Tata lectures on theta. I: Introduction and motivation: Theta functions in one variable. Basic results on theta functions in several variables. With the assistance of C. Musili, M. Nori, E. Previato, and M. Stillman. (English) Zbl 0509.14049

Progress in Mathematics, Vol. 28. Boston-Basel-Stuttgart: Birkhäuser. xiii, 235 p. SFr. 54.00 (1983).
Theta functions were originated by Jacobi in the one variable case in his study of elliptic functions, and were immensely developed by Riemann further in the several variable case in connection with the study of algebraic functions in one variable. The author gave series of lectures on this subject at several universities during October 1978 to August 1980. This volume is based on notes of the lectures and contains the first two out of four intended chapters.
Chapter I is devoted to theta functions in one variable with both geometric and arithmetic aspects. After the standard theta function \(\theta(z,\tau)\) is discussed, “theta functions with characteristics” \(\theta_{ab}(z,\tau)\) are introduced in terms of the Heisenberg group. Then what are discussed next are projective embeddings of elliptic curves, the equations defining the image curves in the projective space, Riemann relations of thetas. The theta functions \(\theta_{ab}(z,\tau)\) have been regarded as functions in \(z\in\mathbb C\) for a fixed \(\tau\) in the upper half plane so far, and they are concerned with functions theory on an elliptic curve. Now \(\theta_{ab}(z,\tau)\) are functions in two variables \((z,\tau)\) and we find the functional equation for \(\theta\) with respect to \(\mathrm{SL}(2,\mathbb Z)\), and the moduli space of elliptic curves, realized as an algebraic curve. After this, Jacobi’s identity of the \(z\)-derivative of \(\theta\) is proved, (and we remark that possible generalizations of this kind of identity attract attention of some mathematicians). In the rest of the chapter we find three arithmetic applications of theta series: in particular the link between \(\theta\) and \(\zeta\) and a brief introduction to Hecke’s theory relating modular forms and Dirichlet series as the last topic. These are the contents of Chapter I.

In Chapter II, the generalization of the geometric results of Chapter I (but not the arithmetic ones) to theta functions in several variables is carried out. The topics covered are the following: after projective embedding of the \(g\)-dimensional complex tori, the theory of Jacobian varieties associated with algebraic function fields of genus \(g\); the functional equation for \(\theta\); Riemann’s theta relations with applications such as explicit equations for abelian varieties and for certain modular varieties; finally constructing a large class of modular forms from theta functions via pluriharmonic polynomials and quadratic forms. Each chapter ends with some open problems.
The description in this book is not very formal but a sort of natural; and likely to give good stimuli to the readers. On the other hand it has been very common for books on thetas to contain quite a few typographical errors because of the complexity of notation, and unfortunately the present book inherits this tradition faithfully.


14K25 Theta functions and abelian varieties
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
11F11 Holomorphic modular forms of integral weight
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
33E05 Elliptic functions and integrals
14K10 Algebraic moduli of abelian varieties, classification
14H10 Families, moduli of curves (algebraic)
14K30 Picard schemes, higher Jacobians
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