LOGARITHMIC FUNCTIONAL AND THE WEIL RECIPROCITY LAW

Transcription

1 85 LOGARITHMIC FUNCTIONAL AND THE WEIL RECIPROCITY LAW KHOVANSKII A. Department o Mathematics, University o Toronto, Toronto, Ontario, Canada askold@math.toronto.edu Dedicated to my old riend Sergei Abramov This article is devoted to the (one-dimensional) logarithmic unctional with argument being one-dimensional cycle in the group (C ) 2. This unctional generalizes usual logarithm, that can be viewed as zero-dimensional logarithmic unctional. Logarithmic unctional inherits multiplicative property o the logarithm. It generalizes the unctional introduced by Beilinson or topological proo o the Weil reciprocity law. Beilinson s proo is discussed in details in this article. Logarithmic unctional can be easily generalized or multidimensional case. It s multidimensional analog (see Re. 5) proves multidimensional reciprocity laws o Parshin. I plan to return to this topic in upcoming publications. 1. Introduction We start introduction with a brie discussion o classic results related to this paper: reciprocity law, its topological proo and multidimensional generalization. Further we comment on the logarithmic unctional and its relation to the Beilinson unctional. We end introduction with layout o material The Weil reciprocity law Given two polynomials o degrees n and m with leading coeicients equal to one, the ollowing equality holds: the product o values o the irst polynomial over the roots o the second one is equal to the product o values o the second polynomial over the roots o the irst one, multiplied by ( 1) mn. André Weil has ound urther generalization o this equality called the reciprocity law. The Weil reciprocity law applies to any pair o non-zero rational unctions, g on arbitrary irreducible algebraic curve X over an

2 86 A. Khovanskii algebraically closed ield K. In this paper we will consider only the case o the ield K being the ield o complex numbers C. We will give exact statement o the reciprocity law or K = C in Section 2. Now we just give general comments. The law is as ollows. For each point a X some non-zero element [, g] a o the ield K is deined, that is called the Weil symbol o unctions, g at the point a. The Weil symbol depends on unctions, g skew-symmetrically, i.e. [, g] a = [g, ] 1 a. Besides the Weil symbol is multiplicative in each argument, i.e. or any triplet o non-zero rational unctions, g, ϕ and any point a X the ollowing equalities hold [ϕ, g] a = [, g] a [ϕ, g] a and [, ϕg] a = [, g] a [, ϕ] a. Let O be the union o supports o divisors o unctions and g. Weil symbol o unctions, g can dier orm 1 only at points a o the inite set O. Thereore the product o Weil symbols over all points a o curve X is well deined: it is equal to the product o symbols [, g] a over inite set o points a O. The Weil reciprocity law states that the product o Weil symbols [, g] a over all point o the curve X is equal to one Topological explanation o the reciprocity law over the ield C In the case o K = C Beilinson 1 proved the Weil reciprocity law topologically and has ound topological generalization o the Weil symbols. Let, g be non-zero rational unctions on connected complex algebraic curve X and let O X be inite set containing support o divisors o unctions and g. Consider piecewise-smooth oriented closed curve γ : [0, 1] X \ O, γ(0) = γ(1) on the maniold X that does not contain points o set O. Beilnson introduced a unctional that associates a pair, g and curve γ with an element B γ (, g) o the group C/Z (i.e. some complex number deined up to an integer additive term). He showed that: 1) For ixed, g the unctional B γ (, g) considered as a unction o cycle γ gives an element o one-dimensional cohomology group o the maniold X \ O with coeicients in group C/Z, i.e. or homologous to zero in X \ O integer-valued linear combination k i γ i o oriented curves γ i the equality ki B γ (, g) = 0 holds. 2) The unctional skew-symmetrically depends on and g, i.e. B γ (, g) = B γ (g, ). 3) The unctional has the ollowing property o multiplicativity: or any three rational unctions, g, ϕ not having zeros and poles in the set X \ O and or any curve γ X \ O, equalities B γ (ϕ, g) = B γ (, g) + B γ (ϕ, g) and B γ (, ϕg) = B γ (, g) + B γ (, ϕ) hold.

3 Logarithmic Functional and the Weil Reciprocity Law 87 4) The unctional B γ (, g) is related to Weil symbols as ollows: or any point a O and small curve γ X \ O running around point a, equality [, g] a = exp B γ (, g) holds. In accordance with 4) Weil symbols correspond to the values o Beilinson unctional on special curves related to the points o the set O. The sum o those curves over all points in O is homologous to zero in X \ O, which gives the Weil reciprocity law (see Section 5). Summarizing, Beilinson unctional explains the Weil reciprocity law over ield C, and also provides an analog o Weil symbols or arbitrary cycles in X \ O, not necessary related to the points o the set O Multi-dimensional reciprocity laws A.N. Parshin (Res. 6,7) has ound remarkable multi-dimensional generalization o the reciprocity law. It applies to an arbitrary collection = { 1,..., n+1 } o n + 1 nonzero rational unctions j on irreducible n-dimensional maniold M over arbitrary algebraically closed ield K. In multi-dimensional generalization instead o point on maniold M one considers lag F = {M 0 M 1 M n = M} consisting o a chain o embedded germs M j o algebraic maniolds o increasing dimensions, dim M j = j, locally irreducible in the neighborhood o the point M 0. Each collection o unctions and lag F is associated with nonzero element [] F o the ield K, which is called Parshin symbol. Reciprocity laws state that or some chosen (precisely described) inite sets o lags L the product [] F F L is equal to 1. Let n = 1 and maniold M be nonsingular algebraic curve. In this case: 1) lag F = {M 0 M} is deined by point a = M 0, 2) Parshin symbol [] F o the pair o unctions = 1, 2 on lag F coincides with Weil symbol [ 1, 2 ] a, 3) the only chosen set o lags L is equal to the union o supports o the divisor o unctions 1, 2, 4) reciprocity law or this set coincides with the Weil reciprocity law. In the case o K = C Brylinski and McLaughlin 2 proved multidimensional multiplicity laws topologically and ound topological generalization o Parshin symbols. Their topological construction heavily uses shea theory and is not intuitive. About 10 years ago I have ound 4 explicit ormula or the product o roots o a system o algebraic equations with general enough set o Newton polyhedrons in the group (C ) n. This ormula (which is multi-dimensional gener-

4 88 A. Khovanskii alization o Vieta ormula) uses Parshin symbols. Its proo however is based on simple geometry and combinatorics and does not use Parshin theory The logarithmic unctional The search or ormula or the product o the roots o a system o equations convinced me that or K = C there should be intuitive geometric explanation o Parshin symbols and reciprocity laws. Such explanation based on multidimensional logarithmic unctional was inally ound in Re. 5. This paper is devoted to one-dimensional case. One-dimensional logarithmic unctional (word one-dimensional we will omit urther) associates each one-dimensional cycle γ in one-dimensional complex X and piecewisesmooth mapping (, g) : X (C ) 2 into group (C ) 2 with a complex number, deined up to an integer additive term (i.e., with an element o the group C/Z). Logarithmic unctional is direct generalization o usual logarithm. Zero-dimensional logarithmic unctional (see Section 7.1) reduces to the usual logarithmic unction. All theorems about logarithmic unctional translate almost automatically to multidimensional case. I started presentation rom one-dimensional case because o the ollowing reasons: 1) The Weil reciprocity law is ormulated much simpler than Parshin reciprocity laws. That s why the introduction o the logarithmic unctional is much clearer in one-dimensional case. 2) Properties o logarithmic unctional and methods o their prove become apparent enough already in one-dimensional case. 3) In one-dimensional case there already exist simple Beilinson unctional giving clear topological proo o the reciprocity law. It has only one drawback: it is not transparent how to generalize it or multi-dimensional case. One o the goals o this publication is to compare logarithmic unctional and Beilinson unctional. The later is deined or a pair o rational unctions, g and one-dimensional cycle γ on complex algebraic curve X. Generalizing Beilinson unctional we deine logarithmic unctional in Beilinson orm, or in short LB-unctional, or a pair o smooth unctions, g and one-dimensional cycle on real maniold M. LB-unctional has many properties o Beilinson unctional, however there are some dierences too: when unctions, g are ixed, LB-unctional not always gives a class o one-dimensional cohomology group o the maniold M. For this it is required that the orm d dg is identical to zero on the maniold M. For the rational unctions, g on complex curve X the identity d dg 0 holds automatically. This act plays a key role in the proo o the reciprocity law with the help o LB-unctional.

5 Logarithmic Functional and the Weil Reciprocity Law 89 We show that logarithmic unctional is always representable as LBunctional. This representation relates logarithmic unctional with Beilinson unctional Organization o material In Section 2 the Weil reciprocity law is given. In section 3 LB-unctional is deined or a pair o complex-valued unction on a circle. In Section 4 LB-unctional is deined or a mapping (, g) : M (C ) 2 o the maniold M into group (C ) 2 and closed oriented curve on maniold M. Section 5 discusses Beilinson s proo o the Weil theorem. In Section 6 LB-unctional is deined or the mapping (, g) : M (C ) 2 and one-cycle on the maniold M, being an image o one-cycle γ in one-dimensional complex X o piecewise-smooth mapping φ : X M. Section 7 deines logarithmic unctional and proves its main properties. Section 8 shows that logarithmic unctional can always be represented in the orm o LB-unctional. 2. Formulation o the Weil reciprocity law Let Γ be a connected compact one-dimensional complex maniold (another words, Γ is irreducible regular complex algebraic curve). Local parameter u near point a Γ is deined as an arbitrary meromorphic unction u with zero o multiplicity 1 at the point a. Local parameter is a coordinate unction in small neighborhood o the point a. Let ϕ be a meromorphic unction on the curve Γ and k m c mu m be a Laurent series with respect to local parameter u near point a. We will call the leading monomial the irst nonzero term o the series, i.e. χ(u) = c k u k. The leading monomial is deined or any meromorphic unction ϕ not identical to zero. For every pair o meromorphic unctions, g not identical to zero on a curve Γ, and every point a Γ the Weil symbol [, g] a is deined. It is nonzero complex number given by [, g] a = ( 1) nm a n mb m n, where a m u m and b n u n are leading monomials on parameter u o unctions and g at the point a. Weil symbol is deined with the help o parameter u but it does not depend on the choice o this parameter. Let v be another local parameter near point a and let cv be the leading monomial o the unction u on parameter v, i.e. u = cv+.... Then a m c m v m and b n c n v n are leading monomials o unctions and g on parameter v.

6 90 A. Khovanskii The equality a n mb m n = (a m c m ) n (b n c n ) m proves the correctness o the deinition o Weil symbol. As it is seen rom the deinition Weil symbol multiplicatively depends on unctions and g. Multiplicativity on means that i = 1 2 then [, g] a = [ 1 g] a [ 2 g] a. Multiplicativity with respect to g is deined analogously. For every pair o meromorphic unctions, g not identical to zero on a curve Γ, the Weil symbol [, g] a diers rom one only in inite set o points a. Indeed Weil symbol can dier rom one only on the union o supports o the divisors o unctions and g. The Weil reciprocity law. (see...) For every pair o meromorphic unctions, g not identical to zero on an irreducible algebraic curve Γ, the equality [, g]a = 1 holds. Here product is taken over all points a o the curve Γ. Ininite product above makes sense since only inite number o terms in it are dierent rom one. Simple algebraic prove o the law can be ound in Re. 3. Simple topological proo based on properties o LB-unctional described in Sections 3 and 4, can be ound in Section 5 (this proo is reormulation o Beilinson s reasoning rom Re. 1). 3. LB-unctional o the pair o complex valued unctions o the segment on real variable Let J be a segment a x b o the real line. For any continuous on segment J unction having non-zero complex values : J C, where C = C \ 0, denote by ln any continuous branch o the logarithm o. Function ln is deined up to an additive term 2kπi, where k is an integer. I (a) = (b) then we can deine an integer number deg J degree o mapping : J/ J S 1 o the circle J/ J, obtained rom segment J by identiying its ends a and b, to the unit circle S 1 C. Obviously, deg J = 1 (ln (b) ln (a)). Consider a pair o piecewise-smooth complex-valued unctions and g, having no zero values on the segment J, and (a) = (b), g(a) = g(b). For such pair o unctions we will call LB-unctional the complex number

7 Logarithmic Functional and the Weil Reciprocity Law 91 LB J (, g) given by ormula LB J (, g) = 1 b () 2 ln dg g 1 deg J a ln g(b). Lemma 3.1. LB-unctional is deined up to an integer additive term and is well deined element o the group C/Z. Proo. Function ln is deined up to an additive term 2kπi, dg deg J g, and deg J dg g 1 () 2 b a ln dg g b a dg g = is an integer number. This means that the number is deined up to an integer additive term. The value ln g(b) is deined up to an additive term 2mπi and deg J d means that the number 1 deg J is an integer number. This ln g(b) is deined up to an integer. Now we give slightly more general ormula or LB-unctional. Let a = x 0 < x 1 < x n = b be an increasing sequence o points on the segment [a, b]. Given a continuous unction : J/ J C we construct a discontinuous unction φ, which is equal to one o continuous branches ln j o the logarithm o on each interval J j deined by inequalities x j < x < x j+1. Let jump o unction φ at point x j be an integer number m φ (x j ) = 1 ( lim t x + j φ(x) lim t x j φ(x)) or j = 1,..., n 1, and at point x n an integer number m φ (x n ) = 1 ( lim φ(x) lim φ(x)). t x + n t x 0 Lemma 3.2. Let or unctions, g and segment J LB-unctional be deined. Then the equality LB J (, g) = 1 () 2 φ dg g 1 n m φ (x i ) ln g(x i ) holds. J i=1 Proo. First show that LB J (, g) = 1 n n 1 dg () 2 ln i g + ln i (x i ) ln g(x i ) i=0 J i i=0 n ln i (x i+1 ) ln g(x i ). Let s change unction φ on interval J j by adding to the branch o ln j number 2kπi leaving φ without changes on other intervals. As the result the value o LB-unctional above is incremented by the number i=1

8 92 A. Khovanskii 2kπi () 2 ( J i dg g + ln g(x i) ln g(x i+1 ) ), which is equal to zero. Since the change o the branch o the logarithm on any o the intervals J j does not aect the result, we can assume that the unction φ is taken as a continuous branch o the logarithm o unction on the whole segment J = [a, b]. In this case the ormula or the LB-unctional holds (it coincides with the deinition o the LB-unctional). The claim o lemma ollows, since m φ (x j ) = ln j (x j ) ln j+1 (x j ) or j = 1,..., n 1, and m φ (x n )(φ) = ln n (x n ) ln 0 (x 0 ) I one o unctions or g is constant then it is easy to compute LBunctional. The ollowing is obvious Lemma 3.3. Let or unctions, g and segment J LB-unctional be deined. I C then LB J (, g) = 1 ln C deg J g g. I g C then LB J (, g) = 1 ln C deg J. The ollowing obvious lemma shows that under change o variable LBunctional behaves as an integral o a dierential orm. Lemma 3.4. Let or unctions, g and segment J unctional LB J (, g) be deined, and let φ : J 1 J be a piecewise-smooth homeomorphism o segment J 1 into segment J. Then unctional LB J1 ( φ, g φ) is deined. Additionally, i φ preserves orientation then LB J1 ( φ, g φ) = LB J (, g), and i φ changes orientation then LB J1 ( φ, g φ) = LB J (, g). Now we discuss how the LB-unctional changes under homotopy o the pair o unctions, g. Let I be the unit segment 0 t 1 and F : I J C, G : I J C be piecewise-smooth unctions such, that F (t, a) = F (t, b), G(t, a) = G(t, b) or every ixed t. Let 0 (x) = F (0, x), 1 (x) = F (1, x), g 0 (x) = G(0, x), g 1 (x) = G(1, x). Theorem 3.1. The equality holds: LB J ( 1, g 1 ) LB J ( 0, g 0 ) = 1 () 2 I J df F dg G. Proo. Dierential o the orm ln F dg G is equal to df F dg G. Using Stokes

9 ormula we get I J df F dg b G = a ln 1 dg 1 g 1 Logarithmic Functional and the Weil Reciprocity Law 93 b a ln 0 dg 0 g (ln F (t, b) ln F (t, a)) dg G. (1) Since the degree o the mapping is homotopy invariant, the dierence ln F (t, b) ln F (t, a) does not depend on parameter t and is equal to both numbers deg 1 J 1 and deg J 0 0. Thereore 1 0 (ln F (t, b) ln F (t, a)) dg G = deg J and the statement o the theorem ollows. 1 1 ln g 1(a) deg J 0 0 ln g 0(a) Corollary 3.1. Let and g be piecewise-smooth unctions on real line periodic with period A = b a with values in C and let J c be a segment a + c x b + c o the length A. Then LB Jc (, g) does not depend on the choice o the point c. Proo. Consider homotopy F (t, x) = (t + x), G(t, x) = g(t + x). This homotopy preserves LB-unctional since df dg 0. From previous theorem the proo ollows. Lemma 3.5. The equality LB J (, g) = LB J (g, ) holds. Proo. Using equalities ln(b) = ln(a) + deg J deg J and Newton-Leibnitz ormula we get: g g b a On the other hand b d[ln ln g] = deg J a () 2 deg J d[ln ln g] = Two expressions or the integral ln g(a) + deg J b a b a deg J g g. ln g d + a b (2), ln g(b) = ln g(a) + ln dg g. d[ln ln g] must coincide. g ln (a)+ (3) g

10 94 A. Khovanskii Lemma 3.6. For any piecewise-smooth unction : J C, having equal values at the end-points a and b o the segment J, LB J (, ) = 1 2 deg J Proo. Substituting g = in (3) we obtain b 2 a 1 ln d = 2 deg J The claim ollows, since deg 2 J deg J ln (c) + deg2 J mod 2... Lemma 3.7. For any three piecewise-smooth unctions, ϕ, g with values in the group C on segment J with end-points a and b, such that (a) = (b), ϕ(a) = ϕ(b), g(a) = g(b), the equality holds. LB J (ϕ, g) = LB J (, g) + LB J (ϕ, g) Proo. In order to prove this it is enough to use the ollowing acts: 1) in group C/Z the equality 1 ln(ϕ) = 1 ln + 1 ln ϕ holds. 2) or any pair o continuous unctions and ϕ that do not have zero values ϕ and (a) = (b), ϕ(a) = ϕ(b), the equality deg J ϕ = deg J + deg J ϕ ϕ holds. The ollowing lemma is obvious and we only give a ormulation o it. Lemma 3.8. Let segment J with end-points a and b is split by a point c (a < c < b) into two segments: J 1 with end-points a, c and J 2 with endpoints c, b. Let, g be a pair o piecewise-smooth unctions on J with values in group C such, that (a) = (c) = (b), g(a) = g(c) = g(b). Then the unctionals LB J (, g), LB J1 (, g), LB J2 (, g) are deined and LB J (, g) = LB J1 (, g) + LB J2 (, g). 4. LB-unctional o the pair o complex valued unctions and one-dimensional cycle on real maniold Let M be a smooth real maniold and K(M) multiplicative group with elements being smooth complex-valued unctions on M not having values o 0. Let γ : J M be piecewise-smooth closed curve on M. Element I J (γ, γ g) o the group C/Z will be called LB-unctional o the pair o unctions, g K(M) and oriented closed curve γ and will be denoted as LB γ (, g).

11 Logarithmic Functional and the Weil Reciprocity Law 95 Lemma 4.1. LB-unctional o the pair o unctions, g K(M) and oriented closed curve γ does not change under orientation preserving reparametrization o the curve γ. Proo. For re-parametrization o the curve γ preserving the end-point γ(a) = γ(b) the statement is proved in Lemma 3.4. Independence o the LB-unctional rom the choice o point γ(a) = γ(b) is proved in Corollary 3.1. An element LB γ (, g) o the group C/Z, deined by ormula LB γ (, g) = ki LB γi (, g), is called LB-unctional o pair o unctions, g K(M) and cycle γ = k i γ i, where k i Z and γ i parameterized closed curves on maniold M. For any cycle γ in maniold M and any unction K(M) denote by deg γ the degree o mapping : γ S1 o the cycle γ into unit circle S 1. Theorem 4.1. For any cycle γ in the maniold M the ollowing equalities hold: 1) LB γ (, g) = LB γ (g, ) or any pair o unctions, g K(M). 2) LB γ (ϕ, g) = LB γ (, g) + LB γ (ϕ, g) or any, ϕ, g K(M). 3) LB γ (, ) = 1 2 deg γ or any K(M). 4) LB γ (C, ) = LB γ (, C) = ln C deg γ or any unction K(M) and any non-zero constant C. Proo. Statement 1) ollows rom Lemma 3.5, statement 2) ollows rom Lemma 3.7, statement 3) rom Lemma 3.6 and statement 4) rom Lemma 3.3. Theorem 4.2. I or a pair o unctions, g K(M) on maniold M the equality d dg 0 holds, then LB-unctional LB γ (, g) depends only on homology class o the cycle γ = k i γ i. Proo. According to Theorem 3.1 under conditions o Theorem 4.2 LBunctional along closed curve does not change under homotopy o the curve. By homotopying i necessary every component o the cycle γ it is possible to assume that cycle consists rom closed curves, passing through the ixed point c. Consider the undamental group π 1 (M, c) o the maniold M with basis point c. For unctions and g satisying condition o the theorem, the mapping I : π 1 (M, c) C/Z o the closed curve γ π 1 (M, c) into element

12 96 A. Khovanskii I γ (, g) o group C/Z is deined correctly and is group homomorphism. Any homomorphism o the undamental group into Abel group is passed trough the homomorphism o the undamental group into group o one-dimensional homologies. 5. Topological proo o the Weil reciprocity law Beilinson s unctional is a particular case o LB-unctional, in which, g is a pair o analytic unctions on one-dimensional complex maniold, and γ is one-dimensional real cycle on this maniold. Under this restrictions we will call LB-unctional Beilinson s unctional and write B γ (, g) instead o LB γ (, g). Example 5.1. (Beilinson, see Re. 1) Let M be one-dimensional complex maniold and K a (M) K(M) be a subgroup o group K(M) consisting o analytic unctions not equal to 0 anywhere. For any two unctions, g K a (M) the equality d dg 0 holds. In accordance with Theorem 4.2, or any pair o unctions, g K a (M) one-dimensional co-chain on maniold M, that associates with a cycle γ element B γ (, g) C/Z, is an element o H 1 (M a, C/Z). According to Theorem 4.1, this class has the ollowing properties: 1) B γ (, g) = B γ (g, ) or any pair o unctions, g K a (M a ). 2) B γ (ϕ, g) = B γ (, g) + B γ (ϕ, g) or any, ϕ, g K a (M a ). 3) B γ (, ) = 1 2 deg γ or any K a(m a ). 4)B γ (C, ) = B γ (, C) = ln C deg γ or any unction K a(m a ) and arbitrary nonzero constant C. Let X be a connected compact one-dimensional complex maniold with a boundary, and, g be nonzero meromorphic unctions on the maniold X, that are regular on the boundary γ = X o maniold X and not equal to zero in the points o boundary X. Under these assumptions the ollowing lemma holds. Lemma 5.1. Let a single valued branch o the unction ln exists on the maniold X. Then B-unctional B γ (, g) o unctions, g and o boundary o the maniold X is equal to ord a g 1 ln (a), where ord ag is order o p X meromorphic unction g at the point a. (The ininite sum appearing above is well deined since only initely many terms in this sum are not equal to zero.)

13 Logarithmic Functional and the Weil Reciprocity Law 97 Proo. The statement ollows rom the equality 1 () 2 ln dg g = 1 (ord p g) ln (p). X Let U C be a simply-connected domain with smooth boundary γ, containing point 0 U. Let, g be meromorphic unctions in the domain U, such that their restrictions on punctured domain U \ 0 are analytic unctions, not taking values o 0. Let a 1 z k, a 2 z m be leading terms o Laurent series o unctions and g at the point 0: = a 1 z k +..., and g = a 2 z m p X Lemma 5.2. Under the listed above assumptions B γ (, g) = km 2 + m ln a 1 k ln a 2. Proo. Represent, g as = 1 z k, g = g 1 z m, where 1, g 1 are analytic in the domain U. Observe, that 1 (0) = a 1 0, g 1 (0) = a 2 0. Using multiplicativity property o B-unctional we get B γ (, g) = kmb γ (z, z) + mb γ ( 1, z)+kb γ (z, g 1 )+B γ ( 1, g 1 ). Using Lemma 5.1 and skew-symmetric property o B-unctional we get B γ ( 1, z) = ln a 1, B γ (z, g 1 ) = ln a 2 and B γ ( 1, g 1 ) = 0. Further B γ (z, z) = 1 2 deg γ z z = 1 2. Topological proo o the Weil theorem is actually completed. We just need to reormulate obtained results. Group C/Z is isomorphic to the multiplicative group C o the ield o the complex numbers. Required isomorphism is given by mapping τ : C/Z C deined by ormula τ(a) = exp(a). For any one-dimensional cycle γ in the maniold M and or any pair o unctions, g K(M) we call exponential B-unctional Bγ (, g) an element o the group C, deined by ormula B γ (, g) = exp(b γ (, g)). Using the notion o exponential B-unctional the last Lemma can be reormulated as ollows Lemma 5.3. Under assumptions o Lemma 5.2, exponential B-unctional B γ (, g) coincides with the Weil symbol [, g] 0 o unctions and g at point 0. Let Γ be a compact complex curve,, g meromorphic unctions on Γ such that they are not identical to zero on any connected component o Γ, D a union o supports o divisors o unctions and g. Let U p be a small open disk containing point p D, and γ p = U p be the border o disk U p, U = U p, γ = γ p. Consider maniold W = M \ U. Cycle γ homologically p D p D

14 98 A. Khovanskii equals to zero on maniold W, since W = γ. Thereore, the equality B γ (, g) = 0 holds and exp(b γ (, g)) = exp(b γp (, g)) = 1. By Lemma 5.3 we have exp(b γp (, g)) = [, g] p which proves the Weil theorem. p P 6. Generalized LB-unctional In this section we will give another (more general, but in act equivalent) deinition o LB-unctional. It will be easier to generalize to multidimensional case. Any one-dimensional cycle in (C ) 2 (as in any other maniold) can be viewed as integer combination o oriented closed curves, i.e. as integer combination o images o oriented circle. Thereore, it is suicient to deine LB-unctional or a pair o unctions on a standard circle, as it was done above. When n > 1 n-dimensional cycle in maniold M can be viewed as an image o the mapping F : X M o some n-dimensional cycle γ in some n-dimensional simplicial complex X. Here we give a deinition o LBunctional or an image in the group (C ) 2 o one-dimensional cycle, laying in one-dimensional simplicial complex X under mapping F : X (C ) 2. Let X be one-dimensional simplicial complex. Let us ix orientation on each edge o complex X. Denote by S( j, Q p ) the incidence coeicient between edge j and vertex Q p. It is equal to zero i and only i vertex Q p does not belong to the edge j, and it is equal to +1 or -1 depending on the sign o Q p as a boundary o oriented edge j. For any continuous unction : X C on edges o complex X it is possible to choose single-valued branches o unction ln. We will need a deinition o jump unction or such a collection o branches o unction ln on one-dimensional cycle γ = k j j o complex X. Let a singlevalued branch ln j : j C o the multi-valued unction ln be ixed on the edge j o complex X. Denote by φ a collection o branches ln j. We will view collection φ as discontinuous unction on complex X (the collection φ deines a unction on the complement o complex X to the set o its vertices, but at vertices o complex X this unction is multi-valued). Deine on vertices o complex X jump unction or unction φ and cycle γ. I vertex Q p is adjacent to the edge j then on the vertex Q p a restriction o unction ln j given on the edge j is deined. We deine the value o the jump unction m φ,γ or unction φ and cycle γ = k j j at the vertex Q p o complex X by ormula m φ,γ (Q p ) = 1 S( j, Q p )k j ln j (Q p ),

15 Logarithmic Functional and the Weil Reciprocity Law 99 where summation is done over all edges j o complex X. Lemma 6.1. The jump unction is an integer valued unction on the vertices o complex X. Proo. On every edge j adjacent to the vertex Q p we can choose branches ln j so they are equal at Q p. With such choice o the unction φ the number m φ,γ (Q p ) is equal to zero, because the chain γ = k j j is a cycle. With the change o branches ln j the change in the value o jump unction is integer number. Consider piecewise-smooth mapping (, g) : X (C ) 2 o one-dimensional complex X and one-dimensional cycle γ = k j j in X. Let φ = {ln j } be a collection o single-valued branches o unction ln on edges j o complex X. For pair o unctions, g and cycle γ we will call LB-unctional the complex number LB γ (, g) deined by ormula LB γ (, g) = 1 () 2 φ dg g 1 m φ,γ (Q) ln g(q), γ Q V where summation is over set V o all vertices o complex X and m φ,γ is jump unction or unction φ and cycle γ. The ollowing lemmas a proved similarly to Lemma 3.1 and 3.2. Lemma 6.2. LB-unctional LB γ (, g) is deined up to an integer additive term and is correctly deined element o the group C/Z. Lemma 6.3. LB-unctional LB γ (, g) or the mapping (, g) : X (C ) 2 does not change under a simplicial subdivision o the complex X. Let discuss the change o LB-unctional under homotopy o the mapping (, g). Let I be a unit segment 0 t 1 and (F, G) : I X (C ) 2 be a piecewise-smooth mapping, which coincides with the mapping ( 0, g 0 ) : X (C ) 2 when t = 0 and with the mapping ( 1, g 1 ) : X (C ) 2 when t = 1. Similarly to the Theorem 3.1 we can prove Theorem 6.1. The equality holds. LB γ ( 1, g 1 ) LB γ ( 0, g 0 ) = 1 () 2 I γ df F dg G

16 100 A. Khovanskii 7. Logarithmic unction and logarithmic unctional In this section we deine logarithmic unctional, ormulate and prove its major properties. Deinitions and results o this section can be easily extended to the multi-dimensional case and we plan to return to this topic in uture publications Zero-dimensional logarithmic unctional and logarithm In order to highlight an analogy between logarithm and logarithmic unctional we will deine zero-dimensional unctional here. Consider group C with coordinate z. Group C is homotopy-equivalent to the circle T 1, deined by the equation z = 1. Denote by e = 1 the unit element o the group C. Let X be inite set o points and K(X) multiplicative group with elements being complex-valued unctions on X having no 0 values. Thus element o the set K(X) is a mapping : X C. Let D be zero-dimensional cycle rom the group H 0 (X, Z), i.e. D is a linear combination o points rom the set X with integer coeicients: D = k j x j, where k j Z i x j X. Any mapping : X (C ) 2 o the inite set X into group C is homotopic to the mapping o X into point e since group C is connected. We will call zero-dimensional logarithmic unctional a unctor that associates with the pair, D, consisting o unction K(M) and zerodimensional cycle D = k j x j, the complex number ln(, D) = 1 where γ is arbitrary piecewise-smooth curve in group C with the boundary γ equal to k j (x j ) ( k j )e. Obviously, the number ln(, D) is deined up to an additive integer term and is correctly deined element o the group C/Z. The ollowing ormula holds ln(, D) = 1 kj ln (x j ), where ln (x j ) is any o values o multi-valued unction ln at point x j. Thus zero-dimensional logarithmic unctional reduces to the usual logarithmic unction. It has the ollowing multiplicativity property: or any pair o unctions, g K(X) and any zero-dimensional cycle D the equality ln( g, D) = ln(, D) + ln(g, D) holds. For any ixed unction γ dz z,

17 Logarithmic Functional and the Weil Reciprocity Law 101 K(X) and any pair o zero-dimensional cycles D 1, D 2 the equality ln(, (D 1 + D 2 )) = ln(, D 1 ) + ln(, D 2 ) holds. Another words, or ixed unction unctional ln(, D) is zero-dimensional co-chain with the values in group C/Z. Let M be real maniold and : X C be a continuous mapping. Let s associate with every zero-dimensional cycle D = k j x j (k j Z, x j M) on the maniold M an element ln(, D) o the group C/Z deined by ormula ln(, D) = 1 kj ln (x j ). It is obvious that co-chain ln(, D) deines an element o zero-dimensional cohomology group o maniold M with coeicients in group C/Z i and only i d 0 (i.e. i and only i the unction is constant on each connected component o the maniold M) Properties o one-dimensional logarithmic unctional In this subsection we deine one-dimensional logarithmic unctional and ormulate its important properties. As beore we will use the term logarithmic unctional skipping word one-dimensional. Consider group (C ) 2 with co-ordinate unctions z 1 and z 2. Group (C ) 2 is homotopy equivalent to the torus T 2 (C ) 2, deined by equations z 1 = z 2 = 1. On group (C ) 2 there is remarkable 2-orm ω = 1 () 2 dz 1 z 1 dz 2 z 2. Restriction o orm ω to the torus T 2 is real 2-orm ω T 2 = 1 4π 2 d(arg z 1) d(arg z 2 ), that is not equal to zero nowhere. Integral o orm ω over the torus T 2, oriented by the orm d(arg z 1 ) d(arg z 2 ), is equal to 1. Deine by Id a subset o group (C ) 2 consisting o points (z 1, z 2 ) such, that one o coordinates is equal to 1, i.e. Id = {z 1 = 1} {z 2 = 1}. Let X be one-dimensional simplicial complex and let γ = k j j be integer linear combination o its oriented edges j which constitutes a cycle, i.e. γ = 0. Let (, g) : X (C ) 2 be a piecewise-smooth mapping o complex X into the group (C ) 2. Logarithmic unctional is a unctor, that associates with the mapping (, g) : X (C ) 2 and one-dimensional cycle γ on X an element ln(, g, γ) o the group C/Z, deined by ormula ln(, g, γ) = 1 () 2 σ dz 1 dz 2 = ω, z 1 z 2 σ

18 102 A. Khovanskii where σ is 2-chain in (C ) 2, with the boundary σ equal to the dierence o the image (γ) o cycle γ under mapping = (, g) and some cycle γ 1 lying in the set Id. We list the major properties o the logarithmic unctional: 1) The value o the logarithmic unctional is correctly deined element o the group C/Z. 2) Logarithmic unctional skew-symmetrically depends on components and g, i.e. ln(, g, γ) = ln(g,, γ). 3) Logarithmic unctional multiplicatively depends on components and g, i.e. ln(φ, g, γ) = ln(, g, γ) + ln(φ, g, γ), ln(, gφ, γ) = ln(, g, γ) + ln(, φ, γ). 4) Logarithmic unctional has the ollowing topological property. Let M be a real maniold and (, g) : M C be a smooth mapping. One-dimensional cycle γ on maniold M can be viewed as an image under some piecewisesmooth mapping φ : X M o some one-dimensional cycle γ = k j j on some one-dimensional complex X, where k j Z and j are edges o complex X. Associate with the cycle γ on maniold M an element ln( φ, g φ, γ) o group C/Z. Obtained with this association one-dimensional co-chain gives a class o one-dimensional cohomology group o maniold M with coeicients in group C/Z i and only i d dg 0 (i.e. i and only i the dierential o the mapping (, g) : M (C ) 2 degenerates at each point o maniold M). 5) The equality ln(, g, γ) = LB γ (, g) holds. One one hand, this equality gives a ormula or logarithmic unctional that does not use auxiliary 2-chain σ. On the other hand, this equality gives geometric sense to LB-unctional and shows that LB-unctional is analog o logarithm Prove o properties o logarithmic unctional This subsection will give prove o properties 1) 4). Property 5) will be proved in next section. It is easy to see that sets Id (deined in previous subsection) and (C ) 2 have the ollowing topological properties: 1) The set Id is homotopy equivalent to the boucquet o two circles, π 1 (Id) ree group with two generators and H 1 (Id, Z) Z + Z. 2) The set (C ) 2 is homotopy equivalent to the real torus T 2 and π 1 ((C ) 2 ) H 1 ((C ) 2, Z) Z + Z.

19 Logarithmic Functional and the Weil Reciprocity Law 103 3) Enclosure π : Id (C ) 2 induces isomorphism o one-dimensional homologies and mapping onto o the undamental groups o sets Id and (C ) 2. It ollows rom item 3) that any continuous mapping : X (C ) 2 o one-dimensional simplicial complex X into group (C ) 2 is homotopic to the mapping o complex X into the set Id (C ) 2. We now prove more precise statement. Let V be a set o vertices o one-dimensional complex X and E be the set o its edges. Fix inside each edge τ j E two dierent points A j and B j. Points A j, B j split edge τ j into three segments, which intersect at the ends only central segment with endpoints A j and B j and two boundary segments. Fix also in each edge τ j point O j, laying inside central segment [A j, B j ]. Denote by X 1 the union o all central segments on all edges τ j, and by X 0 the union o all extreme segments on all edges τ j. As it is seen rom this construction, every connected component o the set X 1 is a central segment o one o the edges τ j. Every connected component o the set X 0 contains one o the vertices o complex X and is the union o all extreme segments, containing this vertex. So, every connected component o the sets X 0 and X 1 is contractible. Lemma 7.1. Any continuous (piecewise-smooth) mapping (, g) : X (C ) 2 o one-dimensional complex X into group (C ) 2 is homotopic (piecewise-smoothly homotopic) to the mapping ( 1, g 1 ) : X (C ) 2 such, that unction 1 is equal to 1 on the set X 1 and unction g 1 is equal to 1 on the set X 0. Proo. First we will homotopically change unction without changing unction g. Since the set C is connected we can assume that the ollowing condition is satisied: (O j ) = 1 at the chosen point O j on every edge τ j. I this is not so, then we can homotopically change unction in such a way, that in the process o homotopy t values t (O j ) move along the curves connecting points O j with 1. Let this condition holds. It is possible to construct a homotopy φ t : X X o the identity mapping o the complex X into itsel such, that it leaves intact every vertex o the complex, translates every edge τ j into itsel and contracts every central segment [A j, B j ] into the point O j. Homotopy t = φ t translates unction into unction 1 which is equal to 1 on the set X 1. Analogously, without changing unction, we can homotopically change unction g into the unction g 1 having required properties. We can assume, that unction g is equal to 1 at every

20 104 A. Khovanskii vertex (otherwise it can be homotopically changed into the unction having this property). It is possible to construct homotopy ψ t : X X o the identity mapping o the complex X into itsel such, that it leaves intact every vertex o the complex, translates every edge τ j into itsel and contracts each o boundary segments into the vertex that is contained in this segment. Homotopy g t = g ψ t translates unction g into unction g 1 which is equal to 1 on the set X 0. On group (C ) 2 there is remarkable 2-orm ω = 1 () 2 dz 1 z 1 dz 2 z 2. Lemma 7.2. For any 2-cycle σ in (C ) 2 integral ω is an integer number. σ Proo. Consider projection ρ : (C ) 2 T 2 o the group (C ) 2 into torus T 2, given by ormula ρ(z 1, z 2 ) = ( z1 z, z2 1 z ). It is clear, that integral 2 σ ω is equal to the degree o the mapping ρ : σ T 2, which is restriction o projection ρ onto cycle σ. Thus, this integral is an integer number. Theorem 7.1. The value ln(, g, γ) o the logarithmic unctional is correctly deined element o the group C/Z (i.e. when 2-chain σ in the deinition o unctional changes, the value can be incremented by integer number only). Proo. Since embedding π : Id (C ) 2 induces isomorphism π o groups H 1 (Id, Z) and H 1 ((C ) 2, Z), or the image (, g) γ o cycle γ there exists homological cycle γ 1 laying in the set Id, i.e. there exists 2-chain σ 1 such, that (, g) γ γ 1 = σ 1. Let γ 2 be another cycle in Id that is homological to the cycle (, g) γ and σ 2 be such 2-chain, that (, g) γ γ 2 = σ 2. Since isomorphism π has no kernel, cycles γ 1 and γ 2 are homological in the set Id, i.e. there exists a chain σ 3 laying in Id such, that γ 1 γ 2 = σ 3. By the construction the chain σ = σ 2 σ 1 σ 3 has zero boundary and thereore is a 2-cycle in (C ) 2. By Lemma 7.2 integral ω is an integer number. On σ the other hand σ ω = σ 2 ω σ 1 ω σ 3 ω. The chain σ 3 is laying in the set Id. Restriction o the orm ω on Id is identically equal to zero, thereore σ 3 ω = 0. Thus, the dierence σ 2 ω σ 1 ω is an integer number. Theorem 7.2. Logarithmic unctional is skew-symmetrical, i.e. ln(, g, γ) = ln(g,, γ).

21 Logarithmic Functional and the Weil Reciprocity Law 105 Proo. Let R : (C ) 2 (C ) 2 be a mapping that swaps co-ordinates, i.e. let R(z 1, z 2 ) = (z 2, z 1 ). Under this mapping the orm ω changes the sign, i.e. R ω = ω. Mapping R translates the set Id into itsel. Let σ be a 2-chain in (C ) 2 such, that the cycle ( σ) (, g) (γ) is contained in the set Id. Then ln(, g, γ) = σ ω. Further, ln(g,, γ) = R(σ) ω = σ R ω = ω = ln(, g, γ). σ Theorem 7.3. Logarithmic unctional has the ollowing multiplicative properties: ln(ϕ, g, γ) = ln(, g, γ) + ln(ϕ, g, γ), ln(, gϕ, γ) = ln(, g, γ) + ln(, ϕ, γ). Proo. Since the unctional is skew-symmetrical, it is suicient to prove the irst equality ln(ϕ, g, γ) = ln(, g, γ) + ln(ϕ, g, γ). For evaluation o ln(, g, γ) and ln(ϕ, g, γ) we will choose speciic 2-chains σ 1 and σ 2. Let X = X 0 X 1 be a covering o the complex X by two closed sets with contractible connected components the same as in Lemma 7.1. We describe the choice o the chain σ 1 irst. Let I = [0, 1] be a unit segment, W = I X and (F, G) : W (C ) 2 be a piecewise-smooth homotopy o the mapping (, g) discussed in Lemma 7.1. Another words, let (F, G) be such mapping, that: 1) restriction (F, G) {1} X o the mapping (F, G) on the set {1} X coincides with the mapping (, g) when sets X and {1} X are identiied, 2) restriction F {0} X1 o unction F on the set {0} X 1 identically equals to 1, 3) restriction G {0} X0 o unction G on the set {0} X 0 identically equals to 1. For the cycle γ = k i i take σ 1 = k i (F, G) (I i ). By construction the boundary o the chain σ 1 is equal to (, g) (γ) γ 1, where γ 1 is a cycle laying in the set Id. Thus ln(, g, γ) = σ 1 ω. Chain σ 2 is constructed analogously. Let (Φ, G) : W (C ) 2 be piecewisesmooth mapping, with component G the same as in above described homotopy, and component Φ having the ollowing properties: 1) restriction Φ {1} X coincides with unction ϕ, 2) restriction Φ {0} X1 o unction Φ on the set {0} X 1 identically equals to 1. For cycle γ = k i i take σ 2 = k i (Φ, G) (I i ). By construction the boundary o the chain σ 2 is equal to (ϕ, g) (γ) γ 2, where γ 2 is a cycle laying in the set Id. Thus ln(ϕ, g, γ) = ω. σ 2 Now we construct chain σ 3. Consider the mapping (F Φ, G) : W (C ) 2 with irst component being the product o unctions F and Φ, and second component equal to unction G. Take σ 3 = k i (F Φ, G) (I i ). By

22 106 A. Khovanskii construction the boundary o the chain σ 3 is equal to the dierence o the cycle (ϕ, g) (γ) and the cycle laying in the set Id. Thereore, ln(ϕ, g, γ) = σ 3 ω. Thus we have the ollowing equalities ln(, g, γ) = 1 ki ln(ϕ, g, γ) = 1 ki ln(ϕ, g, γ) = 1 ki I i I i I i df F dg G, dφ Φ dg G, d(f Φ) (F Φ) dg G. Equality ln(ϕ, g, γ) = ln(, g, γ) + ln(ϕ, g, γ) now ollows rom the identity d(f Φ) (F Φ) = df F + dφ Φ. Let real maniold M and smooth mapping (, g) : M (C ) 2 be given. With each triple, consisting o one-dimensional complex X, one-dimensional cycle γ on X and piecewise-smooth mapping φ : X M, we will associate an element ln( φ, g φ, γ) o group C/Z. In order to prove topological property 4) o the logarithmic unctional described in previous subsection it is suicient to prove the ollowing theorem. Theorem 7.4. Let orm d dg be identically equal to zero on the maniold M and let cycle φ γ homologically equals to zero. Then ln( φ, g φ, γ) = 0. Conversely, i rom homological equality to zero o the cycle φ γ ollows equality ln( φ, g φ, γ) = 0 then on the maniold M orm d dg is identically equal to zero. Proo. I cycle φ γ homologically equals to zero, then on the maniold M there exists piecewise-smooth 2-chain σ such that σ = φ γ. By deinition ln( φ, g φ, γ) = 1 d () 2 dg g. I orm d dg is identically equal to zero, then integrand is equal to zero. Thus, ln( φ, g φ, γ) = 0. Conversely, let on some bi-vector v at some point a o maniold M orm d dg is not equal to zero. Consider arbitrary smooth mapping φ : (R 2, 0) σ

23 Logarithmic Functional and the Weil Reciprocity Law 107 (M, a) o the standard plane R 2 into maniold M that translates point 0 into point a M, and dierential dφ is nonsingular at point 0 and maps plane R 2 into subspace o the tangent space T M a at the point a, containing bi-vector v. Let S ε be a square ε x ε, ε y ε on the plane R 2, and X ε = S ε be its boundary. Let γ be a cycle geometrically coinciding with X ε and oriented as boundary o the square S ε. Obviously, or small enough ε element ln( φ, g φ, γ) o group C/Z is not equal to zero, however cycle φ γ by construction is homologically equal to zero. 8. Logarithmic unctional and generalized LB-unctional In this section we prove the ollowing theorem. Theorem 8.1. The equality ln(, g, γ) = LB γ (, g) holds. We irst prove auxiliary result. Lemma 8.1. Let X 0, X 1 X be subsets o complex X deined in Lemma 7.1, (, g) : X (C ) 2 piecewise-smooth mapping such, that restriction o unction on X 1 and restriction o unction g on X 0 identically equal to 1. Then or any one-dimensional cycle γ in complex X the equality LB γ (, g) = 0 holds. Proo. Consider the ollowing subpartition o complex X: every edge τ j o complex X is split into three edges by points A j, B j belonging to the boundaries o sets X 0 and X 1. Choose collection φ o the branches o unction ln on the edges o sub-partitioned complex X that satisies the ollowing two conditions: 1) on every edge that is connected component o set X 1, the branch o unction ln is identically equal to zero; 2) branches ln on the union o edges, that are connected components o the set X 0, deine on this component continuous unction. Condition 2) can be satisied because every connected component o the set X 0 is contractible. By the deinition o LB-unctional we have LB γ (, g) = 1 () 2 γ φ dg g 1 m φ,γ (P ) ln g(p ), where summation is over set Ṽ o all vertices o sub-partitioned complex X. Now, φ dg g = 0, because integrand is equal to zero. Indeed, on every γ P Ṽ edge rom set X 1 unction φ is equal to zero. On every edge rom set X 0 unction g is identically equal to 1 and thus dg/g = 0. At the vertex P o

24 108 A. Khovanskii the original complex X (beore subpartition), coeicient m φ,γ is equal to 0, because unction φ is continuous at such point P. At every vertex P, belonging to the boundary o sets X 0 and X 1, the number ln g is equal to 2kπ, where k Z, because unction g is equal to 1 on the set X 0. Thereore, 1 the number m φ,γ (P ) ln g(p ) is integer. From all this it ollows that P Ṽ element LB γ (, g) o group C/Z equals to zero. Proo. (o Theorem 8.1). By Lemma 7.1 the mapping (, g) : X (C ) 2 is homotopic to the mapping ( 1, g 1 ) : X (C ) 2, or which restriction o unction 1 on X 1 and restriction o unction g 1 on X 0 are identically equal to 1. Let W = I X and (F, G) : W (C ) 2 be homotopy connecting mappings (, g) and ( 1, g 1 ). By Theorem 6.1 LB γ (, g) LB γ ( 1, g 1 ) = ω, where σ 2-chain on W that is equal to I γ. By Lemma 8.1 LB γ ( 1, g 1 ) = 0. By construction σ = γ γ 1, where cycle γ 1 is laying in the set Id. Thereore, LB γ (, g) = ln γ (, g). The author would like to thank Eugene Zima or his help during preparation o this paper. σ Reerences 1. Beilinson A.A. Higher regulators and values o L-unctions o curves. Functional Analysis and its Applications, 14(2): , McLaughlin D.A. Brylinski J.-L. Multidimensional reciprocity laws. J. reine angew. Math., 481: , Khovanskii A. Newton polytopes, curves on toric suraces, and inversion o weil s theorem. Russian Math. Surveys, 52(6): , Khovanskii A. Newton polyhedrons, a new ormula or mixed volume, product o roots o a system o equations. In Proceed. o a Con. in Honour o V.I. Arnold, Fields Inst. Comm. vol 24, Amer. Math. Soc., pages , USA, Khovanskii A. A Function Log in the Space o n-dimensional Cycles and Parshin Kato Theory. International Conerence on Toric Topology, Osaka City University, Osaka, Japan, May-June Parshin A. N. Local class ield theory, Trudy Mat. Inst. Steklov, volume Parshin A. N. Galois cohomology and Brauer group o local ields, Trudy Mat. Inst. Steklov, volume