Hershel M. Farkas

Theta constants, Riemann surfaces and the modular group : an introduction with applications to uniformization theorems, partition identities and combinatorial number theory

The modular group and elliptic function theory Theta functions with characteristics Function theory for the modular group

Automorphisms of compact Riemann surfaces and the vanishing of theta constants

\Gamma

On the Schottky Relation and Its Generalization to Arbitrary Genus

and its subgroups Theta constant identities Partition theory: Ramanujan congruences and generalizations Identities related to partition functions Combinatorial and number theoretic applications Bibliography Bibliographical notes Index.

New theta constant identities

THEOREM 1. Let S be a compact Riemann surface of genus 2g—l, g à 2, which permits a conformai fixed point free involution 1\ Let 7i, * • • , 72o-i ; Si, • • • , Ô20_i be a canonical dissection of S and let T be such that T(yi) is homologous to yi9 T(8i) is homologous to Si, T(yi) is homologous to y0+i-i and T(ôi) is homologous to S0+»_i, i = 2, • • • , g. Then, there exist at least 2°~~(2~-~1) half integer theta characteristics €i, • • • such that 0€1(O) = 0€2(O) = • • • = 0 and the order of the zero is ^ 2 .

AUTOMORPHIC FORMS FOR SUBGROUPS OF THE MODULAR GROUP

If S is a compact Riemann surface of genus g>2 together with a canonical homology basis (F, A) r = at, *.., y, A = al .**, 3g and if de1, **,9 denotes the basis of abelian differentials of first kind on S dual to (F, A); i.e., 4 there must be these relations. Another way of looking at the problem is the following: The totality of g x g matrices which are symmetric and have positive definite imaginary part is called the Siegel upper half plane of degree g, denoted by ?Rg. Not all elements of g9 are Riemann matrices for some (S, r, A). As a matter of fact for g > 4 the elements of 9g which are Riemannmat rices for (S, F, A) form a set of positive codimension. The problem we are considering is to determine this set. The first person to make a break-through in this direction was F. Schottky [12]. In the case g = 4, Schottky showed that for the set in question in e, the associated even theta constants satisfy a special relation of the form 1/ r, ?+ Vr2 + V r3 = 0 where ri is a product of 8 theta constants. Rationalizing this expression we have an explicit homogeneous polynomial in the Riemann theta constants which of course gives the one relation for g = 4 among the 10 periods. Subsequently, Schottky and Jung in a joint note [13] indicated a way of re-deriving the genus 4 result and generalizing it to arbitrary g. Their idea was to establish certain relations between the Riemann theta constants and what was referred to in [10] as the Schottky theta constants; however, to our knowledge they never establish these relations. These relations were established for the case g = 2 by Rauch and the author in [10, 11] and the relations for g = 2 were seen to be a consequence of the vanishing

Generalizations of Thomae's formula for Zn curves

The residue theorem is employed to obtain new identities amongpthe powers of theta constants with rational characteristics. The technique is then used to derive some known identities of Ramanujan.

Theta Functions in Complex Analysis and Number Theory

The theory of theta constants with rational characteristics is developed from the point of view of automorphic functions for the principal congruence subgroups of the modular group PSL(2, ℤ). New identities are derived and particular emphasis is given to the level 3 case where a striking generalization of the classical λ-function is obtained.

From Fourier Analysis and Number Theory to Radon Transforms and Geometry

- Introduction.- 1. Riemann Surfaces.- 2. Zn Curves.- 3. Examples of Thomae Formulae.- 4. Thomae Formulae for Nonsingular Zn Curves.- 5. Thomae Formulae for Singular Zn Curves.-6. Some More Singular Zn Curves.-Appendix A. Constructions and Generalizations for the Nonsingular and Singular Cases.-Appendix B. The Construction and Basepoint Change Formulae for the Symmetric Equation Case.-References.-List of Symbols.-Index.

Elliptic Functions and Modular Forms

In these notes we try to demonstrate the utility of the theory of theta functions in combinatorial number theory and complex analysis. The main idea is to use identities among theta functions to deduce either useful number-theoretic information related to representations as sums of squares and triangular numbers, statements concerning congruences, or statements concerning partitions of sets of integers. In complex analysis the main utility is in the theory of compact Riemann surfaces, with which we do not deal. We do show how identities among theta functions yield proofs of Picard’s theorem and a conformal map of the rectangle onto the disk.

Period relations for hyperelliptic riemann surfaces

Preface. Biographical Sketch of Leon Ehrenpreis (Yael Ehrenpreis Meyer).- Differences of Partition Functions - The Anti-Telescoping Method(G.E. Andrews).- The Extremal Plurisubharmonic Function for Linear Growth (D. Bainbridge).- Mahonian Partition Identities Via Polyhedral Geometry (M. Beck, B. Braun, N. Le).- Second Order Modular Forms with Characters (T. Blann, N. Diamantis).- Disjointness of Moebius From Horocycle Flows (J. Bourgain, P. Sarnak, T. Zeigler).- Duality and Differential Operators for Harmonic MAASS Forms (K. Bringmann, B. Kane, R.C. Rhoades).- Function Theory Related to the Group PSL2(R) (R. Bruggeman, J. Lewis, D. Zagier).- Analysis of Degenerate Diffusion Operators Arising in Population Biology (C.L. Epstein, R. Mazzeo).- A Matrix Related to the Theorem of Fermat and the Goldbach Conjecture (H.M. Farkas).- Continuous Solutions of Linear Equations (C. Fefferman, J. Kollar).- Recurrence for Stationary Group Actions (H. Furstenberg, E. Glasner).- On the Honda-Kaneko Congruences (P. Guerzhoy).- Some Intrinsic Constructions on Compact Riemann Surfaces (Robert C. Gunning).- The Parallel Refractor (C.E. Gutierrez, F. Tournier).- On a Theorem of N. Katz and Bases in Irreducible Representations (D. Kazhdan).- Vector-valued Modular Forms with an Unnatural Boundary (M. Knopp, G. Mason).- Loss of Derivatives (J.J. Kohn).- On an Oscillatory Result for the Coefficients of General Dirichlet Series (W. Kohnen, W. de Azevedo Pribitkin).- Representation Varieties of Fucsian Groups (M. Larsen, Alexander Lubotzky).- Two Embedding Theorems (G.A. Mendoza).- Cubature Formulas and Discrete Fourier Transform on Compact Manifolds (I. Z. Pesenson, D. Geller).- The Moment Zeta Function and Applications (I. Rivin).- A Transcendence Criterion for CM on Some Families of Calabi-Yau Manifolds (P. Tretkoff).- Ehrenpreis and the Fundamental Principle (F. Treves).- Minimal Entire Functions (B. Weiss).- A Conjecture by Leon Ehrenpreis about Zeroes of Exponential Polynomials (A. Yger).- The Discrete Analog of the Malgrange-Ehrenpreis Theorem (D. Zeilberger).- The Legacy of Leon Ehrenpreis.- PhD Students.- Publications of Leon Ehrenpreis.