Because the product of two harmonic maass forms is a harmonic maass. Bruggeman, we study the mock theta functions, that were introduced by ramanujan. A thesis in mathematics submitted to the graduate faculty of texas tec. This is a try to solve a problem of landweber in la1. To complete the proof it suffices to prove that is not a mock theta function according to the modern definition. These famous series play a prominent role in the study of integer partition congruences for. I am studying about mock modular forms and mock theta functions. Those interested in chernsimons theory will find here an introduction using the simplest case, that of abelian chernsimons theory. 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. The periods of an arbitrary abelian function in complex variables satisfy the riemannfrobenius relations, which yield convergence for the series defining the thetafunctions with. I wonder how zwegers connected mock theta functions with harmonic maass forms. 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. The proof of these and others for mock theta functions of seventh order were given by hickerson 21, 22. On ramanujans definition of mock theta function ncbi.
Watsons 1936 presidential address \ramanujans discovery of the mock theta functions makes it obvious that his skill and ingenuity did not desert him at the oncoming of his untimely end. The first examples of mock theta functions were described by srinivasa ramanujan in his last 1920 letter to g. 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. Zagier the modular group takes its name from the fact that the points of the quotient space.
Ramanujan emphasized that a mock theta function should not have the same singularities. 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. 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. Some generalizations of the rigidity theorems and some general. In this dissertation, i consider several of the examples that ramanujan gave of mock theta functions, and relate them to realanalytic. In 17, lubicz and robert generalized the tate pairing over any abelian variety and more precisely over theta functions. This is the motivation for the study of the modularity of fourier coe. In this context they play a proming role specifically in the quantization of higher. Theta functions and knots can be read in two perspectives. Theta functions article about theta functions by the. For further bibliographic reading see mckean and moll and armitage and eberlein. The main references used in writing this chapter are whittaker and watson, lawden, and walker.
The theory of modular forms has important applications to many areas of mathematics, e. One of our exercises emphasizes their relation with eld theory and galois theory. False, partial, and mock jacobi theta functions as qhypergeometric series kathrin bringmann, amanda folsom, and robert c. The mock theta functions were invented by the indian mathematician srinivasa ramanujan, who lived from 1887 until 1920. These are in fact specializations with of the general theta functions. As we said then, the extent to which we can express modular forms in terms of.
The second part of the chapter takes up general jacobian theta functions i. Mock theta functions have a long history but recent work establishes surprising connections with different areas of mathematics and physics. Introduction the goal of this article is to provide an overview on mock theta functions and their connection to weak maass forms. Theta functions, gaussian series, and spatially periodic solutions of the kortewegde vries equation article pdf available in journal of mathematical physics 233 march 1982 with 146 reads. In chapter 1 we give results for lerch sums also called appell functions, or generalized lambert series. Pdf theta functions, gaussian series, and spatially.
The most famous among these theta relations is a quartic relation 9, 12 on p. His topics include the qbinomial theorem, heines transformation, the jacobi triple product identity, the rogersfine identity, bailey chains, wpbailey pairs and chains, further results on baileywpbailey pairs and chains, bijective proofs of basic hypergeometric identities, qcontinued fractions, lambert series, and mock theta functions. Thanks for contributing an answer to mathematics stack exchange. In this paper we take a radical di erent point of view to 1. The complete elliptic integrals and the special theta functions are related as follows. As alluded to above, to associate modular behavior to a given mock theta function mq, one. Ramanujans mock theta functions and their applications d. The security of the new algorithms is an important issue for the use of practical. Aimed at graduate students, the book synthesizes the classical and modern points of view in theta functions, concentrating on connections to number theory and representation theory. Ramanujan sent to hardy january 12, 1920 defined 17 jacobilike functions fq for complex q lt 1, called mock theta functions since then. S k these are the spaces of modular forms and cusp forms, respectively.
There are several closely related functions called jacobi theta functions, and many different and incompatible systems of notation for them. Watson go camping and pitch their tent under the stars. Introduction on pages 2 and 17 in his lost notebook 23, ramanujan recorded four identities involving. Holomorphic projections and ramanujans mock theta functions. Readers with an interest in theta functions or knot theory can learn how the two are related. In jacobis approach these general theta functions provide the basic building blocks for elliptic functions, as functions of t see. Ds have is expressed by the trisecant identity, due to john fay theta functions on. We show how they can be interpreted in the theory of realanalytic modular forms. Generating functions play a central role throughout number theory. Theta function of four types file exchange matlab central. But avoid asking for help, clarification, or responding to other answers. Finite modular form thetafunktion algebra function geometry proof theorem variable. Ramanujans identities and monotonicity properties of quotients of theta functions by atul abhay dixit, b. On our analytic way to the group structure of an elliptic function we meet so called theta functions.
Type is a type of the theta function, there are four numbered types. 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. 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. 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. For example, they impact the theory of donaldson invariants of that are related to gauge theory for example, refs. They linked ramanujans functions to partitions of a given integerthe ways of writing an integer as a sum of smaller integers. We construct elliptic genera of level 1 in section 4, for both spin manifolds and almost complex manifolds with c 1. Sander zwegers 2001, 2002 discovered that adding certain nonholomorphic. Th thetatype,v,m returns values of theta functions evaluated for corresponding values of argument v and parameter m. Short addition sequences for theta functions halinria. Pdf ramanujans last letter to hardy concerns the asymptotic properties of modular forms and his mock theta functions. This chapter is based in part on abramowitz and stegun 1964, chapter 16, by l. Etaquotients and theta functions 5 consisting of all forms which are holomorphic resp.
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