Stark–Heegner points and the Shimura correspondence

Abstract

Let g = c(D)q D and f = 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 StarkHeegner 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.

Overconvergent generalised eigenforms of weight one and class fields of real quadratic fields

“…Note that although ˜ is always ramified at p, (8) ensures that the trace of ˜ (σ p ) does not depend on the choice of Frobenius element at p.…”

“…The p-adic logarithms of the same Heegner points are realised in [8] as the Fourier coefficients of a "modular form of weight 3/2 + ε" arising as an infinitesimal p-adic deformation of g over weight space. It would be interesting to flesh out the rather tantalising analogy between weak harmonic Maass forms and p-adic deformations of classical eigenforms.…”

“…while a direct calculation of ˜ (σ 2 ) using (8) shows that ψ(σ 2 ) ψ (σ 2 )κ (σ 2 ) · ε ψ(σ 2 )κ(σ 2 ) · ε ψ (σ 2 ) = η(σ )η (σ ) η (σ )(d 1 (σ ) + d 2 (σ )) · ε η (σ )(d 1 (σ ) + d 2 (σ )) · ε η (σ )η (σ ) .…”

The 𝑝-adic variation of the Gross–Kohnen–Zagier theorem

“…However, it should be noticed that the coefficients we are considering in this work and in [LN18] are slightly different. In this paper we consider coefficients of Jacobi forms which are lifts of modular newforms of level prime to p and trivial character, whose ordinary p-stabilisation belong to our Hida family; the discrepancy between newforms and their p-stabilisations is the origin of the Euler factors (denoted E p and defined in (5) below) relating p-adic L-functions with Heegner points and Jacobi-Fourier coefficients of Jacobi forms in a way similar to [DT08]; this should clarify the crucial role played in this paper by the p-adic L-functions from [BD07]. On the other hand, in [LN18] we consider Jacobi forms lifting the members of the Hida family f 2k (or more generally f κ for an arithmetic point κ, as in [Ste94]).…”

Darmon cycles and the Kohnen-Shintani lifting

“…We now introduce a normalization of the Fourier coefficients c(D, k) (See Proposition 1.3 of [13] and Proposition 3.3 of [25]). For D * a fundamental discriminant of either Type and for every k ∈ U cl , define the normalized Fourier coefficient as follows :…”