An algebraic integration for Mahler measure

Transcription

1 An algebraic integration for Mahler measure Matilde N. Lalín Department of Mathematics, Universit of British Columbia 1984 Mathematics Road, Vancouver, British Columbia V6T 1Z, Canada Abstract There are man eamples of several-variable polnomials whose Mahler measure is epressed in terms of special values of pollogarithms. These eamples are epected to be related to computations of regulators, as observed b Deninger [D], and later Rodriguez-Villegas [R-V], 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. Classification Primar 11G55; Secondar 19F99. Kewords Mahler measure, regulator, pollogarithms, Riemann zeta function, L-functions, polnomials 1 Introduction The logarithmic) Mahler measure of a Laurent polnomial P C[ ±1 1,..., ±1 n ] is defined b mp ) := log P e πiθ 1,..., e πiθn ) 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 polnomial. It is natural, then, to wonder what happens with several variables. The problem of finding eplicit closed formulas for Mahler measures of several-variable polnomials is hard. However, several eamples have been found, especiall for cases of two and three variables. Some formulas have been completel proved, and some others have been established numericall and are strongl believed to be true. A remarkable fact is that in most of these eamples, the Mahler measure of polnomials with integral coefficients can be epressed in terms of special values of L-series or pollogarithms that is to sa, Riemann zeta-functions, Dirichlet L-series, L-series of varieties, zeta functions of number fields, etc.). For instance, the first and simplest eample in two variables was discovered b Smth [S]Eample 5): m1 + + ) = 4π Lχ, ) = L χ, 1) ) where χ is the character of conductor. address: mlalin@math.ubc.ca 1

2 Another eample was computed numericall b Bod [Bo] and studied b Deninger [D] and Rodriguez-Villegas [R-V]), m ) + 1? = L E, 0) ) where E is the elliptic curve of conductor 15 which is the projective closure of the curve = 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-V] has clarified this relationship b eplicitl computing the regulator and relating this machiner to the cases alread numericall) known b Bod, proving some of them and deepl understanding the cases with two variables. Recentl, Maillot [M] has sketched how one could continue these ideas for more variables. It is our goal to develop these ideas and to appl them in order to understand the few known eamples with three and more variables involving Dirichlet L-series, Riemann zeta functions, and pollogarithms. In this article, we describe a general situation and illustrate our eplanation with a few eamples. In [L] we will show the computational power of our method b showing how to prove man other formulas of Mahler measures and generalized Mahler measures as well. Background In this section, we describe some ingredients that will be used in our construction..1 Pollogarithms The cases that we are going to stud involve zeta functions or Dirichlet L-series, but the all ma be thought of as special values of pollogarithms. In fact, this common feature seems to be the most appropriate wa of dealing with the interpretation of these formulas. Here, we proceed to recall some definitions and establish some common notation. Definition 1 The nth pollogarithm is the function defined b the power series Li n ) := k=1 k k n C, < 1. 4) This function can be continued analticall to C \ 1, ). We will work with Zagier s modification of the pollogarithm [Z4]), n 1 L n ) := Re n j B j log ) j Li n j ), 5) j! j=0 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 analtic in P 1 C) \ 0, 1, }.

3 L n satisfies ver clean functional equations. The simplest ones are ) 1 L n = 1) n 1 L n ), L n ) = 1) n 1 L n ). For n =, one obtains the Bloch Wigner dilogarithm, D) = ImLi ) log Li 1 )) = ImLi )) + log arg1 ), 6) which satisfies the well-known five-term relation ) ) 1 1 D) + D1 ) + D) + D + D = 0. 7) 1 1 For n =, we obtain L ) = Re Li ) log Li ) + 1 ) log Li 1 ). 8) This modified trilogarithm satisfies more functional equations, such as the Spence Kummer relation: ) ) ) ) ) 1 ) 1 ) 1 ) 1 ) L 1 ) +L )+L L L L 1 ) 1 1 ) 1 L L ) L ) + L 1) = 0. 9) 1. Pollogarithmic motivic complees Given a field F, consider Z[P 1 F ], the free abelian group generated b the elements of P1 F. For each n we are interested in working with this group modulo the rational) functional equations of the nth pollogarithm. Unfortunatel, the functional equations of higher pollogarithms are not known eplicitl. For X an algebraic variet, Goncharov [G1, G, G], has constructed some groups that conjecturall correspond to the groups in the above paragraph and the fit into pollogarithmic motivic complees whose cohomolog is related to Bloch groups and is conjectured to be the motivic cohomolog of X. A regulator can be defined in these complees and is conjectured to coincide with Beilinson s regulator. From now on, we will follow [G1, G, G]. We state definitions and results; the proofs ma be found in the mentioned works. Given a field F one defines inductivel some subgroups R n F ), then lets B n F ) := Z[P 1 F ]/R nf ). 10) The classes of in Z[P 1 F ] and in B nf ) will be denoted b } and } n respectivel. We begin b setting R 1 F ) := } + } };, F, 0}, }. 11) Thus B 1 F ) = F. Now, we proceed to construct a famil of morphisms: Z[P 1 F ] δn Bn 1 F ) F if n, F if n = ;

4 Then, one defines } n 1 if n, δ n }) = 1 ) if n =, 0 if } = 0}, 1}, }. 1) A n F ) := ker δ n. 1) Note that an 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 ever 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-pollogarithm in C. As stated at the beginning of Section., the philosoph is that R n F ) should be the subgroup of all the rational functional equations for the n- pollogarithm 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, δ : B F ) F. One obtains the comple B F n) : B n F ) δ B n 1 F ) F δ Bn F ) n F δ δ B F ) F δ n F, where n p n p δ : } p i δ p } p ) i. i=1 The following conjecture relates the cohomolog of the comple B F n) to motivic cohomolog. Conjecture [G] Formula )) We have i=1 H i B F n) Q) = gr γ nk n i F ) Q. 15) Evidence supporting this conjecture is found, for instance, in the cases n = 1,. First, it is clear that H 1 B F 1)) = F = K 1 F ). For n = it is known that where B F ) = Z[P 1 F ]/ R, );, F, 0}, }, R, ) := } + } + 1 } + } } is the five-term relation of the dilogarithm. Besides, H 1 B F )) Q = K ind F ) Q 16) H B F )) = K F ) 17) H n B F n)) = K M n F ) 18) The first assertion was proved b Suslin. The second one is Matsumoto s theorem, and the last one corresponds to the definition of Milnor s K-theor. 4

5 . Regulators Deninger [D] observed that the Mahler measure can be seen as a regulator evaluated in a ccle that ma or ma not have trivial boundar. More precisel, mp ) = mp 1 ) + πi) n 1 η n n) 1,..., n ). 19) We have to eplain the ingredients in this formula. In this section, we are concerned with η n n) 1,..., n ). This form will be described in the contet of Goncharov s construction of the regulator on the pollogarithmic motivic complees. Let us establish some notation: For an integers p 1 and k 0, define β k,p := 1) p p 1)! k + p + 1)! where the B i are Bernoulli numbers. Ln z) n > 1 odd, L n z) := il n z) n even. [ p 1 ] j=0 Definition We consider the following 1-forms: Recall that G ) k + p + 1 k+p j B k+p j j + 1 L p,q ) := L p ) log q 1 d log, p, 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 t σ1),..., t σm) ). Now, we are read to describe the differential forms. Definition 4 Let, i rational functions on a comple variet X. + η n+m m + 1) : } n 1 m L n )Alt m 1 p m d log j di arg j p + 1)!m p)! p 0 j=1 j=p+1 β k,p Ln k,k ) Alt m log 1 p m d log j di arg j 0) p 1)!m p)! 1 k, 1 p m j= j=p+1 η m m) : 1 m Alt m p+1 log 1 d log j p + 1)!m p 1)! p 0 j= m j=p+ di arg j 1) 5

6 These differential forms tpicall will 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. Let u ū the projection U Fv, and π a uniformizer for v. There is a homomorphism n n 1 θ : F F v defined b 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 ] b 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 ) defines a morphism of complees Observation 5 The induced morphism v : B F n) B Fv n 1)[ 1]. ) v : H n B F n)) H n 1 B Fv n 1)) coincides with the tame smbol defined b Milnor v : K M n F ) K M n 1F v ). Let X be a comple variet. Let X 1) denote the set of the codimension one closed irreducible subvarieties. Let A j X)k) denote the space of smooth j-forms with values in πi) k R. Let d be the de Rham differential on A j X) and let D be the de Rham differential on distributions. So d d arg ) = 0 D d arg ) = πδ) The difference D d is the de Rham residue homomorphism. Goncharov [G] proves the following, Theorem 6 We have that η n m) induces a homomorphism of complees B n CX)) δ B n 1 CX)) CX) δ... δ n CX) η n 1) η n ) η n n) A 0 X)n 1) d A 1 X)n 1) d... d A n 1 X)n 1) such that: η n 1)} n ) = L n ); dη n n) 1 n ) = β Re d1 n 1 is even; dn n ), where β = 1 if n is odd and i if n η n m) ) defines a distribution on XC); 6

7 The morphism η n m) is compatible with residues D η n m) η n m + 1) δ = πi η n 1 m 1) vy, m < n, 4) Y X 1) D η n n) βre n d1 1 d n n ) = πi where v Y is the valuation defined b the divisor Y. Y X 1) η n 1 n 1) vy, 5) The relation of η n ) to the regulator is roughl as follows. As we mentioned in Conjecture, the cohomolog of the first comple corresponds to the Adams filtration which is the absolute cohomolog. On the other hand, a slight modification of the second comple leads to Deligne cohomolog. 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 [G]). We should also remark that the final goal is not to work with CX), but with X itself. For X a regular projective variet over a field F Goncharov [G] describes the difficulties for defining the comple B X n) as opposed to B F X) n). Basicall, it is known how to define B X n) when X = SpecF ) for an arbitrar field F, X is a regular curve over an arbitrar field F, or X is an arbitrar regular scheme but n. 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 [Z] Conjecture 1), [ZG]). The two-variable case Rodriguez-Villegas [R-V] has performed the eplicit construction of the regulator and applied the ideas of Deninger [D] to eplain man eamples in two variables. This work was later continued b Bod and Rodriguez-Villegas [BR-V1, BR-V]. Let P C[, ]. Then we ma write B Jensen s formula, Here mp ) = ma d ) + 1 πi P, ) = a d ) d + + a 0 ), P, ) = a d ) d n=1 d α n )). n=1 T 1 log + α n ) d = mp ) 1 π η, ) := iη ) ) = log d arg log d arg γ η, ). 6) is the regulator in this case, defined in the set C = P, ) = 0} minus the set Z of zeros ans poles of and. Also, P = a d ), and γ is the union of paths in C where = 1 and 1. Finall, note that γ = P, ) = 0} = = 1}. In general, η, ) is closed in C \ Z, since dη, ) = Im d Now, the computation can be performed if we arrive to one of these two situations: 7 ) d see Theorem 6).

8 1. η is eact, and γ 0. In this case we can integrate using the Stokes theorem.. η is not eact, and γ = 0. In this case, we can compute the integral b using the Residue theorem. Eamples for the second case are found, for instance, in the famil of Laurent polnomials k studied b Bod [Bo], Deninger [D], and Rodriguez-Villegas [R-V]. Technicall 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, η, 1 ) = dd). 7) Thus, η is eact when = i r i i 1 i ) 8) in CC) ) Q. This condition ma be rephrased as the smbol, } is trivial in K CC)). In fact, if condition 8) is satisfied, we obtain η, ) = ) r i dd i ) = dd r i i }. 9) i i Finall, we write γ = k ɛ k [w k ], ɛ k = ±1, where w k CC), w k ) = w k ) = 1. Thus, we have the following. Theorem 7 [BR-V1, BR-V] Let P C[, ] be irreducible and such that, satisfies equation 8). Then πlog a d mp )) = Dξ) for ξ = ɛ k r i i w k )}. k i Bod and Rodriguez-Villegas prove even more. Under certain assumptions, it is possible to appl Zagier s Theorem [Z] and relate the Mahler measure of P to a rational combination of terms of the form 1 ζ F ) π[f :Q] for certain number fields F which depend on P or more specificall, on w k )..1 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 Smth s formula, [S] Eample 5): For this case, πm + 1) = 4 Lχ, ). = 1 ). 8

9 ξ6 γ) = 1 = 1 Figure 1: Integration path for + 1 Then, Here πmp ) = η, ) = η, 1 ) = D γ). γ γ γ =, ) = 1, 1 1} = e πiθ, 1 e πiθ ) θ [1/6 ; 5/6]}. Figure 1 shows the integration path γ. Then γ = [ ξ 6 ] [ξ 6 ] where ξ 6 = 1+ i ) and we obtain πm + 1) = Dξ 6 ) D ξ 6 ) = Dξ 6 ) = Lχ, ). 4 The three-variable case Our goal is to etend this situation to three variables. Let P C[,, z]. We will take 1 η,, z) := η ) z) = log 1 + log ) d log d log z d arg d arg z ) d log z d log d arg z d arg 1 + log z d log d log d arg d arg ). 0) This differential form is defined in the surface S = P,, z) = 0} minus the set Z of poles and zeros of, and z. We can epress the Mahler measure of P as mp ) = mp ) 1 π) Γ η,, z). 1) Where P, following the previous notation, is the principal coefficient of the polnomial P C[, ][z] and Γ = P,, z) = 0} = = 1, z 1}. d ) dz z. Tpicall, Recall η in closed in S \ Z since it verifies dη,, z) = Re d one epects that integral 1) can be computed if we are in one of the two ideal situations 9

10 that we described before. Either the form η,, z) is not eact, the set Γ consists of closed subsets and the integral is computed b residues, or the form η,, z) is eact and the set Γ 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 K surfaces to Eisenstein-Kronecker series in a similar wa as Rodriguez-Villegas does for two-variable cases [R-V]. In the second case, we need that η,, 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 cohomolog to be defined everwhere in the surface S, we need the residues to be zero. This situation is fulfilled when the tame smbols are zero see Section.). This condition will not be a problem for us because when η is eact the tame smbols are zero. As in the two-variable case, Theorem 6 implies where η, 1, ) = d ω, ), ) ω, ) := η )} ) = D) d arg + 1 log log 1 d log log d log 1 ). ) Thus, in order to appl the Stokes theorem, we need to require that z = r i i 1 i ) i 4) in CS) ) Q for η to be eact. An equivalent wa of epressing this condition is that,, z} is trivial in K MCS)). In this case, η,, z) = r i η i, 1 i, i ) = r i ω i, i ), where Γ Γ Γ = P,, z) = 0} = = z = 1}. This set Γ seems to have no boundar. However, Γ as described above ma contain singularities which ma give rise to a boundar when desingularized. We thus change our point of view. Namel, assume that P R[,, z] and is nonreciprocal this condition is true for all the eamples we stud); then, P,, z) = P, ȳ, z). This propert, together with the condition = = z = 1, allows us to write Γ = P,, z) = P 1, 1, z 1 ) = 0} = = z = 1}. This idea was proposed b Maillot [M]). Observe that we are integrating now on a path = = z = 1} inside the curve C = Res z P,, z), P 1, 1, z 1 )) = 0}. 10 Γ

11 In order to easil compute Γ ω, ), we have again the two possibilities that we had before. We are going to concentrate, as usual, in the case when ω, ) is eact. The differential form ω is defined in this new curve C. As before, to ensure that it is defined everwhere, we need to ask that the residues are trivial. This fact is guaranteed b the trivialit of tame smbols. This last condition is satisfied if ω is eact. Indeed, we have changed our ambient variet, and we now wonder when ω is eact in C ω is not eact in S since that would impl that η is zero). Fortunatel, we have ω, ) = dl ) 5) b Theorem 6. The condition for ω to be eact is not as easil 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 b the five-term relation. We ma epress the condition we need as } = r i i } i 6) in B CC)) CC) ) Q. Assuming Conjecture, this is equivalent to saing that a certain smbol for and is trivial in gr γ K 4CC)) Q. Then, we have ω, ) = r i L i ) γ. γ where γ = C T. Now assume that γ = ɛ k [w k ], k ɛ k = ±1 where w k CC), w k ) = w k ) = 1. Thus we have proved the following. Theorem 8 Let P,, z) R[,, z] be irreducible and nonreciprocal, let S = P,, z) = 0}, and let C = Res z P,, z), P 1, 1, z 1 )) = 0}. Assume that z = i r i i 1 i ) i 7) in CS) ) Q, and assume that i } i = j r i,j i,j } i,j 8) in B CC)) CC) ) Q for all i. Then 4π mp ) mp )) = L ξ) for ξ = k ɛ k r i r i,j i,j w k )}. 9) B using Zagier s conjecture see Zagier [Z], Zagier and Gangl [ZG]), it is possible to formulate a conjecture that would impl, under certain additional circumstances, a relationship with ζ F ) in a similar fashion as Bod and Rodriguez-Villegas have done for the two-variable case. We will illustrate this phenomenon at the end of Section i,j

12 4.1 The case of Res 0,m,m+n} We will proceed to the stud of a famil of three-variable polnomials which comes from the world of resultants, namel, Res 0,m,m+n}. This famil was computed in [DL], and the computation is quite involved, though elementar. The Mahler measure of Res 0,m,m+n} is the same as the Mahler measure of a certain rational function. More precisel, we have the following. Theorem 9 [DL] Theorem 6) We have m z 1 )m 1 ) n ) 1 ) m+n = n π L φ m ) L φ m 1 )) + m π L φ n 1 ) L φ n )) 40) where φ 1 is the root of m+n + n 1 = 0 which lies in the interval [0, 1] and φ is the root of m+n n 1 = 0 which lies in [1, ). Proof. Since we would like to see that η,, z) is eact, we need to solve equation 7) for this case. The equation for the wedge product becomes z = m 1 ) + n 1 ) m + n) 1 ) = m 1 ) + n 1 ) +m 1 ) n 1 ). After performing the Stokes theorem for the first time, we evaluate the form ω in the following element of B CC)) CC) : = m} } ) n} } ). We need to compute the corresponding curve C. We take advantage of the fact that our equation has the shape z = R, ). In order to compute C, we simpl need to consider For this case Let us denote R, )R 1, 1 ) = z z 1 = 1. 41) 1 ) m 1 ) n 1 1 ) m 1 1 ) n 1 ) m+n ) m+n = 1. 1 = = 1 1 then we ma rewrite the equation for C as Now we use the five-term relation: Then we obtain m 1 n 1 n 1 ŷm 1 = 1. 1 = 1 1 ŷ 1 = 1 1. } + } + 1 } + 1 } + 1 } = 0. = m} + 1 } + 1 } ) n} + 1 } + 1 } ). 1

13 Observe that = b 1 1, = b 1 Thus, we ma write 1. = m} + 1 } ŷ 1 1 } } ŷ 1 1 } 1 ) n} + 1 } 1 1 } } 1 1 } 1 ) = m} + 1 } ŷ1 m m 1 } 1 mŷ 1 } ŷ 1 1 } m 1 n} + n 1 } } 1 n 1 } n 1 + n 1 } 1. Because of the equation for C, we obtain, 1 } n 1 ŷ m 1 1 } m 1 n 1 = 1 } m 1 n } n 1 ŷ m 1 = m 1 } 1 + n 1 } 1 + n 1 } 1 mŷ 1 } ŷ 1, = m} ŷ 1 } ŷ 1 1 } 1 ŷ 1 } ŷ 1 1 } 1 ) n} 1 } 1 1 } 1 1 } 1 1 } 1 ). = m} ŷ 1 } ŷ 1 1 } 1 ) n} 1 } 1 1 } 1 ). We now need to stud the path of integration. First write = e iα, = e iβ for π α, β π. Then, and Let a = sin α sinα+β) 1 = e iβ sin α sinα + β), 1 = e iα sin β sinα + β), 1 = e iα sin β sinα + β), ŷ 1 = e iβ sin α sinα + β). sin β, b =. Then, we ma write sinα+β) 1 = ±ae iβ, 1 = ±be iα, 1 = ±be iα, ŷ 1 = ±ae iβ. B means of the Sine theorem, we ma think of a, b, and 1 as the sides of a triangle with the additional condition a m b n = 1. The triangle determines the angles α and β, which are opposite to the sides a, b respectivel. We need to be careful and take the complement of an angle if it happens to be greater than π. This corresponds to the cases when the sines are negatives.) However, we need to be cautious. In fact, the problem of constructing the triangle given the sides has alwas two smmetric solutions. We count each triangle once, so we multipl our final result b 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, and a + b = 1. Let φ 1 be the root of m+n + n 1 = 0 with 0 φ 1 1, 1

14 a b β α β α α β 1) ) ) Figure : We are integrating over all the possible triangles. The angles are measured negativel if the are greater than π, as α in the case ). We do not count the triangles pointing down, as in ). and let φ be the root of m+n n 1 = 0 with 1 φ. Then the first two conditions are translated as a = φ1 n, b = φm 1, α = 0, β = 0, a = φ n, b = φm, α = 0, β = 0. The third condition is inconsequential since it requires both a, b 1 but the can not be both equal to 1 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, θ 1 α π, π β θ, π α 0, θ β 0. Here, θ 1 is the angle that is opposite to the side a when the triangle is right-angled with hpotenuse b and θ is opposite to b when a is the hpotenuse. We do not need to compute those angles. In fact, we ma describe the integration path as either or 0 α π, π α 0, 0 β π, π β 0. It is appropriate to think of it in this wa, because 1 } + ŷ 1 } and 1 } + 1 } change continuousl around the right-angled triangles. Moreover, because of this propert, everthing reduces to evaluating L in Ω = m} ŷ 1 } 1 } ) n} 1 } 1 } ) 14

15 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 = 1, but these conditions correspond to = m} and = n}. The lead to Ω = m} and Ω = n}, and integrate to zero when the variables move in the unit circle. The poles are at = 1, which corresponds to = m n)}. Integrating, we obtain Ω = m n)}, which leads to zero when moves in the unit circle. We obtain Finall, 4π mp ) = 4nL φ m ) L φ m 1 )) + 4mL φ n 1 ) L φ n ))). mp ) = n π L φ m ) L φ m 1 )) + m π L φ n 1 ) L φ n )), so we recover the result of [DL]. The case with m = n = 1 is especiall elegant. Here the rational function has the form z = 1 )1 ) 1 ), and mp ) = 4 π L φ) L φ)) where φ + φ 1 = 0 and 0 φ 1 in other words, φ = 1+ 5 ). Moreover, we ma use Zagier s conjecture see [Z] 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) )) has rank. We ma take 1}, φ} } as basis. In order to see this, we need to check that φ} φ is trivial. That is the case because φ} = 1 φ } = φ } = φ} φ} = 1+φ} φ} = φ 1 } φ} = 0, which implies φ} φ = 0. Then the conjecture predicts ζ L Q 5) ) Q 5 φ) L 1) L φ 1 ) L 1) = 5ζ)L φ) L φ)). Indeed, which allows us to write ζ Q 5) ) = ζ) 5 L φ) L φ)), mres 0,1,} ) = 4 5ζ Q 5) ) π. ζ) 15

16 5 A few words about the four-variable case Unfortunatel, we do not have a general sstematic method to describe algebraicall 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 should be as follows: η,, w, z) := iη 4 4),, w, z) = 1 4 d log z Im d d d log Im dw w dz ) + log Im z dw ) d + log w Im w dw w dz ) z +η,, w) d arg z η,, z) d arg w + η, w, z) d arg η, w, z) d arg ) 4) where η,, z) denotes the differential previousl defined for three variables. We have, η, 1,, w) = dω,, w), 4) where ω,, w) := iη 4 ),, w) = D) ) 1 d log d log w d arg d arg w + 1 η, w) log d log 1 log 1 d log ). 44) d dz ) z with Net, Finall, ω,, ) = dµ, ) 45) µ, ) := iη 4 ), ) = L ) d arg 1 D) log d log. 46) µ, ) = dl 4 ). 47) 5.1 An eample in four variables In spite of the fact that we do not know how to treat the integration domains, we ma still be able to do the algebraic integration for some eamples of four-variable polnomials. Here is an eample. We will stud 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 polnomial that is the general determinant. Because of homogeneities, this Mahler measure problem ma be reduced to computing the Mahler measure of a four-variable polnomial. The result is the following. Theorem 10 [DL]Theorem 7) We have m1 )1 ) 1 w)1 z)) = 9 ζ). 48) π 16

17 Proof. First, we have to solve the equation with the wedge product: w z = 1 w z = ) w z ) w z. Now, the first term on the right-hand side is ) w z = w ) w z = 1 + ) ) w z 1 + ) ) w z. w Net, we use the formula for z as a function of the other variables: 1 + ) ) w z = + w w = + w ) w + + w w1 w) ) w 1 + ) + ) w 1 w). w w Note that + ) ) w 1 w) = 1 + ) ) w 1 w) ) w 1 w). Hence, w z = ) w z + ) 1 + ) w z1 w) + + w 1 + ) ) w w 1 w) ). w The form ω will be evaluated in the following element: } 1 = w z + } w z1 w) } + + w = } w z + ) w w} ) } w z1 w) z z } ) w w} w ). For appling the Stokes theorem, we still appl the technique that is analogous to the computation of the equation for C in the three-variable case. We can appl the analogue of the equation 41), 1 ) 1 )1 ) )1 1 ) ) 1 w 1 w 1 = 1, 17

18 w π 0 π π =w =1 =w=1 ==w ==w =w=1 =w =1 =w =w =w =1 ==1 ==1 =w =1 =1 =1 =w=1 =w=1 Figure : Integration set for Res 0,0),1,0),0,1)} 18

19 which can be simplified as = 1, = 1, w =, or w =. The above conditions correspond to two pramids in the torus T, as seen in Figure. We will make the computation over the lower pramid and then multipl the result b. In the case when = 1, we have w = 1 or z = 1. If w = 1, = 0. If z = 1, = 1 1 } w. w Then µ is evaluated on Ω = w}, and Ω is integrated on the boundar, which is = 1, w = 1 and = w. If = 1, Ω = 0. If w = 1, Ω = 1}, which ields πζ). If = w, Ω = }, whose integral is zero. In the case when = 1, we have w = 1 or z = 1. If w = 1, = 0. If z = 1, = 1 1 } w 1 w) 1 1 } 1 1 ) w w} 1 1 ). w Onl the term in the middle ields a nonzero differential form. In fact, the term in the middle ields Ω = w} 1 1 ), and Ω is integrated on the boundar, which is = 1, w = 1 and = w. If = 1, Ω = 0. If w = 1, Ω = 1} 1 1 ). This integration is equal to πζ). If = w, Ω = } 1 1 ), which integrates to zero. When w =, in this case, z = unless = 1), = } + } } Then, ) } 1 ) ). Ω = } } } 1 1 ). 19

20 Now, Ω is to be integrated on the boundar, which is = 1, = 1, and = see Figure ). If = 1, Ω = 1}, which gives 4πζ). If = 1, Ω = 1 1 } } 1 1 ). Now use the fact that } + 1 } } = 1} and the fact that = 1 to conclude 1 1 } = } 1}. The total integration in this case is πζ). If =, Ω = } 1} } 1 1 ) which leads to µ 1, ). We do not need to compute this integral for the final result). When w = in this case, z = unless = 1), B the five-term relation, 1 1 } + } + = } + } ) } ) } ) = } + } } 1 1 ) } ) } ) )} + } 1 + } + } } + } Then, we obtain = } + } } 1 1 ) } ) } ) ++ } ) + } ) 0 } = 0. } = 0

21 } ) } ) ++ } ) + } ) = } + } } 1 1 ) } 1 )1 ) } 1 )1 ) + Now, + + } 1 )1 ) + is zero in the differential form. Therefore, } } } 1 1 ) } 1 ) Ω = } + } + 1 } + } 1 )1 ) + ) ). 1 )1 )} + }, and Ω will be integrated on the boundar, which is = 1, = 1, and =. If = 1, Ω = 1} + } 1 ) }, whose integral is πζ). If = 1, Ω = 1} + 1} 1 ), which gives πζ). If =, } Ω = } + 1 } + } 1 ) + 1} 1 ) } = } 1 } + } 1 ) 1}, which ields πζ) + µ 1, ). The poles are with w = 1, but = 0 in this case. On the other hand, if z = 0, then w = +. But w = 1 implies that = 1, = 1, or =. In the first two cases, w = 1 and = 0. In the third case, = } 1 ) } 1 ), which corresponds to zero if = 1. Thus, 8π mp ) = 6πζ). Finall, mp ) = 9 π ζ). 1

22 6 The n-variable case. The usual application of Jensen s formula as in equation 6) ) allows us to write, for P C[ 1,..., n ], mp ) = mp 1 ) + πi) n 1 η 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 eas 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 an general algebraic wa of describing the successive sets that we obtain b taking boundaries. Suppose that we do have a good description of the boundaries inside certain algebraic varieties, sa 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 } Y i1 = i r i1,i z i1,i } z i1,i Y i1,i 51) in B CS )) n CS ) ) Q for all i 1. More generall, assume that for k = 4,..., n 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 5) in B k CS k )) n k CS k ) ) Q for all i 1,..., i k 1. Finall, assume z i1,...,i n } n 1 Y i1,...,i n = i n 1 r i1,...,i n,i n 1 z i1,...,i n,i n 1 } n 1 z i1,...,i n,i n 1 5) in B n 1 CS n 1 )) CS n 1 ) ) Q for all i 1,..., i n. Then we ma write π) 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 z i1,...,i n 1 w k )} n

23 Here we have written Y i1, Y i1,i,... to denote elements in n CS 1 ) ) Q, n CS ) ) Q,... A solution to equation 50) determines the Y i1 s. Once the Y i1 s are defined, we solve equation 51) and obtain the Y i1,i s. The procedure continues in this fashion until we reach the Y i1,...,i n s. Ideall, we epect that this setting eplains the nature of the n-variable eamples described in [L1]. 6.1 The case of an n-variable famil. Let us consider the case of the famil 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 ) i+1 i+n 1 ), i=1 with the cclical convention that i+n = i. Thus, we proved that η = η n+1 n + 1) 1,..., n, z) is eact. integrate η n+1 n) evaluated on the following element: = The net step is to n 1) in 1) i } i+1 i+n 1 i } i+1 i+n 1 ). i=1 We now prove that ω = η n+1 n) ) is eact. This form is defined in the variet Z which is the projective closure of the algebraic set determined b ) 1 ) 1) n 1 1 n = n We wish to show that ω is trivial in H n 1 DR Z). First, observe that Z = Z + Z, where Z ± is given b the equation ) ) 1 ±z 1 1 n = n and z 1 n even, = i n odd.

24 In general, consider the variet given b the projective closure of the zeros of ) ) n α = n with α a nonzero comple number. This variet is birational to P n 1. This is eas to see b setting i = 1 i 1+ i ). Hence we ma think of each Z ± as a cop 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 = k. If we prove that ω can be etended to the whole Z ± and that this etension is consistent with the birationalit of Z ±, it implies that ω is closed and that it ma be seen as a class in H k 1 DR Pk 1 ) = 0 and then ω 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 B CZ)) n 1 CZ) v η n+1 n) A n 1 Z)n) Res v B CZ) v ) n CZ) v η nn 1) A n Z v )n 1), which describes the relation between the tame smbol and the residue morphism. We would like to see that Res v η n+1 n) 1 } n )) = 0, where v is the valuation defined b i = ±1, 0, for some i. Instead, we will see that η n n 1) v 1 } n )) = 0. First, suppose i = 1. Then, if i 1, reducing modulo i 1 implies that i = 1, and the onl term that is possibl nonzero in v 1 } n ) is v i ) 1 } i n. However, i is clearl not a uniformizer for i 1. Then, the tame smbol is zero. If i = 1, 1 } reduces to 1}, which corresponds to zero in η n n 1), so we get zero again. The case i = 1 is analogous. Now, consider the case with i = 0 for some i. Then, it is eas to see that 1 } i n if i 1, v 1 } n ) = 55) 0 if i = 1. Since i = 0, we are now in the variet defined b the projective closure of the zeros of the equation ) 1 ) 1 ) 1 i 1 n 1 = i 1 + n We are in a situation that is analogous to the initial one. In other words, we are in the projective space P k. We proceed b induction. In order to prove that η n 1 n ) 1 } 4

25 i n ) is trivial, we can prove that the tame smbols w 1 } i n ) are trivial b induction. However, H k DR Pk ) = R, so even if the smbols are trivial, we are not able to conclude that the form η n 1 n ) 1 } i n ) is eact. What we can conclude is that it is either a generator for H k DR Pk ) or trivial. We eliminate the first possibilit. Suppose, in order to make notation easier, that n = i. Assume that η n 1 n ) 1 } n 1 ) is a generator for H k DR Pk ). B Poincaré dualit, the integral I = « k 1 η k 1 k ) 1 } k 1 ) 1= k 1 must be nonzero. Now the transformation i 1 i does not change the orientation of the variet 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 k DR Pk ), and it must be eact. The case when i = is analogous. Now suppose that n = k + 1 is odd. Then we ma proceed as before. We have that ω can be seen as a class in HDR k Pk ) = R and we can conclude that is eact b 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)Ω). 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 would like to epress m deepest gratitude to m Ph.D. supervisor, Fernando Rodriguez- Villegas, for his invaluable guidance and encouragement along this and other projects. I am grateful to Vincent Maillot for man enlightening discussions about cohomolog and the n-variable case. I am also thankful to Herbert Gangl for helpful discussions about pollogarithms and Bloch groups. I wish to epress m appreciation for the careful work of the referees, which has certainl improved the clarit and eposition of this article, and to Amir Jafari for his comments in the final version. This work is part of m Ph.D. dissertation at the Department of Mathematics at the Universit of Teas at Austin. I am indebted to John Tate and the Harrington fellowship for their generous financial support during m graduate studies. This research was partiall completed while I was a visitor at the Institut des Hautes Études Scientifiques. I am grateful for their support. References [B] M. J. Bertin, Mesure de Mahler d hpersurfaces K, preprint, Januar 005). [Bo] [BR-V1] D. W. Bod, Mahler s measure and special values of L-functions, Eperiment. Math ), 7 8. D. W. Bod, F. Rodriguez-Villegas, Mahler s measure and the dilogarithm I), Canad. J. Math ),

26 [BR-V] [DL] [D] [G1] [G] [G] [L1] [L] [M] [R-V] [S] [Z1] [Z] [Z] [Z4] D. W. Bod, F. Rodriguez-Villegas, with an appendi b N. M. Dunfield, Mahler s measure and the dilogarithm II), preprint, Jul 00). C. D Andrea, M. N. Lalín, On The Mahler measure of resultants in small dimensions, J. Pure Appl. Algebra ) no., C. Deninger, Deligne periods of mied motives, K-theor and the entrop of certain Z n -actions, J. Amer. Math. Soc ), no., A. B. Goncharov, Geometr of Configurations, Pollogarithms, and Motivic Cohomolog, Adv. Math ), no., A. B. Goncharov, Eplicit regulator maps on pollogarithmic motivic complees. Motives, pollogarithms and Hodge theor, Part I Irvine, CA, 1998), 45 76, Int. Press Lect. Ser.,, I, Int. Press, Somerville, MA, 00. A. B. Goncharov, Regulators. Handbook of K-theor. Vol. 1,, 95 49, Springer, Berlin, 005. M. N. Lalín, Mahler measure of some n-variable polnomial families, J. Number Theor ), no. 1, M. N. Lalín, Mahler measures and computations with regulators, to appear in J. Number Theor V. Maillot, Mahler measure in Arakelov geometr, workshop lecture at The man aspects of Mahler s measure, Banff International Research Station, Banff, Canada, 00. F. Rodriguez-Villegas, Modular Mahler measures I, Topics in number theor Universit Park, PA 1997), 17 48, Math. Appl., 467, Kluwer Acad. Publ. Dordrecht, C. J. Smth, On measures of polnomials in several variables, Bull. Austral. Math. Soc. Ser. A 1981), Corrigendum with G. Merson): Bull. Austral. Math. Soc ), D. Zagier, Hperbolic manifolds and special values of Dedekind zeta-functions, Invent. Math ), D. Zagier, The Bloch-Wigner-Ramakrishnan pollogarithm function. Math. Ann ), no. 1, D. Zagier, Special values and functional equations of pollogarithms. Structural properties of pollogarithms, , Math. Surves Monogr., 7, Amer. Math. Soc., Providence, RI, 1991 D. Zagier, Pollogarithms, Dedekind Zeta functions, and the Algebraic K-theor of Fields, Arithmetic algebraic geometr Teel, 1989), 91 40, Progr. Math., 89, Birkhäuser Boston, Boston, MA,

27 [ZG] D. Zagier, H. Gangl, Classical and elliptic pollogarithms and special values of L-series, The arithmetic and geometr of algebraic ccles Banff, AB, 1998), , NATO Sci. Ser. C Math. Phs. Sci., 548, Kluwer Acad. Publ., Dordrecht,