Theta blocks

Theta blocks are products of the Dedekind eta function and powers of Jacobi
theta functions. These are meromorphic Jacobi forms with easily calculable weight, index,
character, and divisor. Holomorphic theta blocks provide examples of weak Jacobi forms.
The generalized valuation, ord, characterizes Jacobi forms from among the weak Jacobi forms
and spaces of Jacobi forms are often efficiently spanned by theta blocks with nonnegative
valuation.

Theta blocks were recently introduced by Gritsenko, Skoruppa, and Zagier, and have both
theoretical and computational applications. Even when dimension formulas are known for
spaces of Jacobi forms, it may not be easy to compute the initial Fourier expansions of a
basis. Theta blocks are the method of choice for spanning spaces of Jacobi forms on a large
scale. Theta blocks also help classify families of Borcherds products and have connections
to root systems.

The mini-course will give proofs of the basic properties of theta blocks and examples as well.
Topics will include:

1. The generalized valuation, ord, from weak Jacobi forms to a (partially) ordered
   abelian group of piecewise quadratic periodic functions.
2. The injectivity of the generalized valuation ord on theta blocks.
3. The remark that the leading Fourier Jacobi coefficient of a Borcherds product for a
   paramodular group in degree two is a theta block.
4. Use of theta blocks to construct holomorphic Borcherds products.
5. Use of theta blocks and Hecke operators to span spaces of Jacobi cusp forms for
   weight two and indices up to a few thousand.