Jeehoon Park
Title:
p-adic family of half-integral weight modular forms via overconvergent
Shintani lifting.
Abstract:
The goal of this talk is to construct
a p-adic analytic family of
overconvergent half-integral
weight modular forms using Hecke-equivariant
overconvergent Shintani lifting. The
classical
Shintani map is the Hecke-equivariant map from
the space of cusp forms of integral
weight to the space of cusp forms of half-integral weight.
Glenn Stevens proved that
there is a $\Lambda$-adic lifting of
this map to the Hida family of ordinary cusp
forms of integral weight
via the cohomological description of the Shintani correspondence, and consequently
constructed a
$\Lambda$-adic modular eigenform
of half-integral weight. The natural thing to do is
generalize his result to the non-ordinary case, i.e. Coleman's p-adic analytic
family of overconvergent cusp forms of finite slope. For this
we will use a slope h
decomposition of compact supported cohomology with values in overconvergent
distribution (overconvergent modular
symbol), which can be interpreted as a cohomological description of Coleman's
p-adic family, and
define the p-adic Hecke algebra for the slope h part of this cohomology.
Then we
follow the
idea of Stevens in the ordinary case.
This construction implies that we found a
local rigid analytic
map from Coleman-Mazur integral weight Eigencurve to half-integral weight
eigencurve. The important feature
of overconvergent Shintani lifting is that it is
purely cohomological and algebraic,
which depends only on the arithmetic
of integral indefinite binary quadratic forms,
so that we can describe Hecke
operators explicitly on
q-expansion of universal overconvergent
half-integral weight modular forms using the
Hecke action
on overconvergent modular symbol.