Toric Periods and $p$-adic Families of Modular Forms of Half-Integral Weight

The primary goal of this work is to construct $p$-adic families of modular forms of half-integral weight, by using Waldspurger's automorphic framework to make the results as comprehensive and precise as possible.

A secondary goal is to clarify the role of test vectors as defined by Gross-Prasad in the elucidation of general formulae for the Fourier coefficients of modular forms of half-integral weight in terms of toric periods of the corresponding modular forms of integral weight.

As a consequence of our work, we develop a generalization of a classical formula due to Shintani, and make precise the conditions under which Shintani's lift vanishes.

We also give a number of results on test vectors for ramified representations which are of independent interest.