Modular Mahler measures. I.

*(English)*Zbl 0980.11026
Ahlgren, Scott D. (ed.) et al., Topics in number theory. In honor of B. Gordon and S. Chowla. Proceedings of the conference, Pennsylvania State University, University Park, PA, USA, July 31-August 3, 1997. Dordrecht: Kluwer Academic Publishers. Math. Appl., Dordr. 467, 17-48 (1999).

The logarithmic Mahler measure of a Laurent polynomial \(P\) in \(n\) variables is the average of \(\log|P|\) over the \(n\)-torus. Recently C. Deninger [J. Am. Math. Soc. 10, 259-281 (1997; Zbl 0913.11027)] has shown that in certain cases \(m(P)\) can be interpreted as a Deligne period of a mixed motive. In the same paper, Deninger showed that a general conjecture of Bloch and Beilinson would imply that if \(P_k = x + 1/x + y + 1/y + k\), then \(m(P_1) = r \cdot L'(E,0)\), where \(E\) is the elliptic curve defined by \(P_1 = 0\) and \(r\) is some non-zero rational number. The reviewer verified that \(r = 1\) to 50 decimal places and went on to a numerical study of \(m(P_k)\) for integer \(k\), and to the study of many similar families of polynomials \(P_k\) depending on one parameter [Exp. Math. 7, 37-82 (1998; Zbl 0932.11069)], and such that \(P_k(x,y) = 0\) defines a curve \(X_k\) of genus 1 or 2. Some precise conditions under which a formula of the sort \(L'(E_k,0) = r_k \cdot m(P_k)\) should hold were discovered by numerical experiment and heuristic arguments. (Here \(E_k\) is an elliptic curve which is a factor of the Jacobian of \(X_k\)).

In this expository paper the author shows how these conjectures can be reduced to the conjecture of Bloch-Beilinson, at least in the genus 1 case. Some of the results of this paper have also been proved by H. C. Bornhorn [Preprint 35, Math. Inst. Univ. Münster, January 1999, see also his thesis (1998; Zbl 1109.11315)]. Bornhorn also treats the genus 2 case.

The paper under review begins with an enlightening discussion of special values of \(L\)-functions. Next comes a short introduction to \(K_2\) of fields and elliptic curves, symbols, the Beilinson regulator, and the Bloch-Beilinson conjecture. The reviewer’s conjecture is then presented and the significance of the condition that \(P_k\) be “tempered” is explained. This discussion ends with a sketch of the basic theorem (p. 33) which shows how \(m(P_k)\) is related to the regulator and why this conjecturally implies that \(L'(E_k,0)\) is a non-zero rational multiple of \(m(P_k)\).

The final section of the paper shows how \(m(P_k)\) considered as a function of the complex variable \(k\) can often be expressed in terms of certain modular forms and in certain cases that \(m(P_k)\) is a Kronecker-Eisenstein series. This gives an unconditional proof of the conjectured formulas in some cases in which the elliptic curve \(E_k\) has complex multiplication. In final remarks, the author points out the intriguing possiblity of a connection with “mirror symmetry”.

For the entire collection see [Zbl 0913.00029].

In this expository paper the author shows how these conjectures can be reduced to the conjecture of Bloch-Beilinson, at least in the genus 1 case. Some of the results of this paper have also been proved by H. C. Bornhorn [Preprint 35, Math. Inst. Univ. Münster, January 1999, see also his thesis (1998; Zbl 1109.11315)]. Bornhorn also treats the genus 2 case.

The paper under review begins with an enlightening discussion of special values of \(L\)-functions. Next comes a short introduction to \(K_2\) of fields and elliptic curves, symbols, the Beilinson regulator, and the Bloch-Beilinson conjecture. The reviewer’s conjecture is then presented and the significance of the condition that \(P_k\) be “tempered” is explained. This discussion ends with a sketch of the basic theorem (p. 33) which shows how \(m(P_k)\) is related to the regulator and why this conjecturally implies that \(L'(E_k,0)\) is a non-zero rational multiple of \(m(P_k)\).

The final section of the paper shows how \(m(P_k)\) considered as a function of the complex variable \(k\) can often be expressed in terms of certain modular forms and in certain cases that \(m(P_k)\) is a Kronecker-Eisenstein series. This gives an unconditional proof of the conjectured formulas in some cases in which the elliptic curve \(E_k\) has complex multiplication. In final remarks, the author points out the intriguing possiblity of a connection with “mirror symmetry”.

For the entire collection see [Zbl 0913.00029].

Reviewer: D.W.Boyd (Vancouver)

##### MSC:

11F11 | Holomorphic modular forms of integral weight |

14C35 | Applications of methods of algebraic \(K\)-theory in algebraic geometry |

11R09 | Polynomials (irreducibility, etc.) |

11R06 | PV-numbers and generalizations; other special algebraic numbers; Mahler measure |

11R42 | Zeta functions and \(L\)-functions of number fields |

14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |