Joint IAS/Princeton University Number Theory Seminar

Waldspurger has shown that the genuine automorphic cuspidal representations of the metaplectic cover $S$ of $SL_2$ are divided naturally into packets, and that these packets are indexed by the cuspidal automorphic representations of $PGL_2$. We construct packets of Lambda-adic modular forms of half integral weight, indexed by Lambda-adic forms on $PGL_2$. The elements of the Lambda-adic packets are nonzero, but they have specializations that vanish, owing to a trivial zero phenomenon and the sign of a complex root number. This is in contrast to the usual trivial zero phenomenon which arises from the vanishing of a $p$-adic factor.