2009 Number Theory Seminar - 12

I will give an expository talk about half integer weight modular forms. Definitions and basic properties of this space will be given. Then, the Shimura correspondence will be discussed. Roughly speaking, this associates to a modular form of half integer weight some modular form of integer weight. This will be followed by a discussion of several improvements on this result, the most important of which is due to Waldspurger. In his work Waldspurger used representation theory to establish an explicit relation between the square of the coefficients in the q-expansion of a form f of half integer weight k/2 and the central values of 'twist' L-series for a form g of integer weight k-1 corresponding to f via the Shimura map.