There s something about Mahler measure

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1 There s something about Mahler measure Junior Number Theory Seminar University of Texas at Austin March 29th, 2005 Matilde N. Lalín Mahler measure Definition For P C[x ±,...,x± n ], the logarithmic Mahler measure is defined by mp = 2πi n log Px,...,x n dx... dx n. T n x x n This integral is not singular and mp always exists. Because of Jensen s formula: 0 log e 2πiθ α dθ = log + α, 2 we have a simple expression for the Mahler measure of one-variable polynomials: mp = log a d + d d log + α n for Px = a d x α n. n= Examples of Mahler measures in several variables It is in general very hard to find formulas for Mahler measure of several-variable polynomials. For more than three variables, very little is known. Theorem 2 [9] For n we have: π 2n x m + n s n h 2 2,...,2n 2 2 = 2n! For n 0: = n h=0 h= π 2n+ m + x... x2n + x 2n n= π 2n 2h 2h! 22h+ ζ2h x2n+ + x 2n+ s n h 2,...,2n 2 2 2h+ π 2n 2h 2h +!Lχ 4, 2h n! There are analogous but more complicated formulas for x xn m + x y + x n m + x... log + x = log max{, x} for x R 0 xn + x n x + x... xn + x n y

2 Where if l = 0 s l a,...,a k = i <...<i l a i...a il if 0 < l k 0 if k < l are the elementary symmetric polynomials, i. e., 5 k x + a i = i= k s l a,...,a k x k l 6 l=0 For example, π 3 x x2 x3 m + + x 2 + x 3 π 4 x x4 m x 4 π 4 x x2 m + x + + y + x 2 = 24Lχ 4, 4 + π 2 Lχ 4, 2 7 = 62ζ5 + 4π2 ζ3 8 3 = 93ζ5 9 0 Polylogarithms Many examples should be understood in the context of polylogarithms. Definition 3 The kth polylogarithm is the function defined by the power series Li k x := n= x n n k x C, x <. This function can be continued analytically to C \ [,. In order to avoid discontinuities, and to extend polylogarithms to the whole complex plane, several modifications have been proposed. Zagier [5] considers the following version: k P k x := Re k 2 j B j log x j Li k j x, 2 j! j=0 where B j is the jth Bernoulli number, Li 0 x 2 and Re k denotes Re or Im depending on whether k is odd or even. This function is one-valued, real analytic in P C \ {0,, } and continuous in P C. Moreover, P k satisfy very clean functional equations. The simplest ones are P k = k P k x P k x = k P k x. x There are also lots of functional equations which depend on the index k. For instance, for k = 2, we have the Bloch Wigner dilogarithm, Dx := ImLi 2 x + arg xlog x 2

3 which satisfies the well-known five-term relation y x Dx + D xy + Dy + D + D = 0. 3 xy xy Beilinson s conjectures One of the main problems in Number Theory is finding rational or integral solutions of polynomial equations with rational coefficients global solutions. In spite of the failure of the local-global principle in general, there are several theorems and conjectures which predict that one may obtain global information from local information and that that relation is made through values of L-functions. These statements include the Dirichlet class number formula, the Birch Swinnerton-Dyer conjecture, and more generally, Bloch s and Beilinson s conjectures. Typically, there are four elements involved in this setting: an arithmetic-geometric object X typically, an algebraic variety, its L-function which codify local information, a finitely generated abelian group K = HM i X, Qj, and a regulator map r D : H i MX, Qj H i DX, Rj Here HD i X, Ri can be thought as a group of differential forms on the variety. Another function reg : K R, is defined and is also called regulator. Basically. regξ = r D ξ When K has rank, Beilinson s conjectures predict that the L X 0 is, up to a rational number, equal to a value of the regulator reg. For instance, for a number field F, Dirichlet class number formula states that cycle lim s ζ Fs = 2r 2π r2 h F reg F. s ω F DF Here, X = O F the ring of integers, L X = ζ F, and the group is OF. Hence, when F is a real quadratic field, Dirichlet class number formula may be written as ζ F 0 is equal to, up to a rational number, log ɛ, for some ɛ OF. An algebraic integration for Mahler measure The appearance of L-functions in Mahler measures formulas is a common phenomenon. Deninger [5] interpreted the Mahler measure as a Deligne period of a mixed motive. More specifically, in two variables, and under certain conditions, he proved that mp = regξ i, where reg is the determinant of the regulator matrix, which we are evaluating in some class in an appropriate group in K-theory. Rodrigue-Villegas [] has worked out the details for two variables. This was further developed by Boyd and Rodrigue-Villegas [], [2]. 3

4 where More specifically one has mp = mp ηx, y, 4 2π γ ηx, y = log x darg y log y darg x 5 is a differential form that is essentially defined in the curve C determined by the eros of P. This form is essentially the regulator. It is closed since dx dηx, y = Im x dy. y One has a crutial property: Theorem 4 ηx, x = ddx. 6 Because of the above property, there is a condition that tells us when ηx, y is exact, namely: x y = r j j j j in 2 CC Q, in other words, {x, y} = 0 in K 2 CC Q. Under those circunstances, ηx, y = d j r j D j = dd j r j [ j ]. We have γ C such that γ = k ɛ k [w k ] ɛ k = ± where w k CC, xw k = yw k =. Then 2πmP = Dξ for ξ = k r j [ j w k ]. j We could summarie the whole picture as follows:... K 3 Q K 3 γ K 2 C, γ K 2 C... There are two nice situations: γ = C T 2 ηx, y is exact, then {x, y} K 3 γ. In this case we have γ, we use Stokes Theorem and we finish with an element K 3 γ K 3 Q, leading to dilogarithms and eta functions of number fields, due to theorems by Borel, Bloch, Suslim and others. 4

5 γ =, then {x, y} K 2 C. In this case, we have ηx, y is not exact and we get essentially the L-series of a curve, leading to examples of Beilinson s conjectures. In general, we may get combinations of both situations. The three-variable case We are going to extend this situation to three variables. We will take ηx, y, = log x dlog y dlog darg y darg 3 + log y dlog dlog x darg darg x +log dlog x dlog y darg xdarg y 3 3 Then η verifies so it is closed. We can express the Mahler measure of P Where dx dηx, y, = Re x dy y d, mp = mp 2π 2 Γ ηx, y,. Γ = {Px, y, = 0} { x = y =, }. We are integrating on a subset of S = {Px, y, = 0}. The differential form is defined in this surface minus the set of eros and poles of x, y and, but that will not interfere our purposes, since we will be dealing with the cases when ηx, y, is exact and that implies trivial tame symbols thus the element in the cohomology can be extended to S. As in the two-variable case, we would like to apply Stokes Theorem. Let us take a look at Smyth s case, we can express the polynomial as Px, y, = x + y. We get: where mp = m y + 2πi 2 log + T 2 In general, we have x y ηx, x, y = dωx, y, dx dy x y = 2π 2 ηx, y,. Γ ωx, y = Dxdarg y + log y log x dlog x log x dlog x. 3 Suppose we have x y = r i x i x i y i in 3 CS Q. Then ηx, y, = r i Γ Γ ηx i, x i, y i = r i 5 Γ ωx i, y i.

6 In Smyth s case, this corresponds to in other words, x y = x x y y y x, ηx, y, = ηx, x, y ηy, y, x. Back to the general picture, Γ = {Px, y, = 0} { x = y = = }. When P Q[x, y, ], Γ can be thought as γ = {Px, y, = Px, y, = 0} { x = y = }. Note that we are integrating now on a path inside the curve C = {Px, y, = Px, y, = 0}. The differential form ω is defined in this new curve this way of thinking the integral over a new curve has been proposed by Maillot. Now it makes sense to try to apply Stokes Theorem again. We have ωx, x = dp 3 x. Suppose we have [x] 2 y = r i [x i ] 2 x i in B 2 CC CC Q. Then, as before: γ ωx, y = r i P 3 x i γ. Back to Smyth s case, in order to compute C we set x x y y = and we get C = {x = y} {xy = } in this example, and [x] 2 y [y] 2 x = ±2[x] 2 x. We integrate in the set described by the following picture π _ π π _ π Then m x + y = 4π 2 and the second condition is γ ωx, y + ωy, x = 4π 28P 3 P 3 = 7 2π 2ζ3 [x i ] 2 y i = 0 in H 2 B QC 3 Q? = K [] 4 QC Q 6

7 Hence, the conditions can be translated as certain elements in different K-theories must be ero, which is analogous to the two-variable case. We could summarie this picture as follows. We first integrate in this picture... K 4 Γ K 3 S, Γ K 3 S... Γ = S T 3 As before, we have two situations. All the examples we have talked about fit into the situation when ηx, y, is exact and Γ. Then we finish with an element in K 4 Γ. Then we go to... K 5 Q K 5 γ K 4 C, γ K 4 C... γ = C T 2 Again we have two possibilities, but in our context, ωx, y is exact and we finish with an element in K 5 γ K 5 Q leading to trilogarithms and eta functions, due to Zagier s conjecture and Borel s theorem. Studied examples Smyth98: Smyth2002: L 2003: Condon 2003: π 2 m + x + y + = 7 2 ζ3 π 2 m + x + y + + x + y = 4 3 ζ3 π 2 m + π 2 m + π 2 m x x2 x x + = 7ζ3 + x 2 x = 7 2 ζ3 + π2 log 2 2 x + y = 28 + x 5 ζ3 y D Andrea & L 2003: π 2 m xy m+n x m y n = 2nP 3 φ m P 3 φ m 2 + 2mP 3 φ n 2 P 3 φ n π 2 m x y w = 9 2 ζ3 7

8 L 2003: π 2 m + + x x2 + x 2 x x2 + x 2 = 2 4 ζ3 + π2 log 2 2 x y References [] D. W. Boyd, F. Rodrigue Villegas, Mahler s measure and the dilogarithm I, Canad. J. Math , [2] D. W. Boyd, F. Rodrigue Villegas, with an appendix by N. M. Dunfield, Mahler s measure and the dilogarithm II, preprint, July 2003 [3] J. Condon, Calculation of the Mahler measure of a three variable polynomial, preprint, October 2003 [4] C. D Andrea, M. N. Lalín, On The Mahler measure of resultants in small dimensions. in preparation. [5] C. Deninger, Deligne periods of mixed motives, K-theory and the entropy of certain Z n -actions, J. Amer. Math. Soc , no. 2, [6] A. B. Goncharov, Geometry of Configurations, Polylogarithms, and Motivic Cohomology, Adv. Math , no. 2, [7] A. B. Goncharov, The Classical Polylogarithms, Algebraic K-theory and ζ F n, The Gelfand Mathematical Seminars , Birkhauser, Boston 993, [8] A. B. Goncharov, Explicit regulator maps on polylogarithmic motivic complexes, preprint, March [9] M. N. Lalín, Mahler measure of some n-variable polynomial families, September 2004, to appear in J. Number Theory [0] M. N. Lalín, An algebraic integration for Mahler measure in preparation. [] F. Rodrigue Villegas, Modular Mahler measures I, Topics in number theory University Park, PA 997, 7-48, Math. Appl., 467, Kluwer Acad. Publ. Dordrecht, 999. [2] C. J. Smyth, On measures of polynomials in several variables, Bull. Austral. Math. Soc. Ser. A 23 98, Corrigendum with G. Myerson: Bull. Austral. Math. Soc , [3] C. J. Smyth, An explicit formula for the Mahler measure of a family of 3-variable polynomials, J. Th. Nombres Bordeaux , [4] D. Zagier, The Bloch-Wigner-Ramakrishnan polylogarithm function. Math. Ann , no. -3,

9 [5] D. Zagier, Polylogarithms, Dedekind Zeta functions, and the Algebraic K-theory of Fields, Arithmetic algebraic geometry Texel, 989, , Progr. Math., 89, Birkhuser Boston, Boston, MA, 99. 9