Feature Elliptic Functions Functions According According to to Eisenstein and and Kronecker: Kronecker: An An Update Update Newly found notes of lectures by Kronecker Newly found notes of lectures by Kronecker on the work of Eisenstein Pierre Charollois (Université Paris 6, France) and Robert Sczech (Rutgers University, Newark, USA) Pierre Charollois (Université Paris 6, France) and Robert Sczech (Rutgers University, Newark, USA)
This article introduces a set of recently discovered lecture notes from the last course of Leopold Kronecker, delivered a few weeks before his death in December 1891. The notes, written by F. von Dalwigk, elaborate on the late recognition by Kronecker of the importance of the “Eisenstein summation process”, invented by the “companion of his youth” in order to deal with conditionally convergent series that are known today as Eisenstein series. We take this opportunity to give a brief update of the well known book by André Weil (1976) that brought these results of Eisenstein and Kronecker back to light. We believe that Eisenstein’s approach to the theory of elliptic functions was in fact a very important part of Kro- necker’s planned proof of his visionary “Jugendtraum”. Gotthold Eisenstein (1823–1852) and Leopold Kronecker (1823–1891) (left to right) 1 Introduction
Born in 1823, Leopold Kronecker died in Berlin on 29 De- with many appendices, partly written by von Dalwigk, relying cember 18911 at the age of 68, precisely 15 days after deliver- on published papers of Kronecker as well as unpublished pa- ing the last lecture of his university course entitled “On ellip- pers from Kronecker’s “Nachlass”. Dalwigk explicitly men- tic functions depending on two pairs of real variables”. This tions the word “Nachlass” in the manuscript. He is likely to historical information was gathered from recently discovered have had access to that specific document from his colleague lecture notes at the library of the University of Saarbrücken. Hensel in Marburg, who was in possession of all the scientific Before discussing the content and the author of these hand- papers of Kronecker at the time. As we learned from Hasse written notes, we wish to recall the circumstances of their and Edwards [Ed], the personal papers of Kronecker were lost discovery. The original discovery is due to Professor Franz in the chaotic events surrounding World War II. Most proba- Lemmermeyer, who started but did not finish the task of re- bly, they were destroyed by a fire caused by exploding muni- typing the text written in old style German cursive handwrit- tions in an old mine near Göttingen. That mine was used to ing. The manuscript then got lost during a library move. Being store some of the collections of the Göttingen Library in 1945. interested to learn more about Kronecker’s work and gain in- This dramatic event is only part of the long-lasting spell put sight into his so-called “liebster Jugendtraum”, we decided to on the posterity of Eisenstein’s ideas, as predicted by André enlist the help of Simone Schulze at the library in finding it. Weil in his essay [We1]. After a search effort which lasted a few weeks, the complete At any rate, the care and the amount of detail included in set of notes was finally found and the library even produced a the manuscript is extraordinary and leads us to think that it high quality digital copy, which is now available to the public was written with the ultimate intention to publish it as a book. [Cw]. We thank Ms Schulze for her help and we also thank We know from a letter of Kronecker to his friend Georg Franz-Josef Rosselli, who undertook the effort to translate the Cantor, who was the first president of the newly founded Ger- cursive German handwriting (prevailing in the 19th century) man Mathematical Association (DMV) and who invited Kro- into modern German typeface [Cw]. necker to deliver the opening address at the first annual meet- The manuscript was written by Friedrich von Dalwigk, a ing of the DMV in 1891, that Kronecker intended to talk at graduate student at the time, who was just finishing his dis- that meeting about the “forgotten” work of Eisenstein. In that sertation on theta functions of many variables [vD]. Later, letter, Kronecker apologises for not being able to attend the von Dalwigk became a professor of applied mathematics at meeting due to the death of his wife Fanni. It is very likely the University of Marburg (1897–1923). Although the course that the lecture notes in question are an expanded version of only consisted of six lectures (due to the premature death of his intended talk. Kronecker), the whole manuscript is more than 120 pages, Kronecker was indeed a great mathematician who made fundamental contributions to algebra and number theory. We 1 According to the tombstone on his grave, Kronecker died on 23 Decem- mention here only his “Jugendtraum”, which, historically, ber 1891. gave rise to class field theory and to Hilbert’s 12th prob-
8 EMS Newsletter September 2016 Feature lem (the analytic generation of all abelian extensions of a Following Eisenstein, we define given number field), one of the great outstanding problems 1 1 ∞ 1 1 in classical algebraic number theory. The celebrated conjec- φ(u) = lim = + + t + m u u + n u n tures of Stark (published in a sequence of four papers during → ∞ m Z+u, n=1 " − # !∈m EMS Newsletter September 2016 9 Feature and its Cayley transform relates φ(u, ξ) to the Eisenstein function φ. The right side c(u) + i of Equation (4) is essentially the generating function for the e(u) = . c(u) i Bernoulli polynomials. Expanding the left side into a power − series in u, one obtains the Fourier expansion of the Bernoulli The addition formula (2) immediately implies polynomials. The above proof of the addition formula for the e(u)e(v) = e(u + v). cotangent function applies to φ(u, ξ) as well and yields addi- From here, it follows that e(u) = e2πiu and c(u) = cot(πu). tion formulas for the Bernoulli polynomials. Due to the fac- Iterating the addition formula for the cotangent function, i.e. tor e( nξ) in the numerator, the convergence of this series is − c(u)c(v) 1 slightly better than that of the cotangent series but is still con- c(u + v) = − , ditional. A substantial part of the lecture notes is devoted to c(u) + c(v) the study of the elliptic analogue of (4), written in Kronecker’s we obtain the formula notation, Un c(u) c(nu) = , n = 1, 2, 3, 4,..., (3) N M V c(u) e( mξ + nη) n ! " Ser (ξ, η, u, τ) = lim lim − , (5) N + M + u + nτ + m with polynomials Un, Vn Z[t] given explicitly by → ∞ → ∞ n= N m= M ! ∈ " #− #− n n Un(t) = Re (t + i) , Vn(t) = Im (t + i) . where ξ, η is a pair of real variables and τ is a point in the A formula of type (3) was called a “transformation formula” upper half space. The complex variable u must be restricted to Z + τZ C. = στ + ρ in the 19th century. The elliptic analogues of (3) played a the complement of the lattice in Writing u τ ffi σ, ρ, prominent role in the work of Kronecker. Taking u Q with as a linear combination of and 1 with real coe cients ∈ denominator n > 1, we conclude that the numbers c(u) are this series can be viewed as a function of two pairs of real variables (ξ, η), (σ, ρ), the ones referred to in the title of the the roots of Vn(t) = 0, that is, c(u) is a real algebraic number whenever u is a non-integral rational number. One can refine lecture notes. this result and give a more precise proof: Various alternative ways to sum the conditionally conver- gent series (5) are discussed in the manuscript. It is a remark- Proposition 2.2. The number c(u) is an algebraic integer if able fact that, in all cases, the value obtained for the sum is and only if n is not the power of an odd prime. If n = pk, with independent of the limiting process chosen. Kronecker’s main an odd prime p and k > 0, then pc(u) is an algebraic integer. result expresses these series in terms of Jacobi theta series. Let Moreover, c(u) is a unit if and only if neither n nor n/2 is a n2τ 1 power of an odd prime. ϑ(z, τ) = ϑ (z, τ) = e + n z 1 2 − 2 n Z+ 1 $ ' Theorem 2.3 (Kronecker-Weber). The set of real numbers ∈#2 % & 1 n n n c(Q Z) generates the real subfield of the maximal abelian = 2 q 8 sin(πz) (1 q ) 1 q e(z) 1 q e( z) , \ − − − − extension of Q. n 1 (≥ ! "! " Kronecker was interested in generalising these results to with a complex variable z, a point τ in the upper half plane the case of elliptic functions with period lattice being an ideal and q = e(τ). ffi in an imaginary quadratic field. One of the di culties he faced Theorem 2.4 (Kronecker). Suppose 0 < Im u < Im τ and was that the addition formula for elliptic functions is in gen- 0 < ξ < 1. Then, eral algebraic and not rational, as in the case of the cotangent ϑ′(0, τ)ϑ(u + η + ξτ, τ) function. To the best of our knowledge, the proof of the Ju- Ser (ξ, η, u, τ) = e(ξu) . (6) gendtraum as envisioned by Kronecker has never been com- ϑ(u, τ)ϑ(η + ξτ, τ) pleted.2 Except for the case of imaginary quadratic fields and This result is reminiscent of the so-called limit formula the case of the rational number field (both closely related of Kronecker, which is not discussed in the lecture notes but to the work of Kronecker), we do not even know whether deserves to be stated here: let τ, τ′ be two complex numbers Hilbert’s 12th problem has a solution. This is partly related to with Im τ > 0 and Im τ < 0 and let 0 ξ, η < 1 be two ′ ≤ a classical theorem of Weierstrass asserting that every mero- real numbers, not both equal to zero. Writing u = η ξτ, v = − morphic function with algebraic addition law is either elliptic, η ξτ′, the second limit formula is the identity circular or rational. − (τ τ′) e(mξ + nη) Instead of the series defining φ(u), Kronecker preferred to − π τ + τ + study the more general series 2 i , (m n)(m ′ n) #m n 2 ∞ e( nξ) e(ξu) ϑ(u, τ)ϑ(v, τ′) (u v) φ(u, ξ) = − = 2πi , 0 < ξ < 1, (4) = log − πi − , (7) + − η(τ)η( τ ) − τ τ n= u n e(u) 1 ′ ′ #−∞ − − − where u is again a complex number that is not an integer. The where η(τ) refers to the Dedekind eta function and the term introduction of the second variable ξ is very natural and is (m, n) = (0, 0) needs to be excluded from the sum. Again, con- suggested by Fourier analysis. Note that the limiting case vergence is only conditional so a specific order of summation as in (5) needs to be observed. lim φ(u, ξ) = φ(u) iπ ξ 0 − Historically, the notion of complex multiplication of ellip- → ξ>0 tic functions appeared for the first time in the work of Abel. Pages 64–67 of Kronecker’s lecture notes are devoted to the 2 See the comment linked to interpretation (c) in [HH]. task of expressing the elliptic functions used by Abel in terms 10 EMS Newsletter September 2016 Feature of the Kronecker series Ser(ξ, η, u, τ). This is perhaps one of Given a pair of integers a 0, b > 0 and z , w C, the ≥ 0 0 ∈ the highlights of the manuscript and deserves special atten- Eisenstein–Kronecker numbers are defined as tion. It is very likely that Kronecker’s conception of the Ju- e∗ (z , w , τ) = K∗ (z , w , b, τ). gendtraum was the result of a close study of the work of Abel. a,b 0 0 a+b 0 0 When b > a + 2, these numbers include, in particular, the 3 Recent developments values of the absolutely convergent partial Hecke L-series a λ It would be wise to leave the complete discussion of the e∗ (0, 0, τ) = , a,b λb legacy of Eisenstein or Kronecker to serious historians. We λ Z+τZ ∈! offer, instead, a brief survey of several recent developments which should be considered as elliptic analogues of the that feature Eisenstein’s summation process and Kronecker’s Bernoulli numbers. As such, the Eisenstein–Kronecker num- Theorem 2.4 for the series Ser (ξ, η, u, τ) bers can be packaged into a generating series that is the ellip- tic analogue of the cotangent function and its relative φ(u, ξ). Algebraicity and p-adic interpolation of To obtain the nice two-variables generating series, it is enough Eisenstein–Kronecker numbers to translate and slightly alter the Kronecker theta function in It was already observed by Weil in his 1976 book that the order to define combination of the methods of Eisenstein and Kronecker (z0+z)w0+wz0 A gives direct access to Damerell’s classical result (1971) on Θz0,w0 (z, w) = e− Θ(z0 + z, w0 + w). the algebraicity of values of L-functions attached to a Hecke Its Laurent expansion around z = w = 0 displays exactly the Grössencharakter of an imaginary quadratic field. collection of Eisenstein–Kronecker numbers. Weil also anticipated that their methods would extend to Proposition 3.1. Fix z , w C. We have the following Lau- the investigation of the p-adic properties of these algebraic 0 0 ∈ numbers. In the following 10 years, the works of Manin- rent expansion near z = w = 0: Visik,˘ Katz and Yager among others provided the expected δ(z0) δ(w0) p-adic interpolation of this family of special values. To give a Θ , (z, w) = ψ(w z ) + z0 w0 0 0 z w taste of the results in question, we wish to introduce a recent e∗ (z , w , τ) a+b 1 a,b 0 0 b 1 a work by Bannai-Kobayashi (2010), based on Kronecker’s se- + ( 1) − z − w , (9) − a!Aa ries, that enables a similar construction. a 0,b>0 Our first task is to connect the series Ser to a series that ≥! where δ(u) = 1 if u Λ and 0 otherwise. includes an s-parameter pertaining to the style of Hecke. Let ∈ Imτ τ be a complex number in the upper half plane and let A = π If, in addition, τ is a CM point and z0, w0 are torsion points be the area of the fundamental domain of the lattice Λ = Z + over the lattice Λ then these coefficients are algebraic. z z τZ divided by π. Let ψ be the character ψ(z) = e −A . For a 0 ≥ Theorem 3.2. Let Λ = Z + τZ be a lattice in C. Assume that an integer, we introduce the Kronecker-Eisenstein series as the complex torus C/Λ has complex multiplication by the ring ∗ (z + λ)a of integers of an imaginary quadratic field k, and possesses a Ka∗(z, w, s, τ) = ψ(zw), Re(s) > a/2 + 1, 2 3 z + λ 2s Weierstrass model E : y = 4x g2 x g3 defined over a λ Λ | | − − !∈ number field F. Fix N > 1 an integer. For z0, w0, two complex where the means that the summation excludes λ = z ∗ − numbers such that Nz0, Nw0 Λ, the Laurent expansion in (9) if z Λ. It is a continuous function of the parameters ∈ 2 ∈ has coefficients in the number field F(E[4N ]). In particular, z C Λ, w C and it has meromorphic continuation to ∈ \ ∈ the rescaled Eisenstein–Kronecker numbers the whole s-plane, with possible poles only at s = 0 (if a = 0 a and z Λ) and s = 1 (if a = 0 and w Λ). Moreover, it sat- ea∗,b(z0, w0, τ)/A ∈ ∈ isfies a functional equation relating the value at s to the value are algebraic. at a + 1 s. Write w = η + ξτ with real variables ξ, η and ab- − We refer to [BK], Th. 1 and Cor. 2.11 for the proofs. The breviate the central value K1∗(z, w, 1, τ) by K(z, w) when there is no ambiguity on the lattice. This specific function is related above construction enables Bannai and Kobayashi to recover to Kronecker’s series by the identity Damerell’s result along the way. Let p be a prime number. Bernoulli numbers and Bernoulli , = , , , τ = ξ, η, , τ , K(z w) K1∗(z w 1 ) Ser ( z ) polynomials satisfy a whole collection of congruences mod- where w = η + ξτ, at least if z, w ! Λ, using [We1, §5, p. 72]. ulo powers of p, known as “Kummer congruences". These In this new set of notations, Theorem 2.4 can be restated as congruences are incorporated in the construction of the Kubota- Leopoldt p-adic zeta function ζ (s), s Z , which interpo- K(z, w) = ezw/A Θ(z, w), p ∈ p lates the values of the Riemann zeta function at negative inte- where gers, as given by the Bernoulli numbers. ϑ′(0, τ)ϑ(z + w, τ) Θ(z, w) = (8) Similarly, Eisenstein–Kronecker numbers satisfy a col- ϑ(z, τ)ϑ(w, τ) lection of congruences that are the building blocks for the denotes the meromorphic function appearing as the ratio in construction of p-adic L-functions of two-variables. Since Equation (6). It will play a major role in the remainder of this Bannai-Kobayashi also have a generating series at their dis- text so we name it the “Kronecker theta function”, in agree- posal, they can interpolate the Eisenstein–Kronecker numbers ment with the terminology in [BK]. p-adically, not only when p splits in k, like in the work of Katz EMS Newsletter September 2016 11 Feature (ordinary case), but also when p is inert in k (supersingular coefficients in the number field generated by the Fourier co- case). efficients of f. As a consequence, for each integer k > 0, the It seems appropriate to remark in passing that in this p- finite sum adic setting, the Kronecker limit formula (7) also has a coun- 1 Ccusp(X, Y, τ) = R (X, Y) f (τ) terpart. It has been originally obtained by Katz (1976) and k (k 2)! f f cusp proved to be crucial in the study of Euler systems attached to − ∈"Bk elliptic units. belongs to Q[X, Y][[q]]. Zagier starts to complete this cus- To complete this p-adic picture and set the stage for a re- pidal term by a contribution that arises from the Eisenstein curring theme for sections to come, we would like to men- series in Mk, using the following convenient recipe. For any tion a generalisation by Colmez-Schneps [CS] of the above even k > 0, let Bk be the usual k-th Bernoulli number and let construction of Manin-Visik˘ and Katz. Colmez and Schneps 2k k 1 m E (τ) = 1 d − q consider the case where the Hecke character is attached to an k − B k m 1 d m extension F of degree n over the imaginary quadratic field k "≥ # "| $ n = F be the normalised weight k Eisenstein series for SL2(Z). Its (the previous setting thus corresponds to 1). The field 1 might not necessarily be a CM field. Building on the tech- pair of period functions in X− Q[X] is defined by the odd (and niques of [Co], they can construct a p-adic L-function for F even) rational fractions by interpolating the algebraic numbers arising from the Lau- k od Bh Bk h h 1 ev k 2 r (X) = − X − , r (X) = X − 1 rent expansion of certain linear combinations of products of Ek h! (k h)! Ek − n generating series of Kronecker’s type, each of them being "h=0 − evaluated at torsion points over the lattice Λ. The juxtaposi- and they make up the contribution of Eisenstein series by the tion of n copies of Proposition 3.1-Theorem 3.2 allows them rule Eis ev od od ev to bootstrap the case n = 1 to arbitrary n 1 using their C (X, Y, τ) = r (X)r (Y) + r (X)r (Y) Ek(τ). ≥ k Ek Ek Ek Ek identity [CS, Eq. (31)]. − The main identity of# Zagier then establishes a$ remarkable closed formula for the generating series Periods of Hecke eigenforms + ∞ Let (z0, w0) be a fixed pair of N-torsion points over the lat- (X Y)(XY 1) cusp Eis k 2 Cτ(X, Y, T) = − + (C + C )T − , Λ. τ X2Y2T 2 k k tice As functions of the variable, the modular forms k=2 e (z , w , τ) are Eisenstein series for the principal congru- " a∗,b 0 0 which factorises as a product of two Kronecker theta func- Γ . ence subgroup (N) In particular, the Laurent expansion (9) tions: for the translated Kronecker theta function at z = w = 0 is a 2 generating series for Eisenstein series of increasing weight Theorem 3.3 (Zagier [Za1]). In (XYT)− Q[X, Y][[q, T]], we and fixed level. Its decomposition under the action of the have 2 Hecke algebra thus possesses only Eisenstein components. Cτ(X, Y, T) = Θ(XT′, YT′) Θ(T ′, XYT′)/ω , (10) From this perspective, interesting new phenomena start to ap- − where ω = 2πi, T = T/ω. pear when one considers a product of two Kronecker theta ′ functions. Equation (10) shows that complete information on Hecke Such a product encodes all period polynomials of modular eigenforms of any desired weight for SL2(Z) and their period forms of all weights, at least in the level one case.3 This is the polynomial is encoded in the Laurent expansion at T = 0 of content of the main result of Zagier’s paper [Za1], which we the right side. To further support that claim, Zagier explains now describe. in the sequel paper [Za2] how to deduce from Equation (10) Let Mk be the C-vector space of modular forms of weight an elementary proof of the Eichler-Selberg formula for traces k 4 on SL (Z) and let Sk Mk be the subspace of cusp of Hecke operators on SL (Z). ≥ 2 ⊂ 2 forms, equipped with the Petersson scalar product ( f, g) and The period polynomials satisfy a collection of linear rela- cusp its basis of normalised Hecke eigenforms . The period tions under the action of SL (Z). These cocycle relations are Bk 2 polynomial attached to f S k is the polynomial of degree reflected using Equation (10) by relations satisfied by Kro- ∈ k 2 defined by necker’s theta functions, e.g., [Za1, p. 461]. A typical exam- ≤ − i ple is ∞ k 2 r f (X) = f (τ)(τ X) − dτ. 0 − 1 1 ! Cτ(X, Y, T) + Cτ 1 , Y, TX + Cτ , Y, T(1 X) = 0, − X 1 X − The Eichler-Shimura-Manin theory implies that the maps % & % − & f rev and f rod assigning to f the even and the odd (11) (→ f (→ f which is the counterpart of the classical relation for the period part of r f are both injective. Moreover, if f is a normalised Hecke eigenform then the two-variables polynomial polynomial ev od od ev 2 1 1 r f (X)r f (Y) + r f (X)r f (Y) r f 1 + U + U = 0, U = − . R (X, Y) = C[X, Y] | 10 f (2i)k 3( f, f ) ∈ % & − In the remainder of this paper, we explain how the method of transforms under σ Gal(C/Q) as R = σ(R ), so R has ∈ σ( f ) f f Eisenstein–Schellbach, when properly modulated, is the ade- quate tool to produce systematically general (n 1)-cocycle − 3 A very recent preprint [CPZ] indicates that a similar result also holds for relations for GLn(Z) involving products of n Kronecker theta arbitrary squarefree level. functions, including Equation (11) as a very special case. 12 EMS Newsletter September 2016 Feature Trigonometric cocycles on GLn where the role of the function φ(u, ξ) is now played by Kro- Let 0 < a, b < 1 be rational numbers and x, y C Z be necker’s theta function. It will make it clear that the trigono- ∈ \ complex parameters. We use the shorthand 0 < t < 1 for the metric relations we have encountered so far are a specialisa- { } fractional part of a non-integral real number t. From a direct tion of the elliptic ones when q 0, i.e. τ i . → → ∞ computation or a mild generalisation of Lemma 2.1, one de- Kronecker has not been given the opportunity to imple- duces the following relation, which amounts to the addition ment the Eisenstein–Schellbach method in his own elliptic formula for the function φ(u, ξ): investigations. Let us now proceed by fixing τ in the upper half-plane and first perform in Equation (12) the change of e(xa)e(yb) e (x + y)a e y b a { − } variables e(x) 1 e(y) 1 − e(x + y) 1 e(y) 1 − − ! −" ! − " e (x + y)b e x a b x nτ + x, ! "! " ! "! { − }" = 0. (12) ← − e(x + y) 1 e(x) 1 y n′τ + y. ! −" ! − " ← As pointed out by the second author in [Scz2], this iden- ! "! " tity can naturally be recast in terms of the cohomology of We also aim to twist that equation by a pair of roots of unity e(nr)e(n′ s) with r, s Q and then sum over n, n′ Z. We the group SL2(Z). The building blocks are products of two ∈ ∈ copies of the trigonometric function φ(u, ξ). Given two pairs write x0 = aτ + r, y0 = bτ + s, q = e(τ) so that the real and t x, x , y, y u = (u1, u2) and ξ = (ξ1, ξ2) , we set imaginary parts of 0 0 make up four pairs of real vari- ables. The first summand becomes Φ(u, ξ) = φ(u1, ξ1)φ(u2, ξ2). na n b To any matrix A = ab SL (Z) is associated σ = 1 a e(nr)q q ′ e(n′ s) cd ∈ 2 0 c ∈ S := e(xa)e(yb) . Z qn x qn′ y M2( ). If c ! 0, we define n,n Z e( ) 1 e( ) 1 # $ # $ '′∈ − − ab 1 Ψ (u, ξ) = sign(c) Φ(uσ, σ− (µ + ξ)). ! "! a " cd The assumption 0 < a < 1 ensures that 1 > q > q , and % & µ Z2/σZ2 b | | | | ∈' similarly for q , so that this double series is absolutely con- If c = 0 then Ψ(A)(u, ξ) = 0 by definition. The next proposi- vergent. The sum S naturally splits as a product, whose value tion stands for the case n = 2 of a more general (n 1)-cocycle is deduced from two consecutive uses of Kronecker’s Theo- − relation for the group GLn(Z) obtained in [Scz3, Cor. p. 598]. rem 2.4: Proposition 3.4. Let A, B SL (Z) be two matrices. For any S = e(xa)e(yb) Θ(x, x0) Θ(y, y0). ∈ 2 u in a dense open domain in C2 and any non-zero ξ Q2, the ∈ following inhomogenous 1-cocycle relation holds: A similar resummation process can be performed on the sec- 1 ond term and the third term of Equation (12). After simplifi- Ψ(AB)(u, ξ) Ψ(A)(u, ξ) Ψ(B)(uA, A− ξ) = 0. − − cation by the common factor e(xa+yb), we obtain the identity The addition law (12) corresponds to the choice of matri- 0 1 11 ces A = and B = in this proposition. Θ(x, x0)Θ(y, y0) Θ(x + y, x0)Θ(y, y0 x0) 10− 10 − − The coefficients of the− Laurent expansion at u = 0 of the # $ # $ Θ(x + y, y0)Θ(x, x0 y0) = 0, (13) general (n 1)-cocycle Ψ carry a great deal of arithmetic infor- − − − mation. According to the main result of [Scz3, Th. 1], the pair- which can be extended analytically to remove the restrictive ing of Ψ with an (n 1)-cycle, built out of the abelian group assumptions made on the parameters. Equation (13) is just an- − generated by the fundamental units of a totally real number other form of the Riemann theta addition relation, also known field F of degree n, produces the values ζF (k) at non-positive as the Fay trisecant identity. It simultaneously implies Equa- integers k of the Dedekind zeta function of F and thereby es- tion (12) when τ i and Equation (11) under the choice of → ∞ tablishes their rationality. This provides a new proof, deeply parameters x = XT , x = YT , y = T , y = XYT . ′ 0 ′ − ′ 0 ′ rooted in the Eisenstein–Schellbach method, of the Klingen– More generally, the resummation of the trigonometric Siegel rationality result. (n 1)-cocycle Ψ gives rise to an elliptic (n 1)-cocycle that − − An integral avatar of the cocycle Ψ was later introduced we name the Eisenstein–Kronecker cocycle κ on GLn(Z). One by Charollois and Dasgupta to study the integrality proper- recovers Ψ from κ by letting τ i . The coefficients of the → ∞ ties of the values ζF (k), enabling them to deduce a new con- Laurent expansion of κ at zero now display modular forms, struction of the p-adic L-functions of Cassou–Noguès and essentially sums of products of n Eisenstein series of various Deligne–Ribet. weights and levels, that are members of a compatible p-adic Combining their cohomological construction with the re- family. cent work of Spiess, they also show in [CD] that the order of When paired with a (n 1)-cycle built out of the funda- − vanishing of these p-adic L-functions at s = 0 is at least equal mental units of a totally real number field F of degree n, the to the expected one, as conjectured by Gross. This result was elliptic cocycle κ produces a generating series for the pull- already known from Wiles’ proof of the Iwasawa Main Con- backs of Hecke–Eisenstein series over the Hilbert modular jecture. group SL ( ), whose constant terms are the values ζ (k) at 2 OF F negative integers k 0. These classical modular forms have ≤ On elliptic functions depending on four pairs of real already played a prominent role in the original proof by Siegel variables of the Klingen–Siegel theorem. More details on the construc- Our goal in this last section is to propose a q-deformation of tion of the Eisenstein–Kronecker cocycle κ and its properties the trigonometric cocycle Ψ to construct an elliptic cocycle, will be given in [Ch]. EMS Newsletter September 2016 13 Feature References [Scha] N. Schappacher. On the history of Hilbert’s twelfth prob- lem. in Matériaux pour l’histoire des mathématiques au [BK] K. Bannai, S. Kobayashi. Algebraic theta functions and XXème siècle, Nice 1996. SMF, Paris (1998) 243–273. the p-adic interpolation of Eisenstein–Kronecker numbers. [Sche] K. H. Schellbach. Die einfachsten periodischen Funktio- Duke Math. J. 153 no. 2 (2010) 229–295. nen. J. Reine Angew. Math. 48 (1854), 207–236. [Cw] P. Charollois. Professional webpage: https: [Scz1] R. Sczech. Dedekindsummen mit elliptischen Funktionen. //webusers.imj-prg.fr/~pierre.charollois/Kronecker.html. Invent. Math. 76 (1984), 523–551. Eventually these documents should be available on the [Scz2] R. Sczech. Eisenstein group cocycles for GL2(Q) and val- arXiv. ues of L-functions in real quadratic fields. Comment. Math. [Ch] P. Charollois. The Eisenstein–Kronecker cocycle on GLn. Helv. 67 (1992), 363–382. In preparation. [Scz3] R. Sczech. Eisenstein group cocycles for GLn and values [CD] P. Charollois, S. Dasgupta. Integral Eisenstein cocycles on of L-functions. Invent. Math. 113 (1993), no. 3, 581–616. GLn, I: Sczech’s cocycle and p-adic L-functions of totally [Si] C. L. Siegel. Advanced Analytic Number Theory. Tata In- real fields. Cambridge J. of Math. 2 no.1 (2014), 49–90. stitute of Fundamental Research, Bombay: 1980. [CPZ] Y. Choie, Y. K. Park, D. Zagier. Periods of modular [V] S. G. Vladut.˘ Kronecker’s Jugendtraum and modular func- forms on Γ0(N) and products of Jacobi theta functions. tions. Translated from Russian by M. Tsfasman. Studies in Preprint (2016), available at http://people.mpim-bonn. the Dev. of Modern Math 2, Gordon and Breach Sc. Pub., mpg.de/zagier/files/tex/CPZ/PERIODS.pdf. New-York, 1991. [Co] P. Colmez. Algébricité de valeurs spéciales de fonctions L. [We1] A. Weil. Book review: Gotthold Eisenstein, Mathematis- Invent. Math. 95 (1989), 161-205. che Werke. Bulletin of the AMS 82, vol. 5 (1976), 658– [CS] P. Colmez, L. Schneps. p-adic interpolation of special val- 663. (and Oeuvres Sc., vol. III, [1976c], 368-403). Also ues of Hecke L-functions. Compositio Math. 82 (1992), included as a Foreword to Gotthold Eisenstein’s Mathema- 143-187. tische Werke in 2nd Edition, Chelsea Pub. Comp., 1989, p. [vD] F. von Dalwigk. Beiträge zur Theorie de Thetafunktionen V-XI. von p Variablen. Dissertation, Halle (1891). [We2] A. Weil. Elliptic functions according to Eisenstein and [Ed] H. M. Edwards. On the Kronecker Nachlass. Historia Kronecker. 2nd Edition, Classics in Math. Springer-Verlag, Mathematica 5 (1978), 419–426. Berlin, 1999. (1st Edition 1976.) [Eis1] G. Eisenstein. Bemerkungen zu den elliptischen und [Za1] D. Zagier. Periods of modular forms and Jacobi theta func- Abelschen Transcendenten. J. Reine Angew. Math. 27 tions. Invent. Math. 104 (1991), 449–465. (1844) 185–191 (and Math. Werke I, p. 28–34 New-York. [Za2] D. Zagier. Hecke operators and periods of modular forms. Chelsea (1975)) (and Remarques sur les transcendantes Israel Math. Conf. Proc. 3 (1990), 321–336. elliptiques et abéliennes. Journal de Liouville 10 (1845), 445–450). [Eis2] G. Eisenstein. Genaue Untersuchung der unendlichen Doppelproducte, aus welchen die elliptischen Functionen Pierre Charollois [pierre.charollois@ als Quotienten zusammengesetzt sind, und der mit ih- nen zusammenhängenden Doppelreihen. J. Reine Angew. imj-prg.fr] received his PhD in Bordeaux Math. 35 (1847), 153–274. (and Math. Werke I, 357–478. in 2004. After a postdoctoral stay at New-York. Chelsea (1975)). McGill University (Montréal), he is now [HH] K. Hensel, H. Hasse. Zusätze, in Kronecker’s Werke, Band maître de conférences at Université Paris 6 V, (1930) 507–515. in Jussieu. His focus is on number theory, [Ha] H. Hasse. History of class field theory. in Cassels-Fröhlich modular forms and L-functions, including (eds), Algebraic Number Theory, London: Academic Press a keen interest in the historical grounds (1967), 266–279. [IO] T. Ishii, T. Oda. A short history on investigation of the spe- regarding these objects of study, notably their connections to cial values of zeta and L-functions of totally real number Eisenstein, Kronecker and Lerch. fields. In Automorphic forms and zeta functions. Proc. of the conf. in memory of Tsuneo Arakawa, World Sci. Publ. Robert Sczech [[email protected]] stud- (2006), 198-233. ied mathematics at Bonn University, where [Ma] B. Mazur. Visions, Dreams and Mathematics. in Cir- he received his PhD in 1981. He is cles disturbed: the interplay of Mathematics and presently working at Rutgers University in Narrative. A. Doxiadis, B. Mazur (eds). Prince- Newark, New Jersey. His field of interest is ton Univ. Press (2012) 243–273. With a compan- in number theory and includes the study ion interview, http://thalesandfriends.org/2007/07/23/ mazur-interviewed-by-corfield. of special values of L-functions attached [Ro] D. E. Rowe. Göttingen’s SUB as repository for the papers to number fields (including cohomological interpretation via of distinguished mathematicians. This volume, 39–44. group cocycles and the Stark conjecture). 14 EMS Newsletter September 2016