Many variations of Mahler measures. A lasting symphony.

*(English)*Zbl 07169774
Australian Mathematical Society Lecture Series 28. Cambridge: Cambridge University Press (ISBN 978-1-108-79445-9/pbk; 978-1-108-88555-3/ebook). xv, 167 p. (2020).

The book under review gives a very well-thought introduction to the Mahler measure of polynomials, through the lenses of its many variations, which range from number theory to probability, passing from algebraic geometry and dynamics. The structure of the book is based on a Masterclass, organised by Fabien Pazuki and the reviewer in August 2018 at the University of Copenhagen, where the authors gave two mini-courses around Mahler’s measure. Thus, the book is perfectly suited for self-study, while also serving as one of the very few introductory references to the topic.

Let us go through the contents of the book in more detail. First of all, the logarithmic Mahler measure of a nonzero Laurent polynomial \(P \in \mathbb{C}[x_1^{\pm 1},\dots,x_n^{\pm 1}]\) is defined as the following improper integral: \[m(P) := \int_{[0,1]^n} \log\lvert P(e^{2 \pi i t_1},\dots,e^{2 \pi i t_n}) \rvert \ d t_1 \cdots d t_n\] which converges thanks to a nontrivial estimate on the Lebesgue measure of the locus where \(\lvert P \rvert\) is small. Originally introduced by K. Mahler [J. Lond. Math. Soc. 37, 341–344 (1962; Zbl 0105.06301)], this invariant specializes, for univariate polynomials, to quantities already studied by T. A. Pierce [Ann. Math. (2) 18, 53–64 (1916; JFM 46.0194.03)] and D. H. Lehmer [Ann. Math. (2) 34, 461–479 (1933; Zbl 0007.19904)], which are related to the problem of finding sequences of integer numbers attaining many prime values.

These univariate Mahler measures are treated in the first chapter of the book under review, where the authors give also a proof of Jensen’s formula and Kronecker’s theorem, which characterizes the polynomials \(P \in \mathbb{Z}[x_1]\) having smallest Mahler measure \(m(P) = 0\), which turn out to be exactly the products of cyclotomic polynomials.

As Lehmer already observed, it is thus interesting to find polynomials with integer coefficients whose Mahler measure is strictly positive but very small. He also provided a polynomial of degree 10, which remains to this day the integral polynomial with the smallest known non-zero Mahler measure. The second chapter of the book is devoted to the various attempts that have been carried out in the last eighty years to prove this assertion rigorously. In particular, Section 2.1 contains a complete proof of C. J. Smyth’s solution to Lehmer’s problem in the non-palindromic case (see [Bull. Lond. Math. Soc. 3, 169–175 (1971; Zbl 0235.12003)]), and Section 2.2 deals with E. Dobrowolski’s lower bound for the Mahler measure in terms of the degree (see [Acta Arith. 34, 391–401 (1979; Zbl 0416.12001)]). A very original part of this chapter is given by Section 2.3, where it is shown that Lehmer’s problem has a surprising relation with the problem of bounding the discriminant of certain, very peculiar kinds of polynomials, and number fields.

The third chapter of the book starts treating general, multivariate Mahler measures, by giving a complete proof of the convergence of the integral defining the Mahler measure of a polynomial, in any number of variables. Moreover, Section 3.4 is devoted to a theorem of D. W. Boyd [Can. Math. Bull. 24, 453–469 (1981; Zbl 0474.12005)], which shows that multivariate Mahler measures can be approximated by univariate ones. The following Section 3.5 is devoted to two particular families of highly symmetric polynomials in two variables, whose Mahler measures are shown to coincide. Finally, let us mention that Section 3.3 explains carefully the surprising result of C. J. Smyth [Bull. Aust. Math. Soc. 23, 49–63 (1981; Zbl 0442.10034)], which shows that \(m(x_1 + x_2 + 1) = L'(\chi,-1)\) for some quadratic Dirichlet character \(\chi\), and \(m(x_1 + x_2 + x_3 + 1) = -14 \, \zeta'(-2)\), where \(\zeta\) is Riemann’s \(\zeta\)-function. This was the pioneer of several results linking Mahler measures of polynomials to special values of \(L\)-functions, which are treated in more depth in the second half of the book.

Moving on, the fourth chapter of the book gives a full proof of V. Maillot’s generalization of Smyth’s theorem [Mém. Soc. Math. Fr., Nouv. Sér. 80, 129 p. (2000; Zbl 0963.14009)], which computes the Mahler measure of a linear form \(a x_1 + b x_2 + c\) in terms of the Bloch-Wiegner dilogarithm of some explicit algebraic number. In order to do this, the authors give a complete and elegant proof of the five-term relation satisfied by the classical dilogarithm, by means of the \(q\)-analogue of the binomial theorem, and also a thorough introduction to the Rogers and Bloch-Wigner dilogarithms.

The fifth chapter of the book studies Mahler measures as functions of a parameter. In particular, it is shown that these functions often satisfy some very peculiar differential equations, and can therefore be evaluated explicitly in terms of hypergeometric functions. To conclude the first part of the book, the sixth chapter explores a fascinating relation between Mahler’s measure and random walks, pioneered by the work of J. M. Borwein et al. [Can. J. Math. 64, No. 5, 961–990 (2012; Zbl 1296.33011)], which allows one to prove some more closed, hypergeometric evaluations of Mahler measures.

Let us move to review the second part of the book, which treats in more depth the links between Mahler measures and special values of \(L\)-functions. First of all, the seventh chapter of the book reviews the regulator map on the second \(K\)-theory group of a curve, which produces a class in the first (singular) cohomology group of the curve, that can therefore be integrated against a close loop.

This map admits in fact a generalization to any \(K\)-group associated to any algebraic variety, which is one of the main protagonists of the eight chapter. Here the level of abstraction increases considerably, as the authors introduce the regulator map between motivic and Deligne-Beilinson’s cohomology. This is notably used to state A. A. Beilinson’s conjectures [J. Sov. Math. 30, 2036–2070 (1985; Zbl 0588.14013)], which asserts that this regulator map should be an isomorphism, and also that its determinant with respect to some natural \(\mathbb{Q}\)-structures should be a rational multiple of a special value of an \(L\)-function.

This is a crucial point, because Beilinson’s conjecture is precisely the theoretical reason why one should believe that Smyth’s surprising link between Mahler measures and special values of \(L\)-functions is by no means an accident. Indeed, the foundational work of C. Deninger [J. Am. Math. Soc. 10, No. 2, 259–281 (1997; Zbl 0913.11027)], which is summarized in Section 8.3 of the book under review, shows that the Mahler measure of a generic polynomial may be obtained by integrating the regulator of a very simple motivic cohomology class against a complicated cycle, defined in terms of the polynomial itself. This gives a conjectural explanation for the vast computational evidence, put forward by the seminal work of D. W. Boyd [Exp. Math. 7, No. 1, 37–82 (1998; Zbl 0932.11069)], between the Mahler measure of some families of polynomials in two variables and special values of \(L\)-functions of elliptic curves. These links, as well as some generalizations to a higher number of variables, are the subject of Section 8.4.

Now, the ninth chapter of the book introduces the language of modular forms, in order to give a complete proof of a theorem by M. Rogers and W. Zudilin [Int. Math. Res. Not. 2014, No. 9, 2305–2326 (2014; Zbl 1378.11091)], which provides the identity \(m(x_1+\frac{1}{x_1} + x_2 + \frac{1}{x_2} + 1) = L'(E,0)\), where \(E\) is a rational elliptic curve of conductor \(15\). This is done by means of a clever trick (nowadays known as Rogers-Zudilin method) to compute the \(L\)-function associated to a product of two Eisenstein series.

The tenth chapter of the book under review generalizes the previous considerations to modular forms of higher weight, therefore giving an account of the work [Compos. Math. 153, No. 6, 1119–1152 (2017; Zbl 1386.19010)] by the first author, as well as of its application provided in [Trans. Am. Math. Soc. 372, No. 1, 119–152 (2019; Zbl 1448.11190)] (joint with M. Neururer), related to the computation of Mahler measures of polynomials in three variables.

The book ends with an appendix, which reviews some needed aspects of motivic and Deligne-Beilinson cohomology. In particular, it is explained how one can represent the latter in Voevodsky’s triangulated category of mixed motives, following a course by J. Wildeshaus at the École normale supérieure de Lyon.

To conclude, the book will serve as a great introduction to the subject of Mahler’s measure, in some of its manifold variations, with a special focus on its links with special values of \(L\)-functions. It is particularly suited for a student or research seminar, as well as for individual work, because of its concise nature, which emphasizes the most important points of the theory, while not leaving out crucial details when needed.

Let us go through the contents of the book in more detail. First of all, the logarithmic Mahler measure of a nonzero Laurent polynomial \(P \in \mathbb{C}[x_1^{\pm 1},\dots,x_n^{\pm 1}]\) is defined as the following improper integral: \[m(P) := \int_{[0,1]^n} \log\lvert P(e^{2 \pi i t_1},\dots,e^{2 \pi i t_n}) \rvert \ d t_1 \cdots d t_n\] which converges thanks to a nontrivial estimate on the Lebesgue measure of the locus where \(\lvert P \rvert\) is small. Originally introduced by K. Mahler [J. Lond. Math. Soc. 37, 341–344 (1962; Zbl 0105.06301)], this invariant specializes, for univariate polynomials, to quantities already studied by T. A. Pierce [Ann. Math. (2) 18, 53–64 (1916; JFM 46.0194.03)] and D. H. Lehmer [Ann. Math. (2) 34, 461–479 (1933; Zbl 0007.19904)], which are related to the problem of finding sequences of integer numbers attaining many prime values.

These univariate Mahler measures are treated in the first chapter of the book under review, where the authors give also a proof of Jensen’s formula and Kronecker’s theorem, which characterizes the polynomials \(P \in \mathbb{Z}[x_1]\) having smallest Mahler measure \(m(P) = 0\), which turn out to be exactly the products of cyclotomic polynomials.

As Lehmer already observed, it is thus interesting to find polynomials with integer coefficients whose Mahler measure is strictly positive but very small. He also provided a polynomial of degree 10, which remains to this day the integral polynomial with the smallest known non-zero Mahler measure. The second chapter of the book is devoted to the various attempts that have been carried out in the last eighty years to prove this assertion rigorously. In particular, Section 2.1 contains a complete proof of C. J. Smyth’s solution to Lehmer’s problem in the non-palindromic case (see [Bull. Lond. Math. Soc. 3, 169–175 (1971; Zbl 0235.12003)]), and Section 2.2 deals with E. Dobrowolski’s lower bound for the Mahler measure in terms of the degree (see [Acta Arith. 34, 391–401 (1979; Zbl 0416.12001)]). A very original part of this chapter is given by Section 2.3, where it is shown that Lehmer’s problem has a surprising relation with the problem of bounding the discriminant of certain, very peculiar kinds of polynomials, and number fields.

The third chapter of the book starts treating general, multivariate Mahler measures, by giving a complete proof of the convergence of the integral defining the Mahler measure of a polynomial, in any number of variables. Moreover, Section 3.4 is devoted to a theorem of D. W. Boyd [Can. Math. Bull. 24, 453–469 (1981; Zbl 0474.12005)], which shows that multivariate Mahler measures can be approximated by univariate ones. The following Section 3.5 is devoted to two particular families of highly symmetric polynomials in two variables, whose Mahler measures are shown to coincide. Finally, let us mention that Section 3.3 explains carefully the surprising result of C. J. Smyth [Bull. Aust. Math. Soc. 23, 49–63 (1981; Zbl 0442.10034)], which shows that \(m(x_1 + x_2 + 1) = L'(\chi,-1)\) for some quadratic Dirichlet character \(\chi\), and \(m(x_1 + x_2 + x_3 + 1) = -14 \, \zeta'(-2)\), where \(\zeta\) is Riemann’s \(\zeta\)-function. This was the pioneer of several results linking Mahler measures of polynomials to special values of \(L\)-functions, which are treated in more depth in the second half of the book.

Moving on, the fourth chapter of the book gives a full proof of V. Maillot’s generalization of Smyth’s theorem [Mém. Soc. Math. Fr., Nouv. Sér. 80, 129 p. (2000; Zbl 0963.14009)], which computes the Mahler measure of a linear form \(a x_1 + b x_2 + c\) in terms of the Bloch-Wiegner dilogarithm of some explicit algebraic number. In order to do this, the authors give a complete and elegant proof of the five-term relation satisfied by the classical dilogarithm, by means of the \(q\)-analogue of the binomial theorem, and also a thorough introduction to the Rogers and Bloch-Wigner dilogarithms.

The fifth chapter of the book studies Mahler measures as functions of a parameter. In particular, it is shown that these functions often satisfy some very peculiar differential equations, and can therefore be evaluated explicitly in terms of hypergeometric functions. To conclude the first part of the book, the sixth chapter explores a fascinating relation between Mahler’s measure and random walks, pioneered by the work of J. M. Borwein et al. [Can. J. Math. 64, No. 5, 961–990 (2012; Zbl 1296.33011)], which allows one to prove some more closed, hypergeometric evaluations of Mahler measures.

Let us move to review the second part of the book, which treats in more depth the links between Mahler measures and special values of \(L\)-functions. First of all, the seventh chapter of the book reviews the regulator map on the second \(K\)-theory group of a curve, which produces a class in the first (singular) cohomology group of the curve, that can therefore be integrated against a close loop.

This map admits in fact a generalization to any \(K\)-group associated to any algebraic variety, which is one of the main protagonists of the eight chapter. Here the level of abstraction increases considerably, as the authors introduce the regulator map between motivic and Deligne-Beilinson’s cohomology. This is notably used to state A. A. Beilinson’s conjectures [J. Sov. Math. 30, 2036–2070 (1985; Zbl 0588.14013)], which asserts that this regulator map should be an isomorphism, and also that its determinant with respect to some natural \(\mathbb{Q}\)-structures should be a rational multiple of a special value of an \(L\)-function.

This is a crucial point, because Beilinson’s conjecture is precisely the theoretical reason why one should believe that Smyth’s surprising link between Mahler measures and special values of \(L\)-functions is by no means an accident. Indeed, the foundational work of C. Deninger [J. Am. Math. Soc. 10, No. 2, 259–281 (1997; Zbl 0913.11027)], which is summarized in Section 8.3 of the book under review, shows that the Mahler measure of a generic polynomial may be obtained by integrating the regulator of a very simple motivic cohomology class against a complicated cycle, defined in terms of the polynomial itself. This gives a conjectural explanation for the vast computational evidence, put forward by the seminal work of D. W. Boyd [Exp. Math. 7, No. 1, 37–82 (1998; Zbl 0932.11069)], between the Mahler measure of some families of polynomials in two variables and special values of \(L\)-functions of elliptic curves. These links, as well as some generalizations to a higher number of variables, are the subject of Section 8.4.

Now, the ninth chapter of the book introduces the language of modular forms, in order to give a complete proof of a theorem by M. Rogers and W. Zudilin [Int. Math. Res. Not. 2014, No. 9, 2305–2326 (2014; Zbl 1378.11091)], which provides the identity \(m(x_1+\frac{1}{x_1} + x_2 + \frac{1}{x_2} + 1) = L'(E,0)\), where \(E\) is a rational elliptic curve of conductor \(15\). This is done by means of a clever trick (nowadays known as Rogers-Zudilin method) to compute the \(L\)-function associated to a product of two Eisenstein series.

The tenth chapter of the book under review generalizes the previous considerations to modular forms of higher weight, therefore giving an account of the work [Compos. Math. 153, No. 6, 1119–1152 (2017; Zbl 1386.19010)] by the first author, as well as of its application provided in [Trans. Am. Math. Soc. 372, No. 1, 119–152 (2019; Zbl 1448.11190)] (joint with M. Neururer), related to the computation of Mahler measures of polynomials in three variables.

The book ends with an appendix, which reviews some needed aspects of motivic and Deligne-Beilinson cohomology. In particular, it is explained how one can represent the latter in Voevodsky’s triangulated category of mixed motives, following a course by J. Wildeshaus at the École normale supérieure de Lyon.

To conclude, the book will serve as a great introduction to the subject of Mahler’s measure, in some of its manifold variations, with a special focus on its links with special values of \(L\)-functions. It is particularly suited for a student or research seminar, as well as for individual work, because of its concise nature, which emphasizes the most important points of the theory, while not leaving out crucial details when needed.

Reviewer: Riccardo Pengo (Lyon)

##### MSC:

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

11R09 | Polynomials (irreducibility, etc.) |

11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |

11G50 | Heights |