-Series, The Arithmetic and geometry of Algebraic Cycles (Banff, AB, 1998), 561-615 (2000), Dordrecht: Kluwer Academic Publisher, Dordrecht · Zbl 0990.11041 |
| [19] | Zudilin, W., Regulator of modular units and Mahler measures, Math. Proc. Camb. Philos. Soc., 156, 2, 313-326 (2014) · Zbl 1386.11129 · doi:10.1017/S0305004113000765 |
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