Shimura correspondence for level p2 and the central values of L-series, II

Given a Hecke eigenform f of weight 2 and square-free level N, by the work of Kohnen, there is a unique weight 3/2 modular form of level 4N mapping to f under the Shimura correspondence. Furthermore, by the work of Waldspurger the Fourier coefficients of such a form are related to the quadratic twists of the form f. Gross gave a construction of the half integral weight form when N is prime, and such construction was later generalized to square-free levels. However, in the non-square free case, the situation is more complicated since the natural construction is vacuous. The problem being that there are too many special points so that there is cancellation while trying to encode the information as a linear combination of theta series. In this paper, we concentrate in the case of level p2, for p > 2 a prime number, and show how the set of special points can be split into subsets (indexed by bilateral ideals for an order of reduced discriminant p2) which gives two weight 3/2 modular forms mapping to f under the Shimura correspondence. Moreover, the splitting has a geometric interpretation which allows to prove that the forms are indeed a linear combination of theta series associated to ternary quadratic forms. Once such interpretation is given, we extend the method of Gross–Zagier to the case where the level and the discriminant are not prime to each other to prove a Gross-type formula in this situation.

Modular forms of half-integral weights on SL(2, ℤ)

In this paper, we prove that, for an integerrwith (r, 6) = 1 and 0< r <24 and a nonnegative even integers, the setis isomorphic toas Hecke modules under the Shimura correspondence. HereMs(1) denotes the space of modular forms of weightis the space of newforms of weight 2kon Γ0(6) that are eigenfunctions with eigenvalues€2and€3for Atkin-Lehner involutionsW2andW3, respectively, and the notation ⊕(12/.) means the twist by the quadratic character (12/-). There is also an analogous result for the cases (r, 6) = 3.

Modular forms of half-integral weights on SL(2, ℤ)

In this paper, we prove that, for an integer r with (r, 6) = 1 and 0 < r < 24 and a nonnegative even integer s, the setis isomorphic toas Hecke modules under the Shimura correspondence. Here Ms(1) denotes the space of modular forms of weight is the space of newforms of weight 2k on Γ0 (6) that are eigenfunctions with eigenvalues €2 and €3 for Atkin-Lehner involutions W2 and W3, respectively, and the notation ⊕(12/.) means the twist by the quadratic character (12/-). There is also an analogous result for the cases (r, 6) = 3.

Ternary quadratic forms and half-integral weight modular forms

Let k be a positive integer such that k≡3 mod 4, and let N be a positive square-free integer. In this paper, we compute a basis for the two-dimensional subspace Sk/2(Γ0(4N),F) of half-integral weight modular forms associated, via the Shimura correspondence, to a newform F∈Sk−1(Γ0(N)), which satisfies $L(F,\frac {1}{2})\neq 0$. This is accomplished by using a result of Waldspurger, which allows one to produce a basis for the forms that correspond to a given F via local considerations, once a form in the Kohnen space has been determined.

Representations of metaplectic groups I: epsilon dichotomy and local Langlands correspondence

Using theta correspondence, we classify the irreducible representations of Mp2n in terms of the irreducible representations of SO2n+1 and determine many properties of this classification. This is a local Shimura correspondence which extends the well-known results of Waldspurger for n=1.

Stark–Heegner points and the Shimura correspondence

Let $g = \sum c(D)q^D$ and $f=\sum a_n q^n$ be modular forms of half-integral weight k+1/2 and integral weight 2k respectively that are associated to each other under the Shimura–Kohnen correspondence. For suitable fundamental discriminants D, a theorem of Waldspurger relates the coefficient c(D) to the central critical value L(f,D,k) of the Hecke L-series of f twisted by the quadratic Dirichlet character of conductor D. This paper establishes a similar kind of relationship for central critical derivatives in the special case k=1, where f is of weight 2. The role of c(D) in our main theorem is played by the first derivative in the weight direction of the Dth Fourier coefficient of a p-adic family of half-integral weight modular forms. This family arises naturally, and is related under the Shimura correspondence to the Hida family interpolating f in weight 2. The proof of our main theorem rests on a variant of the Gross–Kohnen–Zagier formula for Stark–Heegner points attached to real quadratic fields, which may be of some independent interest. We also formulate a more general conjectural formula of Gross–Kohnen–Zagier type for Stark–Heegner points, and present numerical evidence for it in settings that seem inaccessible to our methods of proof based on p-adic deformations of modular forms.