Shaul Zemel

Regularized pairings of meromorphic modular forms and theta lifts

Abstract We show that the meromorphic modular forms recently considered by Bringmann and Kane can be obtained as images of regularized theta lifts of Poincare series under weight raising operators. We use this fact in order to simplify the evaluation of the regularized pairing, also defined by Bringmann and Kane, of these functions with other meromorphic modular forms.

Generalizations of Thomae's formula for Zn curves

- 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.

On quasi-modular forms, almost holomorphic modular forms, and the vector-valued modular forms of Shimura

We extend the relation between quasi-modular forms and modular forms to a wider class of functions. We then relate both forms to vector-valued modular forms with symmetric power representations, and prove a general structure theorem for these vector-valued forms.

A Gross–Kohnen–Zagier type theorem for higher-codimensional Heegner cycles

We prove that the Heegner cycles of codimension m+1 inside Kuga-Sato type varieties of dimension 2m+1 are coefficients of modular forms of weight 3/2+m in the appropriate quotient group. The main technical tool for generating the necessary relations is a Borcherds style theta lift with polynomials. We also show how this lift defines a new singular Shimura-type correspondence from weakly holomorphic modular forms of weight 1/2−m to meromorphic modular forms of weight 2m+2.

Shimura lifts of weakly holomorphic modular forms

We show how to realize the Shimura lift of arbitrary level and character using the vector-valued theta lifts of Borcherds. Using the regularization of Borcherds’ lift we extend the Shimura lift to take weakly holomorphic modular forms of half-integral weight to meromorphic modular forms of even integral weight having poles at CM points.

On lattices over valuation rings of arbitrary rank

Abstract We show how several results about p-adic lattices generalize easily to lattices over valuation ring of arbitrary rank having only the Henselian property for quadratic polynomials. If 2 is invertible we obtain the uniqueness of the Jordan decomposition and the Witt Cancellation Theorem. We show that the isomorphism classes of indecomposable rank 2 lattices over such a ring in which 2 is not invertible are characterized by two invariants, provided that the lattices contain a primitive norm divisible by 2 of maximal valuation.

Thomae Formulae for Nonsingular Z n Curves

In this chapter we shall present a proof of the Thomae formula for the general non singular Z n curve associated to the equation.

Shimura’s vector-valued modular forms, weight changing operators, and Laplacians

We investigate the various types of weight raising and weight lowering operators on quasi-modular forms, or equivalently on Shimura’s vector-valued modular forms involving symmetric power representations. We also present all the eigenfunctions of the two possible Laplacian operators.

Thomae formulae for general fully ramified Z n curves

We generalize the elementary methods presented in several examples in the book [FZ] to obtain the Thomae formulae for general fully ramified Zn curves.

Generalized Riordan groups and operators on polynomials

Abstract We present an approach to generalized Riordan arrays which is based on operations in one large group of lower triangular matrices. This allows for direct proofs of many properties of weighted Sheffer sequences, and shows that all the groups arising from different weights are isomorphic since they are conjugate. We also prove a result about the intersection of two generalized Riordan groups with different weights.