Related papers: Explicit Gross-Zagier and Waldspurger Formulae

The Gross-Zagier formula on singular moduli can be seen as a calculation of the intersection multiplicity of two CM divisors on the integral model of a modular curve. We prove a generalization of this result to a Shimura curve.

We survey recent developments on generalizing the Gross--Zagier formula to high dimensional Shimura varieties, with an emphasis on the Arithmetic Gan--Gross--Prasad conjecture and the relative trace formula approach.

We construct explicit equations of Cartwright-Steger and related surfaces.

Xue proved an equational refinement of the unitary Shimura curve case of the arithmetic Gan-Gross-Prasad conjecture via the Gross-Zagier formula for quaternionic Shimura curves. On the other hand, Rapoport, Smithling and Zhang posed a…

We describe a construction of preimages for the Shimura map on Hilbert modular forms, and give an explicit Waldspurger type formula relating their Fourier coefficients to central values of twisted L-functions. Our construction is inspired…

We present an explicit formula for Witten-Kontsevich tau-function.

The purpose of this paper is to generalize some results of Gross-Zagier [20] and Kolvyvagin [28] to totally real fields.

We give some methods for computing equations for certain Shimura curves, natural maps between them, and special points on them. We then illustrate these methods by working out several examples in varying degrees of detail. For instance, we…

Some special solutions to the multidimensional Lam\'e and Bourlet type equations are constructed in an explicit form.

We describe a construction of preimages for the Shimura map on Hilbert modular forms using generalized theta series, and give an explicit Waldspurger type formula relating their Fourier coefficients to central values of twisted…

The Gross-Kohnen-Zagier theorem describes Heegner points on a modular curve in terms of coefficients of modular forms. We give another proof of this theorem which generalizes to higher dimensions.

We describe an algorithm that computes explicit models of hyperelliptic Shimura curves attached to an indefnite quaternion algebra over Q and Atkin-Lehner quotients of them. It exploits Cerednik-Drinfeld's non-archimedean uniformisation of…

We study the explicit formula of Euler numbers and polynomials of higher order

An explicit closed form expression for 2-correlators of Witten's two dimensional topological gravity is derived in arbitrary genus.

Direct proofs are given of Juhl's formulae for GJMS operators and Q-curvatures starting from the original construction of GJMS.

We extend the work of S. Zhang and Yuan-Zhang-Zhang to obtain a Gross-Zagier formula for modular forms of even weight in terms of an arithmetic intersection pairing of CM-cycles on Kuga-Sato varieties over Shimura curves. Combined with a…

The theorems of Gross-Zagier and Zhang relate the N\'eron-Tate heights of complex multiplication points on the modular curve X_0(N) (and on Shimura curve analogues) with the central derivatives of automorphic L-functions. We extend these…

The goal of this paper is to explicitly compute the Kodaira-Spencer map for a quaternionic Shimura curve over Q and its effect on the metrics of the Hodge bundle. The results are known to experts.

We derive an explicit formula for the action of a geometric Hecke correspondence on special cycles on a Shimura variety in terms of such cycles at a fixed neat level and compare it with another closely related expression sometimes used in…

Given a rational elliptic curve E, a suitable imaginary quadratic field K and a quaternionic Hecke eigenform g of weight 2 obtained from E by level raising such that the sign in the functional equation for L_K(E,s) (respectively, L_K(g,1))…