Λ-adic modular forms of half-integral weight and a Λ-adic Shintani lifting

p-Adic family of half-integral weight modular forms via overconvergent Shintani lifting

“…Having Lemma 3.15 in our hands, the theorem follows from the same argument in the proof of the theorem 5.5 in [St1].…”

“…We then review its cohomological version which is well-suited to p-adic variation of Shintani map, following [St1] whose results we generalize to the non-ordinary case. Let Q = Q(X, Y ) = aX 2 +bXY +cY 2 be an integral (meaning a, b, c ∈ Z) binary quadratic form.…”

“…In this section we define the Hecke-equivariant overconvergent Shintani lifting (see [St1] for Λ-adic Shintani liftings) and describe the Hecke-action explicitly on the image of the lifting which we define to be the universal overconvergent half-integral weight modular form. Let σ :…”

“…Furthermore, he expressed the fourier coefficients of θ(f ) as some period integrals of f , hence can be described in terms of a certain cohomology class associated to f as in Shimura's work (see the chapter 8 of [Shm1]). G. Stevens realized that if we have a Λ-adic cohomology class which interpolates the p-stabilized ordinary newforms then it gives rise to a Λ-adic version of Shintani lifting for Hida's universal ordinary Λ-adic modular forms (see [St1]). The main goal of this paper is to generalize his result to the finite slope non-ordinary case, i.e.…”

p-Adic and analytic properties of period integrals and values of L-functions

“…Factorizations of p-adic L-functions as in question (3) have been obtained in a number of situations in low dimension; see for example [3,18,27,33]. Whenever there is a factorization of the central value of the complex L-function -references [33] and [27] Returning to the situation of Question 4.8, we arrive at a p-adic analogue of Question 2.1.…”