Mock theta functions have a long history but recent work establishes surprising connections with different areas of mathematics and physics. The proof of these and others for mock theta functions of seventh order were given by hickerson 21, 22. Instead of expressing g,z as a quotient of the theta functions f defined in 1. This chapter is based in part on abramowitz and stegun 1964, chapter 16, by l. False, partial, and mock jacobi theta functions as qhypergeometric series kathrin bringmann, amanda folsom, and robert c. On our analytic way to the group structure of an elliptic function we meet so called theta functions. In this context they play a proming role specifically in the quantization of higher.
Ramanujan emphasized that a mock theta function should not have the same singularities at roots of unity as a genuine theta function. In jacobis approach these general theta functions provide the basic building blocks for elliptic functions, as functions of t see. Thanks for contributing an answer to mathematics stack exchange. The main references used in writing this chapter are whittaker and watson, lawden, and walker. For this reason, mathematica includes not only four wellknown.
The following etaquotients are the only ones which are theta functions for an even character. The theory of modular forms has important applications to many areas of mathematics, e. Ramanujans letter the mathematical part of ramanujans letter is reproduced in 1. Ramanujans mock theta functions and their applications d.
Using thetafunctions of order with semiinteger characteristics one can construct meromorphic abelian functions with periods. A qseries is simply a power series in powers of q, whose coecients usually count something of combinatorial interest. For a summary of these notations, see whittaker and watson 1990. In mathematics, particularly qanalog theory, the ramanujan theta function generalizes the form of the jacobi theta functions, while capturing their general properties. See 1 for a wideranging survey and references on such problems i. Chapter 3 deals with fourier coefficients of meromorphic jacobi forms.
For example, they impact the theory of donaldson invariants of that are related to gauge theory for example, refs. Because the product of two harmonic maass forms is a harmonic maass. In this dissertation, i consider several of the examples that ramanujan gave of mock theta functions, and relate them to realanalytic modular forms. The title to section 11 refers to modular functions, although these are not defined. One of our exercises emphasizes their relation with eld theory and galois theory. For example, the product of thetafunctions of order 1 is a thetafunction of order. A more detailed theory of elliptic theta functions was developed by c. The periods of an arbitrary abelian function in complex variables satisfy the riemannfrobenius relations, which yield convergence for the series defining the thetafunctions with. The implied constants in ramanujans claim are not mysterious.
Berndt1, geumlan choi, younseo choi2, heekyoung hahn, boon pin yeap, ae ja yee 3, hamza yesilyurt, and jinhee yi abstract. Jan 01, 20 with the relation between mock theta functions and partial theta functions at roots of unity where the mock theta function does not have singularities. Abelian varieties, nil theta and theta functions 69 78. One jacobi theta function named after carl gustav jacob jacobi is a function defined for two complex variables z and. The period matrices and theta functions of riemann chingli chai version 05182015 this article deals with four notions due to riemann. They linked ramanujans functions to partitions of a given integerthe ways of writing an integer as a sum of smaller integers. But avoid asking for help, clarification, or responding to other answers. May 07, 20 this grouptheoretic structure was discovered by zwegers.
Bolster the view that ramanujan had found something. The theory of theta correspondence gives one of the few general methods of constructing automorphic forms of groups over number. My guess is that, just as trig functions are a sum of exponentials, since elliptic functions are some combination of theta functions on a parallelogram, this is why the elliptic. A survey of classical mock theta functions springerlink. The mock theta functions were invented by the indian mathematician srinivasa ramanujan, who lived from 1887 until 1920. In chapter 4 we use the results from chapter 2 to give explicit results for 8 of the 10 fifth order mock theta functions and all 3 seventh order functions, that were originally defined by ramanujan. Building on the work by andrews and garvan, dean hickerson proved that five identities in each of the two families are equivalent, at the same time confirming that these mock theta functions are truly mock theta. Works by atkin, andrews, dyson, selberg, swinnertondyer, and watson on these 22 series. Introduction the goal of this article is to provide an overview on mock theta functions and their connection to weak maass forms. The first examples of mock theta functions were described by srinivasa ramanujan in his last 1920 letter to g. The magic of theta functions paramanands math notes.
Razvan gelca, alastair hamilton, classical theta functions from a quantum group perspective arxiv. A definition of the order of a mock theta function is given in the gordonmcintosh paper on modular transformation of ramanujans fifth and seventhorder mock theta functions watson while constructing transformation laws for the mock theta function found three further mock theta functions of order 3. False, partial, and mock jacobi theta functions as q. Nil theta functions and jacobi theta functions 48 57. In chapter 2 we consider indefinite theta functions of type r1,1. Introduction on pages 2 and 17 in his lost notebook 23, ramanujan recorded four identities involving the rank generating function. Sander zwegers 2001, 2002 discovered that adding certain nonholomorphic.
Nil theta functions associated with a positive definite hmorphism of an abelian variety 83 92. These famous series play a prominent role in the study of integer partition congruences for. The periods of an arbitrary abelian function in complex variables satisfy the riemannfrobenius relations, which yield convergence. We construct elliptic genera of level 1 in section 4, for both spin manifolds and almost complex manifolds with c 1. My objective here will be to provide a concise account of the stark essentials of some of my recent work as it relates to that wonderful creation of the youthful jacobithe theory of theta functions. As we said then, the extent to which we can express modular forms in terms of. We also point out that partialfalse theta functions are not mock theta functions as studied in z, although there is a connection see za for instance. Watson go camping and pitch their tent under the stars. This is the motivation for the study of the modularity of fourier coe. Theta functions are naturally thought of as being the states in the geometric quantization of the given complex space, the given holomorphic line bundle being the prequantum line bundle and the condition of holomorphicity of the section being the polarization condition. The method has its origin in the classical construction of theta functions which are modular formsperhaps of half integral weight on the upper. To complete the proof it suffices to prove that is not a mock theta function according to the modern definition. We show how they can be interpreted in the theory of realanalytic modular forms. Zwegers 2001 phd thesis was a breakthrough in the study of the mock theta functions.
In this dissertation, i consider several of the examples that ramanujan gave of mock theta functions, and relate them to realanalytic. Many relations in the theory of elliptic functions include derivatives of the theta functions with respect to the variable. While the idea of what siegel is doing in order to compute the volume of the fundamental domain described in the article the very first one, for there are discussed. Some generalizations of the rigidity theorems and some general. We now discuss the contents of the succeeding sections. Ramanujans mock theta functions and their applications dapr es zwegers and bringmannono by don zagier introduction one of the most romantic stories in the history of mathematics is that of the friendship between hardy and ramanujan.
For example, the product of thetafunctions of order 1 is a thetafunction of order using thetafunctions of order with semiinteger characteristics one can construct meromorphic abelian functions with periods. It is therefore natural to ask which etaquotients are theta functions. Weve previously seen connections between modular forms and ramanujans work by verifying that eisenstein series are indeed modular forms, and showing that the discriminant function is a weight 12 cusp form with a product expansion in q. Abelian varieties, theta functions and the fourier transform alexander polishchuk. A general construction and algebraic foundations 69 78.
Holomorphic projections and ramanujans mock theta functions. In particular, the jacobi triple product takes on a particularly elegant form when written in terms of the ramanujan theta. The theta functions are quasidoubly periodic, as illustrated in the following table. Of course, ramanujan would not have used this terminology. Abelian varieties, niltheta and theta functions 69 78. In this paper we take a radical di erent point of view to 1. For further bibliographic reading see mckean and moll and armitage and eberlein. These functions are sometimes denoted or, and a number of indexing conventions have been used.
Applied theta functions of one or several variables nicholas wheeler, reed college physics department october 1997 introduction. We prove ramanujans claim as a special case of a more general result. However elliptic functions should be definable analogously to how trig functions are formulated in terms of exponentials, this is the origin of jacobi theta functions. Our paper focusses on identities for the functions g,z.
The second part of the chapter takes up general jacobian theta functions i. In his last letter to hardy, ramanujan defined 17 functions mq, q 0we have this integral representation of the operator e. Zwegerss breakthrough catalyzed the development of the. These complex functions are entire and quasiperiodic with respect to a lattice in the proof of abels theorem we use their properties to characterise all meromorphic functions f from c. Finite modular form thetafunktion algebra function geometry proof theorem variable. We thus see the same points as above, but in a more complicated setting. Niltheta functions and jacobitheta functions 48 57. Pdf mock theta functions, weak maass forms, and applications.
A brief introduction to theta functions dover books on. Niltheta functions associated with a positive definite hmorphism of an abelian variety 83 92. Mock theta functions and quantum modular forms 5 qhypergeometric series u. A riemann bilinear relations, b riemann forms, c riemann theta functions and d riemanns theta formula, in four parts. Curious qseries identities since rogers 32 introduced the false theta functions, they have played a curious role in the theory of partitions see for instance 1, 2, 12.
In chapter 1 we give results for lerch sums also called appell functions, or generalized lambert series. Bruggeman, we study the mock theta functions, that were introduced by ramanujan. In mathematics, a mock modular form is the holomorphic part of a harmonic weak maass form, and a mock theta function is essentially a mock modular form of weight 12. As a result of his thesis, it is known that all of ramanujans examples are essentially the holomorphic part of weight weak harmonic maass forms whose nonholomorphic parts are period integrals of weight unary theta functions. Ramanujan motivated his mock theta functions by rst describing brie y two \genuine theta functions. But these little blemishes do not seriously detract from a gem of a book, which can profitably be dipped into, time after time. Although ramanujans secrets died with him, we have. The complete elliptic integrals and the special theta functions are related as follows. Ramanujans mock theta functions, dysons rank generating functions, borcherds products, and central values and derivatives of quadratic twists of modular lfunctions. These are in fact specializations with of the general theta functions.
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