Gross-Zagier Formula Research Articles

Heegner points on modular curves

In this paper, we study the Heegner points on more general modular curves other than X 0 ( N ) X_0(N) , which generalizes Gross’ work “Heegner points on X 0 ( N ) X_0(N) ”. The explicit Gross-Zagier formula and the Euler system property are stated in this case. Using such a kind of Heegner points, we construct certain families of quadratic twists of X 0 ( 36 ) X_0(36) , with the ranks of Mordell-Weil groups being zero and one respectively, and show that the 2 2 -part of their BSD conjectures hold.

Cube sum problem and an explicit Gross-Zagier formula

A nonzero rational number is called a {\it cube sum} if it is of the form $a^3+b^3$ with $a,b\in{\Bbb Q}^\times$. In this paper, we prove that for any odd integer $k\geq 1$, there exist infinitely many cube-free odd integers $n$ with exactly $k$ distinct prime factors such that $2n$ is a cube sum (resp. not a cube sum). We present also a general construction of Heegner points and obtain an explicit Gross-Zagier formula which is used to prove the Birch and Swinnerton-Dyer conjecture for certain elliptic curves related to the cube sum problem.

$p$-adic heights of generalized Heegner cycles

We relate the $p$-adic heights of generalized Heegner cycles to the derivative of a $p$-adic $L$-function attached to a pair $(f, \chi)$, where $f$ is an ordinary weight $2r$ newform and $\chi$ is an unramified imaginary quadratic Hecke character of infinity type $(\ell,0)$, with $0 < \ell < 2r$. This generalizes the $p$-adic Gross-Zagier formula in the case $\ell = 0$ due to Perrin-Riou (in weight two) and Nekovaur (in higher weight).

Explicit Gross–Zagier and Waldspurger formulae

We give an explicit form of Gross-Zagier formula on Shimura Curves and an explicit form of Waldspurger formula.

Heights of Kudla–Rapoport divisors and derivatives of $$L$$ L -functions

We study special cycles on integral models of Shimura varieties associated with unitary similitude groups of signature (n-1,1). We construct an arithmetic theta lift from harmonic Maass forms of weight 2-n to the arithmetic Chow group of the integral model of a unitary Shimura variety, by associating to a harmonic Maass form f a linear combination of Kudla-Rapoport divisors, equipped with the Green function given by the regularized theta lift of f. Our main result is an equality of two complex numbers: (1) the height pairing of the arithmetic theta lift of f against a CM cycle, and (2) the central derivative of the convolution L-function of a weight n cusp form (depending on f) and the theta function of a positive definite hermitian lattice of rank n-1. When specialized to the case n=2, this result can be viewed as a variant of the Gross-Zagier formula for Shimura curves associated to unitary groups of signature (1,1). The proof relies on, among other things, a new method for computing improper arithmetic intersections.

Modular parametrizations of certain elliptic curves

AbstractKaneko and Sakai [11] recently observed that certain elliptic curves whose associatednewforms (by the modularity theorem) are given by the eta-quotients can be character-ized by a particular differential equation involving modular forms and Ramanujan-Serredifferential operator.In this paper, we study certain properties of modular parametrization associated tothe elliptic curves over Q, and as a consequence we generalize and explain some of theirfindings.1. IntroductionBy the modularity theorem [4, 8], an elliptic curve Eover Qadmits a modular parametrizationΦ E : X 0 (N) → Efor some integer N. If Nis the smallest such integer, then it is equal to theconductor of Eand the pullback of the N´eron differential of Eunder Φ E is a rational multipleof 2πif E (τ), where f E (τ) ∈ S 2 (Γ 0 (N)) is a newform with rational Fourier coefficients. The factthat the L-function of f E (τ) coincides with the Hasse-Weil zeta function of E(which followsfrom Eichler-Shimura theory) is central to the proof of Fermat’s last theorem, and is related tothe Birch and Swinnerton-Dyer conjecture. In addition to this, modular parametrization is usedfor constructing rational points on elliptic curves, and appears in the Gross-Zagier formula [9].In this paper, we study some general properties of Φ

Diagonal cycles and Euler systems I: A $p$-adic Gross-Zagier formula

This article is the first in a series devoted to studying generalised Gross-KudlaSchoen diagonal cycles in the product of three Kuga-Sato varieties and the Euler system properties of the associated Selmer classes, with special emphasis on their application to the Birch–Swinnerton-Dyer conjecture and the theory of Stark-Heegner points. The basis for the entire study is a p-adic formula of Gross-Zagier type which relates the images of these diagonal cycles under the p-adic Abel-Jacobi map to special values of certain p-adic Lfunctions attached to the Garrett-Rankin triple convolution of three Hida families of modular forms. The main goal of this article is to describe and prove this formula. Cet article est le premier d’une serie consacree aux cycles de Gross-Kudla-Schoen generalises appartenant aux groupes de Chow de produits de trois varietes de Kuga-Sato, et aux systemes d’Euler qui leur sont associes. La serie au complet repose sur une variante p-adique de la formule de Gross-Zagier qui relie l’image des cycles de Gross-Kudla-Schoen par l’application d’Abel-Jacobi p-adique aux valeurs speciales de certaines fonctions L p-adiques attachees a la convolution de Garrett-Rankin de trois familles de Hida de formes modulaires cuspidales. L’objectif principal de cet article est de decrire et de demontrer cette variante. MSC: 11F12, 11G05, 11G35, 11G40.

A Gross–Zagier formula for quaternion algebras over totally real fields

We prove a higher dimensional generalization of Gross and Zagier's theorem on the factoriza- tion of differences of singular moduli. Their result is proved by giving a counting formula for the number of isomorphisms between elliptic curves with complex multiplication by two different imaginary quadratic fields K and K ' , when the curves are reduced modulo a supersingular prime and its powers. Equivalently, the Gross-Zagier formula counts optimal embeddings of the ring of integers of an imaginary quadratic field into particular maximal orders in Bp,∞, the definite quaternion algebra over Q ramified only at p and infinity. Our work gives an analogous counting formula for the number of simultaneous embeddings of the rings of integers of primitive CM fields into superspecial orders in definite quaternion algebras over totally real fields of strict class number 1. Our results can also be viewed as a counting formula for the number of isomorphisms modulo p|p between abelian varieties with CM by different fields. Our counting formula can also be used to determine which superspecial primes appear in the factorizations of differences of values of Siegel modular functions at CM points associated to two different CM fields, and to give a bound on those supersingular primes which can appear. In the special case of Jacobians of genus 2 curves, this provides information about the factorizations of numerators of Igusa invariants, and so is also relevant to the problem of constructing genus 2 curves for use in cryptography.

The p-adic Gross-Zagier formula for elliptic curves at supersingular primes

Let p be a prime number and let E be an elliptic curve defined over ℚ of conductor N. Let K be an imaginary quadratic field with discriminant prime to pN such that all prime factors of N split in K. B. Perrin-Riou established the p-adic Gross-Zagier formula that relates the first derivative of the p-adic L-function of E over K to the p-adic height of the Heegner point for K when E has good ordinary reduction at p. In this article, we prove the p-adic Gross-Zagier formula of E for the cyclotomic ℤ p -extension at good supersingular prime p. Our result has an application for the full Birch and Swinnerton-Dyer conjecture. Suppose that the analytic rank of E over ℚ is 1 and assume that the Iwasawa main conjecture is true for all good primes and the p-adic height pairing is not identically equal to zero for all good ordinary primes, then our result implies the full Birch and Swinnerton-Dyer conjecture up to bad primes. In particular, if E has complex multiplication and of analytic rank 1, the full Birch and Swinnerton-Dyer conjecture is true up to a power of bad primes and 2.

Arithmetic of Shimura curves

This is the note for a series of lectures that the author gave at the Centre de Recerca Matematica (CRM), Bellaterra, Barcelona, Spain on October 19–24, 2009. The aim is to give a comprehensive description of some recent work of the author and his students on generalisations of the Gross-Zagier formula, Euler systems on Shimura curves, and rational points on elliptic curves.

The rationality of Stark-Heegner points over genus fields of real quadratic fields

We study the algebraicity of Stark-Heegner points on a modular elliptic curve E. These objects are /?-adic points on E given by the values of certain /?-adic integrals, but they are conjecturally defined over ring class fields of a real quadratic field K. The present article gives some evidence for this algebraicity conjecture by showing that linear combinations of Stark-Heegner points weighted by certain genus characters of K are defined over the predicted quadratic extensions of K. The non- vanishing of these combinations is also related to the appropriate twisted Hasse-Weil L-series of E over K, in the spirit of the Gross-Zagier formula for classical Heegner points.

Faltings heights of CM cycles and derivatives of L-functions

We study the Faltings height pairing of arithmetic special divisors and CM cycles on Shimura varieties associated to orthogonal groups. We compute the Archimedean contribution to the height pairing and derive a conjecture relating the total pairing to the central derivative of a Rankin L-function. We prove the conjecture in certain cases where the Shimura variety has dimension 0, 1, or 2. In particular, we obtain a new proof of the Gross-Zagier formula.

Geometric non-vanishing

We consider L-functions attached to representations of the Galois group of the function field of a curve over a finite field. Under mild tameness hypotheses, we prove non-vanishing results for twists of these L-functions by characters of order prime to the characteristic of the ground field and more generally by certain representations with solvable image. We also allow local restrictions on the twisting representation at finitely many places. Our methods are geometric, and include the Riemann-Roch theorem, the cohomological interpretation of L-functions, and monodromy calculations of Katz. As an application, we prove a result which allows one to deduce the conjecture of Birch and Swinnerton-Dyer for non-isotrivial elliptic curves over function fields whose L-function vanishes to order at most 1 from a suitable Gross-Zagier formula.

Elliptic Curves and Class Fields of Real Quadratic Fields: Algorithms and Evidence

The article [Darmon 02] proposes a conjectural p-adic analytic construction of points on (modular) elliptic curves, points which are defined over ring class fields of real quadratic fields. These points are related to classical Heegner points in the same way as Stark units to circular or elliptic units. For this reason they are called “Stark-Heegner points,” following a terminology introduced in [Darmon 98]. If K is a real quadratic field, the Stark-Heegner points attached to K are conjectured to satisfy an analogue of the Shimura reciprocity law, so that they can in principle be used to find explicit generators for the ring class fields of K. It is also expected that their heights can be expressed in terms of derivatives of the Rankin L-series attached to E and K, in analogy with the Gross-Zagier formula . The main goal of this paper is to describe algorithms for calculating Stark-Heegner points and supply numerical evidence for the Shimura reciprocity and Gross-Zagier conjectures, focussing primarily on elliptic curves of prime conductor.

On the Central Derivative of Hecke L-Series

The well-known Birch and Swinnerton-Dyer conjecture gives a deep connection between the leading coefficient of the L-series and the arithmetic properties of an abelian variety. Both are very important and subtle. This paper is part of an effort to compute the analytic side explicitly in a special case. Indeed, we are interested in the central derivative of certain algebraic Hecke L-series, related to CM abelian varieties or more precisely pieces of it (CM motives). The result, together with the Gross-Zagier formula proved by Zhang ([Zh]), would also give a new way to compute the height of certain Heegner cycles on a Kuga-Sato variety. Let p ≡ 3 mod 4 be a prime number such that p > 3. Let k ≥ 0 be an integer. Let E = Q( √−p) and view it as a subfield of C such that √−p = i√p. Let hp be the ideal class number of E. A canonical Hecke character of E of weight 2k + 1 is a Hecke character μ satisfying

Euler systems and Jochnowitz congruences

This article relates the Gross-Zagier formula with a simpler formula of Gross for special values of L -series, via the theory of congruences between modular forms. Given two modular forms f and g (of different levels) which are congruent but whose functional equations have sign -1 and 1 respectively, and an imaginary quadratic field K satisfying certain auxiliary conditions, the main result gives a congruence between the algebraic part of L '( f/K , 1) (expressed in terms of Heegner points) and the algebraic part of the special value L ( g / K , 1). Congruences of this type were anticipated by Jochnowitz, and for this reason are referred to as "Jochnowitz congruences."

A Conjectural Extension of the Gross-Zagier Formula on Singular Moduli

The values of the elliptic modular $j$-invariant at imaginary quadratic arguments are algebraic integers, known as singular moduli of level one. If $d_1$ and $d_2$ are imaginary quadratic discriminants, then we may consider a generalized resultant of the class polynomials of the orders of discriminant $d_1$ and $d_2$; this is the norm of the differences of singular moduli of the corresponding orders, denoted here by $J(d_1,d_2)$. These resultants are highly factorizable; Gross-Zagier established a closed formula for $J(d_1,d_2)^2$ when $d_1$ and $d_2$ are fundamental discriminants, with $(d_1,d_2)=1$. In this paper we present a conjectural extension of the Gross-Zagier formula to the case when $d_1$ and $d_2$ are not necessarily fundamental, and $(d_1,d_2)=l^e$, where 1 is a prime not dividing the product of the conductors of $d_1$ and $d_2$.

A Rigid Analytic Gross-Zagier Formula and Arithmetic Applications

Let f be a newform of weight 2 and squarefree level N. Its Fourier coefficients generate a ring Of whose fraction field Kf has finite degree over Q. Fix an imaginary quadratic field K of discriminant prime to N, corresponding to a Dirichlet character E. The L-series L(f /K, s) = L(f, s)L(f 0 E, s) of f over K has an analytic continuation to the whole complex plane and a functional equation relating L(f/K, s) to L(f/K, 2 s). Assume that the sign of this functional equation is 1, so that L(f/K, s) vanishes to even order at s = 1. This is equivalent to saying that the number of prime factors of N which are inert in K is odd. Fix any such prime, say p. The field K determines a factorization N = N+Nof N by taking N+, resp. Nto be the product of all the prime factors of N which are split, resp.