Shimura Lift Research Articles

Shimura lift of real analytic Poincaré series and Hilbert modular Eisenstein series

Shimura lift of real analytic Poincaré series and Hilbert modular Eisenstein series

Newforms of half-integral weight

Two very different definitions of a newform of half-integral weight are present and continued to be developed in the literature. The first definition originated with Serre and Stark for forms of weight 1/2 [5], and is analogous to the definition of newform for integral weight forms, which uses forms of lower level and shifts of such forms to characterize the notion of old-forms. The second definition originated with Kohnen for half-integral weight forms of squarefree level [1], who used forms of lower level and their image under the Um2 operator to define the notion of oldforms. The choice of the Um2 operator over the shift operator Bd seems a propitious one, since the U operator commutes with the action of the Shimura lift, while the shift operator B does not. More to the point, Kohnen was able to develop a newform theory on a distinguished subspace of the full space of cusp forms (now referred to as the Kohnen subspace), and obtained a multiplicity-one result (with respect to Hecke eigenvalues) for half-integral weight newforms in this subspace. Even nicer, the multiplicity-one result was established by showing that there is a one-to-one correspondence between newforms of level AN in the subspace and the newforms of integral weight of level N.

The correspondence between theta series of ternary and quaternary quadratic forms

This article uses orthogonal groups and quaternion algebras to describe the Shimura lifting for theta series of general three-dimensional lattices with spherical coefficients.

On Shintani correspondence

For an even natural numberm we will construct a Shintani lifting fromS 2k (m,ψ 2) toS k+1/2(Γ0(2m),ψ 0), ψ a Dirichlet character modulo2m, $$\psi _0 = \left( {\frac{{\psi ( - 1)}}{ \cdot }} \right)\psi $$ , which is adjoint to the Shimura lifting. Whenm is squarefree we will combine this result with the multiplicity 1 theorem proved in [3] to give a formula for the product $$c(m_1 )\overline {c(m_2 )} $$ (m 2 squarefree) of two Fourier coefficients of a new form inS +1/2 new (Γ0(2m)) in terms of certain cycle integrals of the corresponding new form inS 2 new (m). Ifm 1=m 2 we will get Waldspurger's result expressing the squarec(m 1)2 in terms of anL-series off at the centre of the critical strip. Similar product formulas for the Fourier coefficients of a Hecke-Pizer eigenform inS +1/2 old (Γ0(2m)) andS +1/2 +,old (Γ0(2m)) and of a Hecke eigenform inS +1/2 +,old (Γ0(4q)) (q⦻3(4) is a prime) are also given.

The application of the Shimura lifting to the representation of large numbers of ternary quadratic forms

One obtains an asymptotic formula for the number of the representations of numbers, divisible by a large square, by a positive ternary integral quadratic form. One gives an estimate of the remainder, unimprovable with respect to the quadratic part and uniform with respect to the square-free part of the represented number.

Theta series of weight [formula omitted

We show that a cusp form of weight 3 2 , which corresponds to an Eisenstein series of weight 2 via the Shimura lifting, must be linear combination of theta series attached to quadratic forms of one variable.