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The Mahler measure for arbitrary tori
 Matilde Lalín^{1}Email authorView ORCID ID profile and
 Tushant Mittal^{2}
 Received: 8 August 2017
 Accepted: 15 March 2018
 Published: 23 March 2018
Abstract
We consider a variation of the Mahler measure where the defining integral is performed over a more general torus. We focus our investigation on two particular polynomials related to certain elliptic curve E and we establish new formulas for this variation of the Mahler measure in terms of \(L'(E,0)\).
Keywords
 Mahler measure
 Special values of Lfunctions
 Elliptic curve
 Elliptic regulator
Mathematics Subject Classification
 Primary 11R06
 Secondary 11G05
 11F66
 19F27
 33E05
1 Introduction
This construction, when applied to multivariable polynomials, often yields values of special functions, including some of number theoretic interest, such as the Riemann zetafunction and Lfunctions associated to arithmeticgeometric objects such as elliptic curves.
We remark that the formulas above also apply when the constant coefficient is replaced by a variable, in the sense that \(\mathrm {m}(ax+by+cz)=\mathrm {m}(ax+by+c)\).
In this work, we consider the following extension of Mahler measure.
Definition 1
This idea of considering arbitrary tori in the integration was initially proposed to us by RodriguezVillegas a long time ago.
Given this definition, Cassaigne and Maillot’s formula can be interpreted as \(\mathrm {m}_{a,b,c}(x+y+z)\). Some cases of the formula of Vandervelde [19] for the Mahler measure of \(axy+bx+cy+d\) may be also viewed in this context.
Our goal is to explore this definition for other formulas, in particular those involving elliptic curves. More precisely, we connect this idea with Boyd’s examples in order to prove the following results.
Theorem 2
The results above are, in some sense, quite restricted, due to the technical difficulties involving the study of the integration path. They are similar in generality to some earlier formulas from [8] that involve a single varying parameter and relate Mahler measures to polylogarithms.
By changing variables \(y \rightarrow ay\) and \(x\rightarrow ax\) and dividing by \(a^2\) in the first polynomial, and \(y \rightarrow ay\) and \(x\rightarrow a^2x\) and dividing by a in the second polynomial, we obtain the following corollary which expresses the same results in terms of the classical Mahler measure of nontempered polynomials (as defined in Sect. 2, Definition 4), which are very interesting in their own, since the Ktheory framework does not completely apply to these cases.
Corollary 3
Our method of proof follows several steps. In Sect. 2 we recall the relationship between the Mahler measure of polynomials associated to elliptic curves and the regulator. Then, in Sect. 3 we analyze what happens to the regulator integral when the integration domain is changed according to our new definition. We start working with our particular examples in Sect. 4, where we establish the relationship between the regulators of these two families. After that, it remains to discuss the integration paths and to characterize them as elements in the homology, which is done in Sects. 5 and 6. Section 7 deals with a technical residue computation. The proof of our result is completed in Sect. 8. In the Appendix we include additional information on the homology cycles. While this information is not strictly necessary for the proof of Theorem 2, we believe it provides a valuable perspective to the proof under consideration.
2 The connection between Mahler measure and the regulator
In this section we recall the definition of the regulator on the second Kgroup of an elliptic curve E given by Bloch and Beĭlinson and explain how it can be computed in terms of the elliptic dilogarithm. We then discuss the relationship between Mahler measure and the regulator.
Definition 4
P is said to be tempered if the set of roots of its side polynomials consists of roots of unity only. (See Section III of [13] for more details on these definitions.)
Let \(E/\mathbb {Q}\) be an elliptic curve given by an equation \(P(x,y)=0\). RodriguezVillegas [13] associates the condition that the polynomial P is tempered to the conditions that the tame symbols in Ktheory are trivial. In that case we can think of \(K_2(E)\otimes \mathbb {Q}\subset K_2(\mathbb {Q}(E)) \otimes \mathbb {Q}\).
Definition 5
In the above definition, we take \([\gamma ] \in H_1(E,{\mathbb {Z}})\) and interpret \(H^1(E,{\mathbb {R}})\) as the dual of \(H_1(E,{\mathbb {Z}})\).
We remark that the regulator is actually defined over \(K_2({\mathcal {E}})\), where \({\mathcal {E}}\) is the Néron model of the elliptic curve. \(K_2({\mathcal {E}})\otimes \mathbb {Q}\) is a subgroup of \(K_2(E)\otimes \mathbb {Q}\) determined by finitely many extra conditions as described in [3].
Remark 6
Due to the action of complex conjugation on \(\eta \), the regulator map is trivial for the classes that remain invariant by complex conjugation, denoted by \(H_1(E,{\mathbb {Z}})^+\). It therefore suffices to consider the regulator as a function on \(H_1(E,{\mathbb {Z}})^\).
The next definition is due to Bloch [5].
Definition 7
Theorem 8
When P corresponds to an elliptic curve and when the set \(\{x=1,y_i(x)\ge 1\}\) can be seen as a cycle in \(H_1(E,\mathbb {Z})^\), then we may be able to recover a formula of the type (7). This has to be examined on a case by case basis.
3 The initial analysis with an arbitrary torus
When working with an arbitrary torus, we can still do a similar analysis to the one in the previous section. We continue with the notation that \(P(x,y) \in \mathbb {C}[x,y]\) is a polynomial of degree 2 on y.
Now that we have described the general situation, we will concentrate on the particular polynomials involved in Theorem 2.
4 The connection between the regulator and the Lfunction for our polynomials
In this section we prove a relationship between regulators for \(R_{2}(x,y)\) and \(S_{2,1}(X,Y)\). The goal of this step is to relate the differential forms in the integral (8). This will allow us to use formula (2) in order to evaluate those terms. A substantial part of what we present in this section was done by Touafek [17, 18]. We include it here for the sake of completeness.
To make the notation easier to follow, we write the variables of \(S_{2,1}\) with capital letters.
The torsion group of \(E_\alpha \) has order 6 and is generated by \(P=\left( \alpha +1,\frac{(\alpha 2)(\alpha +1)}{2}\right) \), with \(2P=\left( 1,\frac{\alpha 2}{2}\right) , 3P=(0,0), 4P=\left( 1,\frac{\alpha +2}{2}\right) , 5P=\left( \alpha +1,\frac{(\alpha +2)(\alpha +1)}{2}\right) \).
Proposition 9
Proof
5 The integration path
The goal of this section is to determine conditions for the integration paths in integrals (8) and (9) corresponding to \(S_{2,1}(X,Y)\) and \(R_{2}(x,y)\) to be closed paths. This will allow us to determine their homology class later.
Lemma 10

\(X=a, Y_\ge a\) is a closed path for any \(t \in \mathbb {R}\).

\(X=a, Y_+\ge a\) is a closed path for any \(t\ge 3\).

\(X=a, Y_+\le a\) is a closed path for any \(t\le 1\).
Proof
Lemma 11
Proof
From now on we will assume (20) and we will prove that \({y_1}_\ge 1\) under this condition for any \(0\le \theta <\pi \).
Finally, we can extend condition (20) to the extremes by continuity. \(\square \)
6 The cycle of the integration path
Now that we have given conditions for the integration path in (8) to be a closed path, we need to understand its class in the homology group \(H_1(E,\mathbb {Z})\). More precisely, our goal is to prove that the homology classes \([X=a]\) and \([\varphi _*(x=a^2)]\) in the elliptic curve defined by \(S_{2,1}(X,Y)=0\) are equal. In order to prove that the classes are equal, one can show that the integral of the invariant holomorphic differential for \(\omega \) over each path are equal. Since both paths are closed and they do not selfintersect, they must correspond to generators in the homology. Therefore, it suffices to prove that these integrals are positive multiples of each other. In view of Remark 6, we will show that both of them have a purely imaginary value.
Lemma 12
Proof
Lemma 13
Proof
Lemmas 12 and 13 imply that \([X=a]=\pm [\varphi _*(x=a^2)]\) and this is also independent of the value of the parameter a as long as a satisfies the conditions that we discovered in Sect. 5. This corresponds to the result we need with the exception of a sign to be determined later.
Remark 14
7 The integral over \(\arg y\)
In this section we compute the integrals (9).
Lemma 15
Proof
Lemma 16
Proof
As usual, assume that \(\sqrt{\frac{1+\sqrt{5}\sqrt{2\sqrt{5}+2}}{2}}<a<\sqrt{\frac{1+\sqrt{5}+\sqrt{2\sqrt{5}+2}}{2}}\). The extreme cases follow by continuity.
It can be proven that the second integral in (24) is zero by the same reasoning that we did in Lemma 15.
We get the final result by combining the possible values. \(\square \)
8 The proof of Theorem 2
We have now all the elements to prove Theorem 2.
First consider the case of \(S_{2,1}(X,Y)\). By Lemma 10 from Sect. 5, there are two cases where the integration paths are closed. Either \(t\le 1\), which implies \(aa^{1}\le 1\) and \(a \in \left[ \frac{\sqrt{5}1}{2}, \frac{1+\sqrt{5}}{2}\right] \), or \(t\ge 3\), which implies \(aa^{1}\ge 3\) and \(a\ge \frac{3+\sqrt{13}}{2}\) or \(aa^{1}\le 3\) and \(0\le a\le \frac{3+\sqrt{13}}{2}\).
9 Conclusion
There are several directions for further exploration. The most immediate question that we have is the completion of the statement of Theorem 2, in the sense that we would like to give formulas for \(\mathrm {m}_{a,b}(S_{2,1})\) and \(\mathrm {m}_{a,b}(R_{2})\) for any positive parameters a and b. This is a challenging problem, as it requires to integrate \(\eta (x,y)\) in a path that is not closed and cannot be easily identified as a cycle in the homology group.
A different direction would be to consider other polynomials from Boyd’s families.
Finally, it would be also natural to explore this new definition of Mahler measure over arbitrary tori for arbitrary polynomials in a more general context and to relate it to other constructions, such as the Ronkin function associated to amoebas (see [9] for further details).
10 Appendix: A more precise computation of the homology cycles
Proposition 17
Declarations
Authors' contributions
ML, TM have participated in the whole study and drafted the manuscript, and both authors read and approved the final manuscript.
Acknowlegements
We are thankful to MarieJosé Bertin for providing us a copy of Touafek’s doctoral thesis [18]. We are very grateful to the anonymous referees for their dedicated work and for their several corrections and suggestions that greatly improved the exposition. We would like to specially thank the referee who found a mistake in one of our main formulas. This research was supported by the Natural Sciences and Engineering Research Council of Canada [Discovery Grant 3554122013 to ML] and Mitacs [Globalink Research Internship to TM].
Competing interests
The authors declare that they have no competing interests.
Authors’ Affiliations
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