Experimental Mathematics

Mahler's measure and special values of {$L$}-functions

David W. Boyd

Abstract

If $P(x_1$,\,\dots,\,$x_n)$ is a polynomial with integer coefficients, the Mahler measure $M(P)$ of $P$ is defined to be the geometric mean of $|P|$ over the $n$-torus $\T ^n$. For $n = 1$, $M(P)$ is an algebraic integer, but for $n$\raise.5pt\hbox{\footnotesize\mathversion{bold}${}>{}$}$1$, there is reason to believe that $M(P)$ is usually transcendental. For example, Smyth showed that $\log M(1+x+y)=L'(${\mathversion{normal}$\chi$}$_{-3}$,$\,{-}1)$, where {\mathversion{normal}$\chi$}$_{-3}$ is the odd Dirichlet character of conductor $3$. Here we will describe some examples for which it appears that $\log M(P(x$,$\,y)) = r@@L'(E$,$\,0)$, where $E$ is an elliptic curve and $r$ is a rational number, often either an integer or the reciprocal of an integer. Most of the formulas we discover have been verified numerically to high accuracy but not rigorously proved.

Article information

Source
Experiment. Math., Volume 7, Issue 1 (1998), 37-82.

Dates
First available in Project Euclid: 14 March 2003

Permanent link to this document
https://projecteuclid.org/euclid.em/1047674271

Mathematical Reviews number (MathSciNet)
MR1618282

Zentralblatt MATH identifier
0932.11069

Subjects
Primary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10]
Secondary: 11R06: PV-numbers and generalizations; other special algebraic numbers; Mahler measure 11Y35: Analytic computations

Keywords
Mahler measure polynomials computation $L$-function elliptic curve Beilinson conjectures

Citation

Boyd, David W. Mahler's measure and special values of {$L$}-functions. Experiment. Math. 7 (1998), no. 1, 37--82. https://projecteuclid.org/euclid.em/1047674271


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