AN ALGEBRAIC INTEGRATION FOR MAHLER MEASURE

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1 AN ALGEBRAIC INTEGRATION FOR MAHLER MEASURE MATILDE N. LALÍN Abstract There are many eamples of several variable polynomials whose Mahler measure is epressed in terms of special values of polylogarithms. These eamples are epected to be related to computations of regulators, as observed by Deninger [D] and, later, by Rodriguez-Villegas [R] and Maillot [M]. While Rodriguez-Villegas made this relationship eplicit for the two-variable case, it is our goal to understand the three-variable case and shed some light on the eamples with more variables. 1. Introduction The logarithmic Mahler measure of a Laurent polynomial P C[ ±1 1,...,±1 n ] is defined by mp := log P e 2πiθ 1,...,e 2πiθ n dθ 1 dθ n. 1 Because of Jensen s formula, there is a simple epression for the Mahler measure in the one-variable case as a function on the roots of the polynomial. It is natural, then, to wonder what happens with several variables. The problem of finding eplicit closed formulas for Mahler measures of severalvariable polynomials is hard. However, several eamples have been found, especially for cases of two and three variables. Some formulas have been completely proved, and some others have been established numerically and are strongly believed to be true. A remarkable fact is that in most of these eamples, the Mahler measure of polynomials with integral coefficients can be epressed in terms of special values of L-series or polylogarithms i.e., Riemann zeta functions, Dirichlet L-series, L-series of varieties, zeta functions of number fields, etc.. DUKE MATHEMATICAL JOURNAL Vol. 138, No. 3, c 2007 Received 25 November Revision received 27 September Mathematics Subject Classification. Primary 11G55; Secondary 19F99. Author s work partially supported by the University of Teas at Austin Harrington Fellowship and by the Institut des Hautes Études Scientifiques France. 391

2 392 MATILDE N. LALÍN For instance, the first and simplest eample in two variables was discovered by Smyth [S, Eample 5]: m1 + + y = 3 3 4π Lχ 3, 2 = L χ 3, 1, 2 where χ 3 is the character of conductor 3. Another eample was computed numerically by Boyd [Bo] and studied by Deninger [D] and Rodriguez-Villegas [R]: m y + 1 y + 1? = L E,0, 3 where E is the elliptic curve of conductor 15 which is the projective closure of the curve + 1/ + y + 1/y + 1 = 0 and LE,s is the L-function of E. Deninger [D] interpreted computations of Mahler measure in terms of Deligne periods of mied motives, eplaining some of the relations to the L-series via Beilinson s conjectures. Rodriguez-Villegas [R] has clarified this relationship by eplicitly computing the regulator and relating this machinery to the cases already numerically known by Boyd, proving some of them and deeply understanding the cases with two variables. Recently, Maillot [M] has sketched how one could continue these ideas for more variables. It is our goal to develop these ideas and to apply them in order to understand the few known eamples with three and more variables involving Dirichlet L-series, Riemann zeta functions, and polylogarithms. In this article, we describe a general situation and illustrate our eplanation with a few eamples. In [L2], we demonstrate the computational power of our method by showing how to prove many other formulas of Mahler measures and generalized Mahler measures as well. 2. Background In this section, we describe some ingredients that are used in our construction Polylogarithms The cases that we are going to study involve zeta functions or Dirichlet L-series, but they all may be thought of as special values of polylogarithms. In fact, this common feature seems to be the most appropriate way of dealing with the interpretation of these formulas. Here, we proceed to recall some definitions and establish some common notation. Definition 1 The nth polylogarithm is the function defined by the power series Li n := k=1 k, C, < 1. 4 kn

3 AN ALGEBRAIC INTEGRATION FOR MAHLER MEASURE 393 This function can be continued analytically to C\1,. We work with Zagier s modification of the polylogarithm see [Z3], n 1 L n := Re n j=0 2 j B j log j Li n j, 5 j! where B j is the jth Bernoulli number and Re k denotes Re or Im, depending on whether n is odd or even. This function is one-valued, continuous in P 1 C, and real analytic in P 1 C\0, 1, }. L n satisfies very clean functional equations. The simplest ones are 1 L n = 1 n 1 L n, L n = 1 n 1 L n. For n = 2, one obtains the Bloch-Wigner dilogarithm, D = Im Li 2 log Li 1 = Im Li 2 + log arg1, 6 which satisfies the well-known five-term relation 1 y 1 D + D1 y + Dy + D + D = y 1 y For n = 3, we obtain L 3 = Re Li 3 log Li log2 Li 1. 8 This modified trilogarithm satisfies more functional equations, such as the Spence- Kummer relation: 1 y 2 1 y L 3 + L y1 2 3 y + L 3 2L 3 y y1 1 y 1 y 2L 3 2L y1 2L 3 y 1 2L 3 2L 3 y + 2L 3 1 = Polylogarithmic motivic complees Given a field F, consider Z[P 1 F ], the free abelian group generated by the elements of P 1 F. For each n, we are interested in working with this group modulo the rational functional equations of the nth polylogarithm. Unfortunately, the functional equations of higher polylogarithms are not known eplicitly. For X an algebraic variety, Goncharov [G1], [G2], [G3] has constructed some groups that conjecturally correspond to the groups in the previous paragraph, and they fit into polylogarithmic motivic complees whose cohomology is related to Bloch

4 394 MATILDE N. LALÍN groups and is conjectured to be the motivic cohomology of X. A regulator can be defined in these complees and is conjectured to coincide with Beilinson s regulator. From now on, we follow [G1], [G2], and [G3]. We state definitions and results; the proofs may be found in the mentioned works. Given a field F, one defines inductively some subgroups R n F and then lets B n F := Z[P 1 F ]/R nf. 10 The classes of in Z[P 1 F ] and in B nf are denoted by } and } n, respectively. We begin by setting R 1 F := }+y} y};,y F, 0}, }. 11 Thus, B 1 F = F. Now, we proceed to construct a family of morphisms: Z[P 1 F ] δ n Bn 1 F F if n 3, 2 F if n = 2, } n 1 if n 3, δ n } = 1 if n = 2, 0 if } =0}, 1}, }. 12 Then one defines A n F := ker δ n. 13 Note that any element αt = n i f i t} Z[P 1 F t ] has a specialization αt 0 = ni f i t 0 } Z[P 1 F ] for every t 0 P 1 F. Thus, R n F := α0 α1; αt A n F t. 14 Goncharov proves that R n C is the subgroup of all the rational functional equations for the n-polylogarithm in C. As stated at the beginning of Section 2.2, the philosophy is that R n F should be the subgroup of all the rational functional equations for the n-polylogarithm in F. Because of δ n R n F = 0, it induces morphisms in the quotients δ n : B n F B n 1 F F, n 3, δ 2 : B 2 F 2 F.

5 AN ALGEBRAIC INTEGRATION FOR MAHLER MEASURE 395 One obtains the comple B F n :B n F δ B n 1 F F δ B n 2 F 2 F δ δ n 2 B 2 F F δ n F, where n p n p δ : } p y i δ p } p y i. i=1 The following conjecture relates the cohomology of the comple B F n to motivic cohomology. CONJECTURE 2 [G2, 3] We have H i B F n Q = gr γ n K 2n if Q. 15 Evidence supporting this conjecture is found, for instance, in the cases n = 1, 2.First, it is clear that H 1 B F 1 = F = K 1 F. For n = 2, it is known that i=1 B 2 F = Z[P 1 F ]/ R 2,y;,y F, 0}, }, where 1 } 1 y } R 2,y:=}+y}+1 y}+ + 1 y 1 y is the five-term relation of the dilogarithm. Besides, H 1 B F 2 Q = K ind 3 F Q, 16 H 2 B F 2 = K2 F, 17 H n B F n = K M n F, 18 The first assertion was proved by Suslin, the second one is Matsumoto s theorem, and the last one corresponds to the definition of Milnor s K-theory.

6 396 MATILDE N. LALÍN 2.3. Regulators Deninger [D] observed that the Mahler measure can be seen as a regulator evaluated in a cycle that may or may not have trivial boundary. More precisely, mp = mp 1 + η 2π i n 1 n n 1,..., n. 19 G We have to eplain the ingredients in this formula. In this section, we are concerned with η n n 1,..., n. This form is described in the contet of Goncharov s construction of the regulator on the polylogarithmic motivic complees. Let us establish some notation: Ln z if n>1 is odd, L n z := i L n z if n is even. For any integers p 1 and k 0,define [p 1/2] β k,p := 1 p p 1! k + p k+p 2j B k+p 2j, k + p + 1! 2j + 1 where the B i are Bernoulli numbers. Definition 3 We consider the following 1-forms: L p,q := L p log q 1 d log, p 2, Recall that j=0 L 1,q := log d log 1 log 1 d log log q 1. Alt m F t 1,...,t m := σ S m 1 σ F σ 1,..., σ m. Now, we are ready to describe the differential forms. Definition 4 Let, i be rational functions on a comple variety X: η n+m m + 1 : } n 1 m L n Alt m + 1 k, 1 p m p 0 1 2p + 1!m 2p! β k,p L n k,k log 1 Alt m p 1!m p! 2p j=2 2p j=1 d log j d log j m j=p+1 m j=2p+1 diarg j diarg j, 20

7 AN ALGEBRAIC INTEGRATION FOR MAHLER MEASURE 397 η m m : 1 m Alt m p 0 2p+1 log 1 2p + 1!m 2p 1! j=2 d log j m j=2p+2 diarg j. 21 These differential forms typically have singularities. In order to work with them, we need to have control of the residues. Let F be a field with discrete valuation v, residue field F v, and group of units U. Letu ū be the projection U Fv,andletπ be a uniformizer for v. There is a homomorphism θ : n F n 1 F v defined by θπ u 1 u n 1 = ū 1 ū n 1, θu 1 u n = 0. Now, define s v : Z[P 1 F ] Z[P1 F v ] by s v } = }. It induces s v : B m F B m F v.then n m n m 1 v := s v θ : B m F F B m F v F v 22 defines a morphism of complees Observation 5 The induced morphism v : B F n B Fv n 1[ 1]. 23 v : H n B F n H n 1 B Fv n 1 coincides with the tame symbol defined by Milnor: v : K M n F KM n 1 F v. Let X be a comple variety. Let X 1 denote the set of the codimension-one, closed, irreducible subvarieties. Let A j Xk denote the space of smooth j-forms with values in 2πi k R. Let d be the de Rham differential on A j X, andletd be the de Rham differential on distributions, so dd arg = 0, Dd arg = 2πδ.

8 398 MATILDE N. LALÍN The difference D d is the de Rham residue homomorphism. Goncharov [G2] proves the following. THEOREM 6 We have that η n m induces a homomorphism of complees such that: η n 1} n = L n ; dη n n 1 n = β Re d1 n 1 d n n,whereβ = 1 if n is odd and i if n is even; η n m defines a distribution on XC; the morphism η n m is compatible with residues D η n m η n m + 1 δ = 2π i η n 1 m 1 vy, Y X 1 m < n, 24 D η n n β Re n d1 1 d n n = 2π i where v Y is the valuation defined by the divisor Y. Y X 1 η n 1 n 1 vy, 25 The relation of η n to the regulator is roughly as follows. As we mentioned in Conjecture 2, the cohomology of the first comple corresponds to the Adams filtration, which is the absolute cohomology. On the other hand, a slight modification of the second comple leads to Deligne cohomology. Now, η n, as seen as a map between the cohomologies of these two complees, is conjectured to have the same image as the regulator see [G3]. We also note that the final goal is not to work with CX but with X itself. For X a regular projective variety over a field F, Goncharov [G3] describes the difficulties for defining the comple B X n as opposed to B F X n. Basically, it is known how to define B X n when X = SpecF for an arbitrary field F, X is a regular curve over an arbitrary field F,orX is an arbitrary regular scheme but n 3. Now, in terms of the relation between η n, the regulator, and Beilinson s conjectures, the picture is much less known. As an illustration, the case X = SpecF for F a number field corresponds to Zagier s conjecture see [Z2, Conjecture 1], [ZG].

9 AN ALGEBRAIC INTEGRATION FOR MAHLER MEASURE The two-variable case Rodriguez-Villegas [R] has performed the eplicit construction of the regulator and applied the ideas of Deninger [D] to eplain many eamples in two variables. This work was later continued by Boyd and Rodriguez-Villegas [BR1], [BR2]. Let P C[,y]. Then we may write By Jensen s formula, mp = ma d + 1 2π i Here, P,y = a d y d + +a 0, d P,y = a d y αn. d n=1 n=1 T 1 log + α n d = mp 1 2π η,y:= i η 2 2 y = log dargy log y darg γ η,y. 26 is the regulator in this case, defined in the set C =P,y = 0} minus the set Z of zeros and poles of and y. Also, P = a d,andγ is the union of paths in C where =1 and y 1. Finally, note that γ =P,y = 0} = y =1}. In general, η,y is closed in C\Z since dη,y = Im d dy y see Theorem 6. Now, the computation can be performed if we arrive at one of these two situations: 1 η is eact, and γ 0: in this case, we can integrate using the Stokes theorem; 2 η is not eact, and γ = 0: in this case, we can compute the integral by using the residue theorem. Eamples for the second case are found, for instance, in the family of Laurent polynomials + 1/ + y + 1/y + k studied by Boyd [Bo], Deninger [D], and Rodriguez- Villegas [R]. Technically, one needs k [ 4, 4] for these eamples to be in the first case; otherwise, γ 0. However, Deninger has given an interpretation that allows an adaptation of the cases of k 4, 4\0} into this frame as well. Following Theorem 6, Thus, η is eact when η,1 = dd. 27 y = i r i i 1 i 28

10 400 MATILDE N. LALÍN in 2 CC Q. This condition may be rephrased as the symbol,y} is trivial in K 2 CC. In fact, if condition 28 is satisfied, we obtain η,y = r i dd i = dd r i i } i i Finally, we write γ = ɛ k [w k ], ɛ k =±1, k where w k CC, w k = yw k =1. Thus, we have the following. THEOREM 7 see [BR1], [BR2] Let P C[,y] be irreducible and such that,y satisfies equation 28. Then 2π log a d mp = Dξ for ξ = ɛ k r i i w k }. 2 k i Boyd and Rodriguez-Villegas prove even more. Under certain assumptions, it is possible to apply Zagier s theorem, [Z4, Theorem 1], and relate the Mahler measure of P to a rational combination of terms of the form 1/2 ζ F 2/π 2[F :Q] 2 for certain number fields F that depend on P or, more specifically, on w k An eample for the two-variable case To be concrete, we are going to eamine the simplest eample for the eact case in two variables. Consider Smyth s formula see [S, Eample 5]: For this case, πm + y 1 = Lχ 3, 2. y = 1. Then 2πmP = η,y = η,1 = D γ. γ γ Here, γ =,y =1, 1 1 } = e 2πiθ, 1 e 2πiθ θ Figure 1 shows the integration path γ. [ 1 6 ; 5 6]}.

11 AN ALGEBRAIC INTEGRATION FOR MAHLER MEASURE 401 ξ 6 y γ = 1 y = 1 Figure 1. Integration path for + y 1 Then γ = [ ξ 6 ] [ξ 6 ] where ξ 6 = 1 + 3i/2, and we obtain 2πm + y 1 = Dξ 6 D ξ 6 = 2Dξ 6 = Lχ 3, The three-variable case Our goal is to etend this situation to three variables. Let P C[,y,z].Wetake η,y,z:= η 3 3 y z 1 = log 3 d log y d log z dargy dargz 1 + log y 3 d log z d log dargz darg 1 + log z 3 d log d log y darg dargy. 30 This differential form is defined in the surface S =P,y,z = 0} minus the set Z of poles and zeros of, y,andz. We can epress the Mahler measure of P as mp = mp 1 η,y,z, 31 2π 2 Ɣ where P, following the previous notation, is the principal coefficient of the polynomial P C[,y][z] and Ɣ = P,y,z = 0 } = y =1, z 1 }. Recall that η is closed in S\Z since it verifies dη,y,z = Re d dy y dz z. Typically, one epects that integral 31 can be computed if we are in one of the two ideal situations that we described before. Either the form η,y,z is not eact, the

12 402 MATILDE N. LALÍN set Ɣ consists of closed subsets, and the integral is computed by residues or the form η,y,z is eact and the set Ɣ has nontrivial boundaries, so the Stokes theorem is used. The first case leads to instances of Beilinson s conjectures and produces special values of L-functions of surfaces. Eamples in this direction can be found in Bertin s work [B]. Bertin relates the Mahler measure of some K3 surfaces to the Eisenstein- Kronecker series in a similar way as Rodriguez-Villegas does for two-variable cases see [R]. In the second case, we need that η,y,z is eact. We are going to concentrate on this case. We are integrating on a subset of the surface S. In order for the element in the cohomology to be defined everywhere in the surface S, we need the residues to be zero. This situation is fulfilled when the tame symbols are zero see Section 2.2. This condition is a problem for us because when η is eact, the tame symbols are zero. As in the two-variable case, Theorem 6 implies where ω,y:= η 3 2} 2 y η,1,y = dω,y, 32 = D dargy + 1 log y log 1 d log log d log Thus, in order to apply the Stokes theorem, we need to require that y z = r i i 1 i y i 34 in 3 CS Q for η to be eact. An equivalent way of epressing this condition is that,y,z} is trivial in K3 MCS. In this case, η,y,z = r i η i, 1 i,y i = r i ω i,y i, where Ɣ Ɣ Ɣ = P,y,z = 0 } = y = z =1 }. The set Ɣ seems to have no boundary. However, Ɣ as described above may contain singularities that may give rise to a boundary when desingularized. We thus change Ɣ

13 AN ALGEBRAIC INTEGRATION FOR MAHLER MEASURE 403 our point of view. Namely, assume that P R[,y,z] and is nonreciprocal this condition is true for all the eamples that we study; then, P,y,z = P,ȳ, z. This property together with the condition = y = z =1 allows us to write Ɣ = P,y,z = P 1,y 1,z 1 = 0 } = y = z =1 }. This idea was proposed by Maillot [M]. Observe that we are integrating now on a path = y = z =1} inside the curve In order to easily compute C = Res z P,y,z,P 1,y 1,z 1 = 0 }. Ɣ ω,y, we have again the two possibilities that we had before. We are going to concentrate, as usual, on the case where ω,y is eact. The differential form ω is defined in the new curve C. As before, to ensure that it is defined everywhere, we need to ask that the residues be trivial. This fact is guaranteed by the triviality of tame symbols. This last condition is satisfied if ω is eact. Indeed, we have changed our ambient variety, and we now wonder when ω is eact in C ω is not eact in S since that would imply that η is zero. Fortunately, we have ω, = dl 3 35 by Theorem 6. The condition for ω to be eact is not as easily established as in the preceding cases because ω is not multiplicative in the first variable. In fact, the first variable behaves as the dilogarithm; in other words, the transformations are ruled by the five-term relation. We may epress the condition we need as } 2 y = r i i } 2 i 36 in B 2 CC CC. Assuming Conjecture 2, this is equivalent to saying that a Q certain symbol for and y is trivial in gr γ 3 K 4CC Q.Thenwehave ω,y = r i L 3 i γ, where γ = C T 2. γ

14 404 MATILDE N. LALÍN Now, assume that γ = k ɛ k [w k ], ɛ k =±1, where w k CC, w k = yw k =1. Thus, we have proved the following. THEOREM 8 Let P,y,z R[,y,z] be irreducible and nonreciprocal, let S =P,y,z = 0}, and let C =Res z P,y,z,P 1,y 1,z 1 = 0}. Assume that y z = i r i i 1 i y i 37 in 3 CS Q, and assume that i } 2 y i = j r i,j i,j } 2 i,j 38 in B 2 CC CC for all i.then Q 4π 2 mp mp = L 3 ξ for ξ = k ɛ k r i r i,j i,j w k } i,j By using Zagier s conjecture see Zagier [Z2], Zagier and Gangl [ZG], it is possible to formulate a conjecture that would imply, under certain additional circumstances, a relationship with ζ F 3 in a similar fashion as Boyd and Rodriguez-Villegas have done for the two-variable case. We illustrate this phenomenon at the end of Section The case of Res 0,m,m+n} We proceed to the study of a family of three-variable polynomials which comes from the world of resultants, namely, Res 0,m,m+n}. This family was computed in [DL], and the computation is quite involved, though elementary. The Mahler measure of Res 0,m,m+n} is the same as the Mahler measure of a certain rational function. More precisely, we have the following. THEOREM 9 [DL, Theorem 6] We have m z 1 m 1 y n = 2n L3 φ m 1 y m+n π 2 2 L 3 φ m 1 + 2m π 2 L3 φ n 1 L 3 φ n 2, 40

15 AN ALGEBRAIC INTEGRATION FOR MAHLER MEASURE 405 where φ 1 is the root of m+n + n 1 = 0 which lies in the interval [0, 1] and φ 2 is the root of m+n n 1 = 0 which lies in [1,. Proof To see that η,y,z is eact, we need to solve equation 37 for this case. The equation for the wedge product becomes y z = m y 1 + n y 1 y m + n y 1 y = m 1 y + ny 1 y + my 1 y y ny 1 y. After performing the Stokes theorem for the first time, we evaluate the form ω in the following element of B 2 CC CC : = my} 2 y } 2 y ny} 2 y} 2. We then compute the corresponding curve C. We take advantage of the fact that our equation has the shape z = R,y. In order to compute C, we simply consider For this case, Let us denote R,yR 1,y 1 = z z 1 = m 1 y n 1 1 m 1 y 1 n 1 y m+n 1 1 y 1 m+n = 1. 1 = 1 1 y, y 1 = 1 y 1 y, 1 = 1 1, ŷ 1 = 1 y 1. Then we may rewrite the equation for C as Now, we use the five-term relation Then we obtain m 1 yn 1 n 1 ŷ m 1 = 1. } 2 +y} 2 +1 y} } 2 +y 1 } 2 = 0. = my} 2 y + 1 } 2 y +y 1 } 2 y n} } 2 +y 1 } 2. Observe that = 1 /y 1, y = ŷ 1 / 1.

16 406 MATILDE N. LALÍN Thus, we may write = my} 2 y + 1 } 2 ŷ 1 1 } 2 1 +y 1 } 2 ŷ 1 y 1 } 2 1 n} } } 2 y 1 +y 1 } 2 1 y 1 } 2 y 1 = my} 2 y + 1 } 2 ŷ m 1 m 1} 2 1 mŷ 1 } 2 ŷ 1 y 1 } 2 m 1 n} 2 + n 1 } } 2 y n 1 y 1} 2 n 1 + ny 1} 2 y 1. Because of the equation for C, we obtain 1 } 2 y n 1 ŷ m 1 y 1} 2 m 1 n 1 = 1 } 2 m 1 n 1 +y 1} 2 y n 1 ŷ m 1 = m 1 } n 1 } ny 1 } 2 y 1 mŷ 1 } 2 ŷ 1, = my} 2 y ŷ 1 } 2 ŷ 1 1 } 2 1 ŷ 1 } 2 ŷ 1 1 } 2 1 n} 2 1 } 2 1 y 1 } 2 y 1 1 } 2 1 y 1 } 2 y 1 = my} 2 y 2ŷ 1 } 2 ŷ } 2 1 n} } 2 1 2y 1 } 2 y 1. We now need to study the path of integration. First, write = e 2iα, y = e 2iβ for π/2 α, β π/2. Then 1 = e iβ sin α sinα + β, y 1 = e iα sin β sinα + β and 1 = e iα sin β sinα + β, ŷ 1 = e iβ sin α sinα + β. Let a = sin α/sinα + β, b = sin β/sinα + β. Then we may write 1 =±ae iβ, y 1 =±be iα, 1 =±be iα, ŷ 1 =±ae iβ. By means of the sine theorem, we may think of a, b, and 1 as the sides of a triangle with the additional condition a m b n = 1 see Figure 2. The triangle determines the angles α and β, which are opposite to the sides a, b, respectively. We need to be careful and take the complement of an angle if it happens to be greater than π/2. This corresponds to the cases when the sines are

17 AN ALGEBRAIC INTEGRATION FOR MAHLER MEASURE 407 a b Figure 2. We are integrating over all the possible triangles. The angles are measured negatively if they are greater than π/2, as α is in the case 2. We do not count the triangles pointing down, as in 3. negatives. However, we need to be cautious. In fact, the problem of constructing the triangle given the sides always has two symmetric solutions. We count each triangle once, so we multiply our final result by two. To sum up, a and b are enough to describe the set where the integration is performed. Now, the boundaries where the triangle degenerates are three: b + 1 = a, a + 1 = b,anda + b = 1.Let and let φ 1 be the root of m+n + n 1 = 0 with 0 φ 1 1, φ 2 be the root of m+n n 1 = 0 with 1 φ 2. Then the first two conditions are translated as a = φ n 1, b = φm 1, α = 0, β = 0, a = φ n 2, b = φm 2, α = 0, β = 0. The third condition is inconsequential since it requires both a,b 1 but they cannot be both equal to one at the same time and a m b n = 1. Hence, the integration path from condition a + 1 = b to b + 1 = a is 0 α θ 1, 0 β π 2, θ 1 α π 2, π 2 β θ 2, π 2 α 0, θ 2 β 0.

18 408 MATILDE N. LALÍN Here, θ 1 is the angle that is opposite to the side a when the triangle is right-angled with hypotenuse b and θ 2 is opposite to b when a is the hypotenuse. We do not need to compute those angles. In fact, we may describe the integration path as either or 0 α π 2, π 2 α 0 0 β π 2, π 2 β 0. It is appropriate to think of it in this way because 1 } 3 +ŷ 1 } 3 and 1 } 3 +y 1 } 3 change continuously around the right-angled triangles. Moreover, because of this property, everything reduces to evaluating L 3 in = my} 3 2ŷ 1 } } 3 n} } 3 2y 1 } 3 in the cases of b + 1 = a and a + 1 = b and computing the difference. One can have problems when z is zero or has a pole. We have that z is zero for = 1 and y = 1, but these conditions correspond to = my} 2 y and = n} 2. They lead to = my} 3 and = n} 3, and they integrate to zero when the variables move in the unit circle. The poles are at y = 1, which corresponds to = m n} 2. Integrating, we obtain = m n} 3, which leads to zero when moves in the unit circle. We obtain 4π 2 mp = 2 4nL 3 φ m 2 L 3 φ m 1 + 4mL 3φ n 1 L 3 φ n 2. Finally, mp = 2n π 2 L3 φ m 2 L 3 φ m 1 + 2m π 2 L3 φ n 1 L 3 φ n 2, so we recover the result of [DL]. The case with m = n = 1 is especially elegant. Here, the rational function has the form and z = 1 1 y 1 y 2, mp = 4 π 2 L3 φ L 3 φ, where φ 2 + φ 1 = 0 and 0 φ 1. In other words, φ = 1 + 5/2.

19 AN ALGEBRAIC INTEGRATION FOR MAHLER MEASURE 409 Moreover, we may use Zagier s conjecture see [Z2, Conjecture 1], [ZG] to describe this result in terms of the zeta function of Q 5. According to the conjecture, H 1 B Q 5 3 has rank 2. We may take 1} 3, φ} 3 } as a basis. In order to see this, we need to check that φ} 2 φ is trivial. That is the case because φ} 2 =1 φ 2 } 2 = φ 2 } 2 = 2 φ} 2 2φ} 2 = 21 + φ} 2 2φ} 2 = 2φ 1 } 2 2φ} 2 = 0, which implies that φ} 2 φ = 0. Then the conjecture predicts ζ L Q 5 3 Q 5 3 φ L 3 1 L 3 φ 1 L 3 1 = 5ζ 3 L 3 φ L 3 φ. Indeed, ζ Q 5 3 = ζ 3 5 L3 φ L 3 φ, which allows us to write mres 0,1,2} = 4 5ζ Q 5 3. π 2 ζ 3 5. A few words about the four-variable case Unfortunately, we do not have a general systematic method to describe algebraically the successive integration domains in more than three variables. Hence, we cannot formulate a precise general result. However, this does not prevent us from using a similar technique for some four-variable cases. In this section, we recall the list of differentials in four variables. The sequence of differentials is as follows: η,y,w,z:= iη 4 4,y,w,z = 1 d log z Im 4 d + log w Im d log y Im dy + log Im dy y dy y dw w dw w dz z y dw w dz z η,y,z dargw dz z + η,y,w dargz + η,w,z dargy ηy,w,z darg, 42 where η,y,z denotes the differential previously defined for three variables.

20 410 MATILDE N. LALÍN where We have η,1,y,w = dω,y,w, 43 ω,y,w:= iη 4 3,y,w 1 = D 3 d log y d log w dargy dargw with Net, + 1 ηy,w log d log 1 log 1 d log ω,,y = dµ,y 45 µ,y:= iη 4 2,y = L 3 dargy 1 D log y d log Finally, µ, = dl An eample in four variables In spite of the fact that we do not know how to treat the integration domains, we may still do the algebraic integration for some eamples of four-variable polynomials. Here is an eample. We study the case of Res 0,0,1,0,0,1}, whose Mahler measure was first computed in [DL]. This is the case of the nine-variable polynomial that is the general 3 3- determinant. Because of homogeneities, this Mahler measure problem may be reduced to computing the Mahler measure of a four-variable polynomial. The result is the following. THEOREM 10 [DL, Theorem 7] We have m 1 1 y 1 w1 z = 9 ζ π 2 Proof First, we have to solve the equation with the wedge product: y w z = 1 y w z = 1 y 1 1 w z w z.

21 AN ALGEBRAIC INTEGRATION FOR MAHLER MEASURE 411 Now, the first term on the right-hand side is 1 y 1 1 w z = w y y w z = 1 y + y y y w z 1 y + y y y w z. w Net, we use the formula for z as a function of the other variables: 1 y + y y y w z w = + y y w = + y y w + y y w y y w + + y y w w1 w y y w y y w 1 w. 1 + y y w Note that + y y y y w 1 w = 1 y + y y y w 1 w y y w 1 w. Hence, y w z = w z + y y 1 y + y w z1 w + + y y 1 + y y w w y y w w 1 w The form ω is evaluated in the following element: 1 } = w z + y y } w z1 w 2 2 } + y y + w = } 2 w z + 2 y y } y y 2 y y. w w} 2 y y w z1 w z z } y y w w} 2 w 2 y y.

22 412 MATILDE N. LALÍN Figure 3. Integration set for Res 0,0,1,0,0,1} For applying the Stokes theorem, we still apply the technique that is analogous to the computation of the equation for C in the three-variable case. We can apply the analogue of the equation 41, 1 which can be simplified as 1 1 y y 1 = 1, 1 w 1 w 1 = 1, y = 1, w =, or w = y. The above conditions correspond to two pyramids in the torus T 3, as seen in Figure 3. We make the computation over the lower pyramid and then multiply the result by 2. In the case when = 1, wehavew = 1 or z = 1.Ifw = 1, = 0.

23 AN ALGEBRAIC INTEGRATION FOR MAHLER MEASURE 413 If z = 1, = 1 1 } y w. w 2 Then µ is evaluated on =w} 3 y, and is integrated on the boundary, which is y = 1, w = 1,andy = w. If y = 1, = 0.Ifw = 1, which yields 2πζ3. If y = w, =1} 3 y, =y} 3 y, whose integral is zero. In the case when y = 1, wehavew = 1 or z = 1. Ifw = 1, = 0. If z = 1, = 1 1 } w 1 w 1 1 } 1 1 w w} w 2 Only the term in the middle yields a nonzero differential form. In fact, the term in the middle yields =w} 3 1 1, and is integrated on the boundary, which is = 1, w = 1,and = w. If = 1, = 0.Ifw = 1, This integration is equal to πζ3. If = w, which integrates to zero. =1} =} 3 1 1,

24 414 MATILDE N. LALÍN When w = in this case, z = y unless = 1, = } 2 y + y y } y1 2 y y } y y } 2 y y. 2 Then = 2} 3 y 2 y y } } Now is integrated on the boundary, which is = 1, y = 1, and = y see Figure 3. If = 1, = 21} 3 y, which gives 4πζ3. If y = 1, = } } Now, use the fact that } 3 +1 } } =1} 3 3 and the fact that =1 to conclude } =} 3 1} 3. 3 The total integration in this case is 2πζ3. If = y, = 2} 3 2 1} 3 } 3 1 1, whichleadsto 2 µ 1,. We do not need to compute this integral for the final result. When w = y in this case, z = unless y = 1, = } 2 y + y y } 2 y y } y y y y} 2 y 2 =} 2 y +y} 2 y y} 2 y y 1 1 y y } y } y y y. 2 y y 2

25 AN ALGEBRAIC INTEGRATION FOR MAHLER MEASURE 415 By the five-term relation, 1 1 } +y} y 1 1 } } 2 +y} 2 Then we obtain y y } + y y} 2 2 =} 2 y +y} 2 y y} 2 1 } + y y 2 y + y} 2 y + + y y} 2 y y y } 2 1 y 1 y + y/ = 0. } 2 = 0, 1 1 } 2 y y + } y } 2 y y y y} 2 y y y y 2 y + + y y} 2 y y y + y y } y y y 2 =} 2 y +y} 2 y y} } y y y} y y + y + y y Now, } } 2 2 y y y y. is zero in the differential form. Therefore, y + + y y} y y y} } 2 1 y y =} 3 y +y} 3 +1 } 3 y +y} y y } y + y y } y, 3 3 and is integrated on the boundary, which is = 1, y = 1,and = y. If = 1, whose integral is 3πζ3. =1} 3 y +y} 3 1 y } y, y 3 } y y 3

26 416 MATILDE N. LALÍN If y = 1, which gives 3πζ3. If = y, =1} 3 +1} 3 1, = 2} 3 +1 } 3 +2} } 3 1 2} 3 = 2} 3 31 } 3 + 2} } 3, which yields 3πζ3 + 2 µ 1,. The poles are with w = 1, but = 0 in this case. On the other hand, if z = 0, then w = + y y.but w =1 implies that = 1, y = 1,or = y. In the first two cases, w = 1 and = 0. In the third case, = } } 2 1, which corresponds to zero if =1. Thus, Finally, 8π 3 mp = 36πζ3. mp = 9 2π 2 ζ The n-variable case The usual application of Jensen s formula as in equation 26 allows us to write, for P C[ 1,..., n ], mp = mp 1 + η 2π i n 1 n n 1,..., n, 49 G where G = P 1,..., n = 0 } 1 = = n 1 =1, n 1 }. Recall that this is due to Deninger [D]. It is easy to see that we can then follow a process that is analogous to the ones we followed for up to four variables. It remains, of course, to find a general algebraic way of describing the successive sets that we obtain by taking boundaries. Suppose that we do have a good description of the boundaries inside certain algebraic varieties,

27 AN ALGEBRAIC INTEGRATION FOR MAHLER MEASURE 417 say, S 1 =P 1,..., n = 0},...,S n 1. Write, as usual, γ = k ɛ k [w k ], ɛ k =±1, where γ is the collection of paths S n 1 1 =1}. In principle, we should epect the following. CONJECTURE 11 Let P 1,..., n R[ 1,..., n ] be nonreciprocal. Assume that the following conditions are satisfied: 1 n = i 1 r i1 z i1 1 z i1 Y i1 50 in n CS 1 Q; z i1 } 2 Y i1 = i 2 r i1,i 2 z i1,i 2 } 2 z i1,i 2 Y i1,i 2 51 in B 2 CS 2 n 2 CS 2 Q for all i 1. More generally, assume that for k = 4,...,n 2, we have z i1,...,i k 1 } k Y i1,...,i k 1 = i k r i1,...,i k 1,i k z i1,...,i k 1,i k } k z i1,...,i k 1,i k Y i1,...,i k 1,i k 52 in B k CS k n k CS k Q for all i 1,...,i k 1. Finally, assume that z i1,...,i n 2 } n 1 Y i1,...,i n 2 = i n 1 r i1,...,i n 2,i n 1 z i1,...,i n 2,i n 1 } n 1 z i1,...,i n 2,i n 1 53 in B n 1 CS n 1 CS n 1 Q for all i 1,...,i n 2. Then we may write 2π n 1 mp mp = L n ξ 54 for ξ = k i 1,...,i n 1 ɛ k r i1 r i1,...,i n 1 zi1,...,i n 1 w k } n. Here, we have written Y i1, Y i1,i 2,... to denote elements in n 2 CS 1 Q, n 3 CS 2 Q,... A solution to equation 50 determines the Y i1 s. Once the Y i1 s

28 418 MATILDE N. LALÍN are defined, we solve equation 51 and obtain the Y i1,i 2 s. The procedure continues in this fashion until we reach the Y i1,...,i n 2 s. Ideally, we epect that this setting eplains the nature of the n-variable eamples described in [L1] The case of an n-variable family Let us consider the case of the family of n + 1-variable rational functions n z = n whose Mahler measure was computed in [L1]. Though we are not able to perform all the steps for general n, we can at least prove that the first two differentials η n+1 n + 1 and η n+1 n are eact. In this case, the wedge product is 1 n z = = n 1 n 1 i 1 n 1 + i i=1 n 1 in 1 i 1 i i+1 i+n 1 i=1 i 1 + i i+1 i+n 1 with the cyclical convention that i+n = i. Thus, we proved that η = η n+1 n + 1 1,..., n,z is eact. The net step is to integrate η n+1 n evaluated on the following element: = n 1 in 1 i } 2 i+1 i+n 1 i } 2 i+1 i+n 1. i=1 We now prove that ω = η n+1 n is eact. This form is defined in the variety Z, which is the projective closure of the algebraic set determined by 1 n = n 1 + n 2. We now show that ω is trivial in H n 1 DR Z. First, observe that where Z ± is given by the equation Z = Z + Z, 1 ±z 1 1 n = n

29 AN ALGEBRAIC INTEGRATION FOR MAHLER MEASURE 419 and 1 if n is even, z = i if n is odd. In general, consider the variety given by the projective closure of the zeros of n α = n with α a nonzero comple number. This variety is birational to P n 1. This is easy to see by setting y i = 1 i /1 + i. Hence, we may think of each Z ± as a copy of P n 1. The singular points for this birational map are when i =±1. Now, suppose that n is even, and suppose that n = 2k. If we prove that ω can be etended to the whole Z ± and that this etension is consistent with the birationality of Z ±, it implies that ω is closed and that it may be seen as a class in H 2k 1 DR P2k 1 = 0, andthenω is eact. In order to etend ω, we consider the points where some i is equal to 1, 1 the points where the equation has singularities and 0, the points where ω is not defined. Consider the diagram which describes the relation between the tame symbol and the residue morphism. We would like to see that Res v ηn+1 n 1 } 2 2 n = 0, where v is the valuation defined by i =±1, 0, for some i. Instead, we see that η n n 1 v 1 } 2 2 n = 0. First, suppose that i = 1. Then, if i 1, reducing modulo i 1 implies that i = 1, and the only term that is possibly nonzero in v 1 } 2 2 n is v i 1 } 2 2 i n.however, i is clearly not a uniformizer for i 1. Then the tame symbol is zero. If i = 1, 1 } 2 reduces to 1} 2, which corresponds to zero in η n n 1, so we get zero again. The case i = 1 is analogous.

30 420 MATILDE N. LALÍN Now, consider the case with i = 0 for some i. Thenitiseasytosee 1 } 2 2 i n if i 1, v 1 } 2 2 n = 0 if i = Since i = 0, we are now in the variety defined by the projective closure of the zeros of the equation i 1 = i 1 n n We are in a situation that is analogous to the initial one. In other words, we are in the projective space P 2k 2. We proceed by induction. In order to prove that η n 1 n 2 1 } 2 2 i n is trivial, we can prove that the tame symbols w 1 } 2 2 i n are trivial by induction. However, H 2k 2 DR P2k 2 = R, so even if the symbols are trivial, we are not able to conclude that the form η n 1 n 2 1 } 2 2 i n is eact. What we can conclude is that it is either a generator for H 2k 2 DR P2k 2 or trivial. We eliminate the first possibility. Suppose, in order to make notation easier, that n = i. Assume that η n 1 n 2 1 } 2 2 n 1 is a generator for H 2k 2 DR P2k 2. By Poincaré duality, the integral I = η 2k 1 2k 2 1 } 2 2 2k 1 1=1 1 / k 1 /1+ 2k 1 2 must be nonzero. Now, the transformation i 1 i does not change the orientation of the variety but changes the sign of the differential ω. Hence, I = I, and that implies that I = 0. Hence, ω cannot be a generator for H 2k 2 DR P2k 2, and it must be eact. The case when i = is analogous. Now, suppose that n = 2k + 1 is odd. Then we may proceed as before. We have the fact that ω can be seen as a class in HDR 2k P2k = R, and we can conclude that is eact by using the same idea that we used for the even case. To conclude, ω = η n+1 n is eact, and it must be the differential of certain µ = η n+1 n 1. However, we were unable to find the precise formula for µ. The results of [L1] suggest that one should be able to continue this process to reach η n+1 1. Acknowledgments. I epress my deepest gratitude to my Ph.D. supervisor, Fernando Rodriguez-Villegas, for his invaluable guidance and encouragement of this and other projects. I am grateful to Vincent Maillot for many enlightening discussions about cohomology and the n-variable case. I am also thankful to Herbert Gangl for helpful

31 AN ALGEBRAIC INTEGRATION FOR MAHLER MEASURE 421 discussions about polylogarithms and Bloch groups. I wish to epress my appreciation for the careful work of the referees, which has certainly improved the clarity and eposition of this article, and to Amir Jafari for his comments on the final version. This work is part of my Ph.D. dissertation at the Department of Mathematics at the University of Teas at Austin. I am indebted to the department and to John Tate for their generous support during my graduate studies. References [B] M. J. BERTIN, Mesure de Mahler d hypersurfaces K3, preprint, arxiv:math/ v1 [math.nt] [Bo] D. W. BOYD, Mahler s measure and special values of L-functions, Eperiment. Math , MR [BR1] D. W. BOYD and F. RODRIGUEZ-VILLEGAS, Mahler s measure and the dilogarithm, I, Canad. J. Math , [BR2], Mahler s measure and the dilogarithm, II, with an appendi by N. M. Dunfield, preprint, arxiv:math/ v2 [math.nt] [DL] C. D ANDREA and M. N. LALÍN, On the Mahler measure of resultants in small dimensions, J. Pure Appl. Algebra , [D] C. DENINGER, Deligne periods of mied motives, K-theory and the entropy of certain Z n -actions, J. Amer. Math. Soc , MR [G1] A. B. GONCHAROV, Geometry of configurations, polylogarithms, and motivic cohomology, Adv. Math , MR [G2], Eplicit regulator maps on polylogarithmic motivic complees in Motives, Polylogarithms and Hodge Theory, Part I Irvine, Calif., 1998, Int. Press Lect. Ser. 3, I, Int. Press, Somerville, Mass., 2002, MR [G3], Regulators in Handbook of K-Theory, Vol. 1, 2, Springer, Berlin, 2005, MR [L1] M. N. LALÍN, Mahler measure of some n-variable polynomial families, J. Number Theory , MR [L2], Mahler measures and computations with regulators, to appear in J. Number Theory, preprint, arxiv:math/ v2 [math.nt] [M] V. MAILLOT, Mahler measure in Arakelov geometry, workshop lecture at The many aspects of Mahler s measure, Banff International Research Station, Banff, Canada, [R] F. RODRIGUEZ-VILLEGAS [VILLEGAS] Modular Mahler Measures, I in Topics in Number Theory University Park, Penn., 1997, Math. Appl. 467, Kluwer, Dordrecht, Netherlands, 1999, MR [S] C. J. SMYTH, On measures of polynomials in several variables, Bull. Austral. Math. Soc , 49 63; Corrigendum, C. J. Smyth and G. Myerson, Bull. Austral. Math. Soc , MR ; MR

32 422 MATILDE N. LALÍN [Z1] [Z2] [Z3] [Z4] [ZG] D. ZAGIER, Hyperbolic manifolds and special values of Dedekind zeta-functions, Invent. Math , MR , The Bloch-Wigner-Ramakrishnan polylogarithm function, Math. Ann , MR , Polylogarithms, Dedekind zeta functions and the algebraic K-theory of fields in Arithmetic Algebraic Geometry Teel, Netherlands, 1989, Progr. Math. 89,Birkhäuser, Boston, 1991, MR , Special values and functional equations of polylogarithms in Structural Properties of Polylogarithms, Math. Surveys Monogr. 37, Amer. Math. Soc., Providence, 1991, MR D. ZAGIER and H. GANGL, Classical and elliptic polylogarithms and special values of L-series in The Arithmetic and Geometry of Algebraic Cycles Banff, Canada, 1998, NATOSci.Ser.CMath.Phys.Sci.548, Kluwer, Dordrecht, Netherlands, 2000, MR Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada;