Sadiah Zahoor

Congruences between modular forms of integer and half integer weight

The theory of half-integral weight modular forms may be adopted to prove 'congruences' between Selmer type groups. In the paper 'Modular Form Congruences and Selmer Groups', McGraw and Ono used this approach by passing on to 'congruences' between modular forms of integer and half-integer weight. Recall the famous 'congruence modulo 11' between the normalised Discriminant function of weight 12 and the newform f of weight 2 attached to Elliptic curve of conductor 11. Using Shimura's correspondence (1973), which connects modular forms of weight 2k with half-integer modular forms of weight k+1/2, our congruence descends to a congruence modulo 11 between half integer modular forms of weight 3/2 and 13/2.

The talk shall begin with a brief introduction to modular forms of integer and half-integer weight leading to congruences modulo an odd prime p between them. I will also give an overview of current progress and generalisation of Theorem of McGraw and Ono to Hilbert modular forms of integer and half-integer weight.