p-adic families of d-th Shintani liftings

TL;DR

This paper constructs a $ ext{Lambda}$-adic $ ext{d}$-th Shintani lifting, generalizes classical formulas relating Fourier coefficients and L-series, and explores connections to Stark--Heegner points.

Contribution

It provides a detailed construction of a $ ext{Lambda}$-adic $ ext{d}$-th Shintani lifting and extends classical formulas to this new setting.

Findings

Derived a $ ext{Lambda}$-adic version of Kohnen's formula.

Established a relation between Fourier coefficients and Stark--Heegner points.

Generalized classical formulas for Fourier coefficients and L-series.

Abstract

In this note we give a detailed construction of a $\Lambda$-adic $\mathfrak{d}$-th Shintani lifting. We derive a $\Lambda$-adic version of Kohnen's formula relating Fourier coefficients of half-integral weight modular forms and special values of twisted L-series. As a by-product, we derive a mild generalization of such classical formulae, and also point out a relation between Fourier coefficients of $\Lambda$-adic $\d$-th Shintani liftings and Stark--Heegner points.