130,697 research outputs found
Classical hurwitz numbers and related combinatorics
Full text linkWe give a polynomial-time algorithm of computing the classical Hurwitz numbers Hg,d, which were defined by Hurwitz 125 years ago. We show that the generating series of Hg,d for any fixed g > 2 lives in a certain subring of the ring of formal power series that we call the Lambert ring. We then define some analogous numbers appearing in enumerations of graphs, ribbon graphs, and in the intersection theory on moduli spaces of algebraic curves, such that their generating series belong to the same Lambert ring. Several asymptotics of these numbers (for large g or for large d) are obtained
Traces of singular moduli
No full textIn the theory of modular forms, the so-called singular moduli (i.e., the values assumed by the modular invariant j(τ) when the argument is a quadratic irrationality) have been studied intensively since the time of Kronecker and Weber. More recently, \\it B. H. Gross and \\it D. B. Zagier [On singular moduli, J. Reine Angew. Math. 355, 191--220 (1985; Zbl 0545.10015)] have given explicit formulas for their norms, and in 1994 analoguous formulas for their traces have been found by D. Zagier.\\par The paper under review, which grew out of the author's lectures delivered at the above-mentioned conference in 1998, provides a detailed account of these new results on the traces of singular moduli, together with a number of further applications and generalizations. In particular, the author shows that his results are closely related to a spectacular theorem of \\it R. Borcherds [Invent. Math. 120, No. 1, 161--213 (1995; Zbl 0932.11028)] of which he gives a new proof as well as a generalization. Other generalizations of the author's approach are given in the last part of the present paper. These generalizations yield other new results describing the periods of holomorphic and non-holomorphic modular forms, culminating in striking analogues of the Shimura correspondence between different kinds of modular forms
Harer-Zagier formula via Fock space
Full text linkThe goal of this note is to provide a very short proof of Harer-Zagier formula for the number of ways of obtaining a genus g Riemann surface by identifying in pairs the sides of a (2d)-gon, using semi-infinite wedge formalism operators. <br
Modular points, modular curves, modular surfaces and modular forms
No full text[For the entire collection see Zbl 0547.00007.] \\par This is the written version of a talk at the Arbeitstagung at Bonn. It is centered around one example: the modular curve X\\sb 0(37). The elliptic curve E:\\quad y(y-1)=(x+1)x(x-1) is a factor of the Jacobian J\\sb 0(37). The article treats special values of L-series attached to E and its twists, Heegner points on E, the Gross-Zagier theorem and illustrates the interplay between classical algebraic geometry over \\bbfC and Arakelov geometry over \\bbfZ. It also gives an extension of the Gross-Zagier result: \\sum P\\sb dq\\sp d\\quad is a modular form of weight 3/2 and level 37. Here P\\sb d is the Heegner point on X\\sb 0(37) associated to d. This has now been proved for arbitrary N (rather than for N=37) by Gross/Kohnen/Zagier. The proof for the special case treated here uses an ad hoc method. This article is written to wet one's appetite and no doubt it will
Periods of Modular forms on Gamma_0(N) and products of Jacobi theta functions
No full textGeneralizing a result of [15] for modular forms of level one, we give a closed formula for the sum of all Hecke eigenforms on 00(N ), multiplied by their odd period polynomials in two variables, as a single product of Jacobi theta series for any squarefree level N. We also show that for N = 2, 3 and 5 this formula completely determines the Fourier expansions of all Hecke eigenforms of all weights on 00(N ).
Modular forms with rational periods
No full textModular forms with rational periods / W. Kohnen ; D. Zagier. - In: Modular forms / ed.: Robert A. Rankin. - Chichester : Horwood, 1985. - S. 197-249
Permutation combinatorics of worldsheet moduli space
Full text link52 pages, 21 figures52 pages, 21 figures; minor corrections, "On the" dropped from title, matches published version52 pages, 21 figures; minor corrections, "On the" dropped from title, matches published versio
Values of zeta functions and their applications
No full textAccording to the author, ``the purpose of this article is neither to prove new results nor to give a survey of a well-defined area of mathematics'', rather this paper aims ``to give a feel of some of the ways in which special values of zeta-functions interrelate with other interesting mathematical questions''. \\par The author starts with a simple proof of rationality of the numbers π\\sp-2k ζ (2k), kgt;0, k\\in \\Bbb Z, for the Riemann zeta-function ζ(s), and, after a brief description of the general conjectures relating to the critical values of motivic L-functions, proceeds to comment on: the recent status of the Birch and Swinnerton-Dyer conjecture; the periods of modular forms; the polylogarithms and algebraic K-theory; his recent work with F. Rodriguez-Villegas (cf. \\it F. Rodriguez-Villegas and \\it D. Zagier in [Gouvêa F. Q. (ed.), Advances in number theory, 81--99 (1993; Zbl 0791.11060)]) concerning the special value of a Hecke L-series of an imaginary quadratic field at the point of symmetry of the functional equation; the invariants of moduli spaces in relation to the special values of classical zeta-functions as well as of some new zeta-functions arising in mathematical physics (Verlinde formulae). He goes on to define ``multiple zeta-values'' as sums \\sideset\\and ^*→\\sum\\sb n \\prod\\sp r\\sbi=1 n\\sb i\\sp-k\\sb i, k\\sb i≥q 1, k\\sb r≥q 2, with \\sideset\\and ^*→\\sum ranging over all the integers under condition n\\sb rgt; ⋅sgt; n\\sb 1gt; 0, and points out that the graded ring of these zeta-values is related to the category of mixed Tate motives, to the values of the Drinfeld (multiple) integrals, and to the structure of the set of Vassiliev knot invariants. \\par The last chapter of this survey provides an annotated list of references for further reading
On the equality of periods of Kontsevich-Zagier
No full textInternational audienceEffective periods were defined by Kontsevich and Zagier as complex numbers whose real and imaginary parts are values of absolutely convergent integrals of \QQ-rational functions over \QQ-semi-algebraic domains in \RR^d. The Kontsevich-Zagier period conjecture states that any two different integral expressions of a period are related by a finite sequence of transformations only using three rules respecting the rationality of functions and domains: integral addition by integrands or domains, change of variables and Stokes' formula.In this paper, we introduce two geometric interpretations of this conjecture, seen as a generalization of Hilbert's third problem involving either compact semi-algebraic sets or rational polyhedra equipped with piece-wise algebraic forms. Based on known partial results for analogous Hilbert's third problems, we study possible geometric schemes to prove this conjecture and their potential obstructions.Les périodes effectives furent définies par Kontsevich et Zagier comme étant les nombres complexes dont les parties réelle et imaginaire sont valeurs d'intégrales absolument convergentes de fonctions \QQ-rationnelles sur des domaines \QQ-semi-algébriques dans \RR^d. La conjecture des périodes de Kontsevich-Zagier affirme que si une période admet deux représentations intégrales, alors elles sont reliées par une suite finie d'opérations en utilisant uniquement trois règles respectant la rationalité des fonctions et domaines: sommes d'intégrales par intégrandes ou domaines, changement de variables et formule de Stokes.Dans cet article, nous introduissons deux interprétations géométriques de cette conjecture, vue comme une généralisation du 3éme problème de Hilbert soit pour des ensembles semi-algébriques compacts soit pour des polyèdres rationnels munis d'une forme volume algébrique par morceaux. Basés sur des résultats partiels connus pour des problèmes de Hilbert analogues, nous étudions des possibles schémas géométriques pour obtenir une preuve de la conjecture et ses obstructions potentielles
Ramanujan to Hardy: from the first to the last letter. (Ramanujan an Hardy: Vom ersten bis zum letzten Brief.)
No full textRecently invited speaker at the cycle `Un texte, un mathématicien', held at the Bibliothèque nationale de France, Don Zagier changed the title to `Three texts, one mathematician', namely Godfrey Hardy. The latter had intended to write three books: I) An introduction to Calculus II) The famous `A mathematician's apology', III) A volume on his collaboration with Ramanujan: Exchange by mail with the Indian mathematician still staying in Madras and then direct common research activities during Ramanujan's stay in Cambridge.\par Reading the unfinished presentation of III) constitutes a fascinating task
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