In this paper, we study the Heegner points on more general modular curves other than $X_0(N)$, which generalizes Gross' work "Heegner points on $X_0(N)$". The explicit Gross-Zagier formula and the Euler system property are stated in this case. Using such kind of Heegner points, we construct certain families of quadratic twists of $X_0(36)$, with the ranks of Mordell-Weil groups being zero and one respectively, and show that the $2$-part of their BSD conjectures hold.
In recent work, Darmon, Pozzi and Vonk explicitly construct a modular form whose spectral coefficients are $p$-adic logarithms of Gross-Stark units and Stark-Heegner points. Here we describe how this construction gives rise to a practical algorithm for explicitly computing these logarithms to specified precision, and how to recover the exact values of the Gross-Stark units and Stark-Heegner points from them. Key tools are overconvergent modular forms, reduction theory of quadratic forms and Newton polygons. As an application, we tabulate Brumer-Stark units in narrow Hilbert class fields of rea…
Generalised Heegner cycles are associated to a pair of an elliptic Hecke eigenform and a Hecke character over an imaginary quadratic extension $K/\Q$. Let $p$ be an odd prime split in $K/\Q$ and $l\neq p$ an odd unramified prime. We prove the non-triviality of the $p$-adic Abel-Jacobi image of generalised Heegner cycles modulo $p$ over the $\Z_l$-anticylotomic extension of $K$. The result is an evidence for the refined Bloch-Beilinson and the Bloch-Kato conjecture. In the case of two, it provides a refinement of the results of Cornut and Vatsal on the non-triviality of Heegner points over the …
The principal goal of this paper is to develop Kolyvagin's descent to apply with the big Heegner point Euler system constructed by Howard for the big Galois representation $\mathbb{T}$ attached to a Hida family $\mathbb{F}$ of elliptic modular forms. In order to achieve this, we interpolate and control the Tamagawa factors attached to each member of the family $\mathbb{F}$ at bad primes, which should be of independent interest. Using this, we then work out the Kolyvagin descent on the big Heegner point Euler system so as to obtain a big Kolyvagin system that interpolates the collection of Koly…
We prove that Heegner cycles of codimension m+1 inside Kuga-Sato type varieties of dimension 2m+1 are coefficients of modular forms of weight 3/2+m in the appropriate quotient group. The main technical tool for generating the necessary relations is a Borcherds style theta lift with polynomials. We also show how this lift defines a new singular Shimura-type correspondence from weakly holomorphic modular forms of weight 1/2-m to meromorphic modular forms of weight 2m+2.
We compute the arithmetic intersection numbers of certain Heegner divisors on integral models of Shimura curves over Q. Our formulas generalize the formulas of Gross-Kohnen-Zagier for intersection numbers of Heegner divisors on integral models of modular curves.
Given a weight two modular form f with associated p-adic Galois representation V_f, for certain quadratic imaginary fields K one can construct canonical classes in the Galois cohomology of V_f by taking the Kummer images of Heegner points on the modular abelian variety attached to f. We show that these classes can be interpolated as f varies in a Hida family, and construct an Euler system of big Heegner points for Hida's universal ordinary deformation of V_f. We show that the specialization of this big Euler system to any form in the Hida family is nontrivial, extending results of Cornut and V…
In this paper we formulate a conjecture which partially generalizes the Gross-Kohnen-Zagier theorem to higher weight modular forms. For f in S_k(N) satisfying certain conditions, we construct a map from the Heegner points of level N to a complex torus defined by f. We define higher weight analogues of Heegner divisors on this torus. We conjecture they all lie on a line, and their positions are given by the coefficients of a certain Jacobi form corresponding to f. In weight 2, our map is the modular parametrization map (restricted to Heegner points), and our conjectures are implied by Gross-Koh…
Iwasawa theory of Heegner points on abelian varieties of GL_2 type has been studied by, among others, Mazur, Perrin-Riou, Bertolini and Howard. The purpose of this paper is to describe extensions of some of their results in which abelian varieties are replaced by the Galois cohomology of Deligne's p-adic representation attached to a modular form of even weight >2. In this setting, the role of Heegner points is played by higher-dimensional Heegner-type cycles that have been recently defined by Bertolini, Darmon and Prasanna. Our results should be compared with those obtained, via deformation-th…
Given a parametrisation of an elliptic curve over Q by a Shimura curve, we show that the images of almost all Heegner points are of infinite order. For parametrisations of elliptic curves by modular curves this was proven earlier by Nekovar and Schappacher by a different method.
We study generalised Heegner cycles, originally introduced by Bertolini-Darmon-Prasanna for modular curves, in the context of Mumford curves. The main result of this paper relates generalized Heegner cycles with the two variable anticyclotomic $p$-adic $L$-function attached to a Coleman family $f_\infty$ and an imaginary quadratic field $K$. Our generalised Heegner cycles allow us to study the restriction of this function to non-central critical lines. The main result expresses the derivative along the weight variable of this anticyclotomic $p$-adic $L$-function restricted to non necessarily c…
We study Heegner points on elliptic curves, or more generally modular abelian varieties, coming from uniformization by Shimura curves attached to a rather general type of quaternionic or- ders closely related to those introduced by Hijikata{Pizer{Shemanske in the 80's. We address several questions arising from the Birch and Swinnerton-Dyer (BSD) conjecture in this general context. In par- ticular, under mild technical conditions, we show the existence of non-torsion Heegner points on elliptic curves in all situations in which the BSD conjecture predicts their existence.
Let E be a modular elliptic curve defined over a rational function field k of odd characteristic. We construct a sequence of Heegner points on E, defined over a $Z_p^{\infty}$-tower of finite extensions of k, and show that these Heegner points generate a group of infinite rank. This is a function field analogue of a result of C.Cornut and V.Vatsal
We construct two-parameter analytic families of Galois cohomology classes interpolating the etale Abel--Jacobi images of generalised Heegner cycles, with both the modular form and Grossencharacter varying in p-adic families.
In this paper we prove a Gross-Zagier type formula for the anticyclotomic p-adic L-function of an elliptic modular form f of higher weight and of multiplicative type at p. For such f we also decribe explicitely the local Galois representation attached to it at p and compare the Fontaine-Mazur and Teitelbaum L-invariants.
We construct an Euler system of generalized Heegner cycles to bound the Selmer group associated to a modular form and an algebraic Hecke character. The main argument is based on Kolyvagin's machinery explained by Gross while the key object of the Euler system, the generalized Heegner cycles, were first considered by Bertolini, Darmon and Prasanna.
Let f be a Bianchi modular form, that is, an automorphic form for GL(2) over an imaginary quadratic field F, and let P be a prime of F at which f is new. Let K be a quadratic extension of F, and L(f/K,s) the L-function of the base-change of f to K. Under certain hypotheses on f and K, the functional equation of L(f/K,s) ensures that it vanishes at the central point. The Bloch--Kato conjecture predicts that this should force the existence of non-trivial classes in an appropriate global Selmer group attached to f and K. In this paper, we use the theory of double integrals developed by Barrera Sa…
Let $E$ be an elliptic curve of conductor $N$, and let $K$ be an imaginary quadratic field such that the root number of $E/K$ is $-1$. Let $O$ be an order in $K$ and assume that there exists an odd prime $p$, such that $p^2 \mid\mid N$, and $p$ is inert in $O$. Although there are no Heegner points on $X_0(N)$ attached to $O$, in this article we construct such points on Cartan non-split curves. In order to do that we give a method to compute Fourier expansions for forms in Cartan non-split curves, and prove that the constructed points form a Heegner system as in the classical case.
We propose a conjectural construction of global points on modular elliptic curves over arbitrary number fields, generalizing both the p-adic construction of Heegner points via Cerednik-Drinfeld uniformization and the definition of classical Stark-Heegner points. In alignment with Nekovar and Scholl's plectic conjectures, we expect the non-triviality of these plectic Stark-Heegner points to control the Mordell-Weil group of higher rank elliptic curves. We provide some indirect evidence for our conjectures by showing that higher order derivatives of anticyclotomic p-adic L-functions compute plec…
In \cite{LV}, Howard's construction of big Heegner points on modular curves was extended to general Shimura curves over the rationals. In this paper, we relate the higher weight specializations of the big Heegner points of \emph{loc.cit.} in the definite setting to certain higher weight analogues of the Bertolini-Darmon theta elements. As a consequence of this relation, some of the conjectures formulated in \cite{LV} are deduced from recent results of Chida-Hsieh.
Let $E/\bbq$ be an elliptic curve defined over $\bbq$ with conductor $N$ and $\gq$ the absolute Galois group of an algebraic closure $\bar{\bbq}$ of $\bbq$. We prove that for every $σ\in \gq$, the Mordell-Weil group $E(\oqs)$ of $E$ over the fixed subfield of $\bar{\bbq}$ under $σ$ has infinite rank. Our approach uses the modularity of $E/\bbq$ and a collection of algebraic points on $E$ -- the so-called {\em Heegner points} -- arising from the theory of complex multiplication.
Generalised Heegner cycles are associated to a pair of an elliptic newform and a Hecke character over an imaginary quadratic extension $K/\Q$. The cycles live in a middle dimensional Chow group of a Kuga-Sato variety arising from an indefinite Shimura curve over the rationals and a self product of a CM abelian surface. Let $p$ be an odd prime split in $K/\Q$. We prove the non-triviality of the $p$-adic Abel-Jacobi image of generalised Heegner cycles modulo $p$ over the $\Z_p$-anticylotomic extension of $K$. The result implies the non-triviality of the generalised Heegner cycles in the top grad…
In an earlier paper the author proved one divisibility of Perrin- Riou's Iwasawa main conjecture for Heegner points on elliptic curves. In the present paper, that result is generalized to abelian varieties of GL2-type (i.e. abelian varieties with real multiplication defined over totally real fields) under the hypothesis that the abelian variety is associated to a Hilbert modular form via a construction of Zhang.
We prove an abstract modularity result for classes of Heegner divisors in the generalized Jacobian of a modular curve associated to a cuspidal modulus. Extending the Gross-Kohnen-Zagier theorem, we prove that the generating series of these classes is a weakly holomorphic modular form of weight 3/2. Moreover, we show that any harmonic Maass forms of weight 0 defines a functional on the generalized Jacobian. Combining these results, we obtain a unifying framework and new proofs for the Gross-Kohnen-Zagier theorem and Zagier's modularity of traces of singular moduli, together with new geometric i…
We establish a congruence formula between $p$-adic logarithms of Heegner points for two elliptic curves with the same mod $p$ Galois representation. As a first application, we use the congruence formula when $p=2$ to explicitly construct many quadratic twists of analytic rank zero (resp. one) for a wide class of elliptic curves $E$. We show that the number of twists of $E$ up to twisting discriminant $X$ of analytic rank zero (resp. one) is $\gg X/\log^{5/6}X$, improving the current best general bound towards Goldfeld's conjecture due to Ono--Skinner (resp. Perelli--Pomykala). We also prove th…
Given an elliptic curve $E$ over $\mathbb{Q}$, a celebrated conjecture of Goldfeld asserts that a positive proportion of its quadratic twists should have analytic rank 0 (resp. 1). We show this conjecture holds whenever $E$ has a rational 3-isogeny. We also prove the analogous result for the sextic twists of $j$-invariant 0 curves (Mordell curves). To prove these results, we establish a general criterion for the non-triviality of the $p$-adic logarithm of Heegner points at an Eisenstein prime $p$, in terms of the relative $p$-class numbers of certain number fields and then apply this criterion…
Kolyvagin used Heegner points to associate a system of cohomology classes to an elliptic curve over $\Q$ and conjectured that the system contains a non-trivial class. His conjecture has profound implications on the structure of Selmer groups. We provide new computational and theoretical evidence for Kolyvagin's conjecture. More precisely, we explicitly compute Heegner points over ring class fields and use these points to verify the conjecture for specific elliptic curves of rank two. We explain how Kolyvagin's conjecture implies that if the analytic rank of an elliptic curve is at least two th…
Building on ideas of Vatsal, Cornut proved a conjecture of Mazur asserting the generic nonvanishing of Heegner points on an elliptic curve E as one ascends the anticyclotomic Z_p-extension of a quadratic imaginary extension K/Q. In the present article Cornut's result is extended by replacing the elliptic curve E with the Galois cohomology of Deligne's 2-dimensional l-adic representation attached to a modular form of weight 2k>2, and replacing the family of Heegner points with an analogous family of special cohomology classes.
Let k be a global field, $\bar{k}$ a separable closure of k, and $G_k$ the absolute Galois group $\Gal(\bar{k}/k)$ of $\bar{k}$ over k. For every g in $G_k$, let $\bar{k}^g$ be the fixed subfield of $\bar{k}$ under g. Let E/k be an elliptic curve over k. We show that for each g in $G_k$, the Mordell-Weil group $E(\bar{k}^g)$ has infinite rank in the following two cases. Firstly when k is a global function field of odd characteristic and E is parametrized by a Drinfeld modular curve, and secondly when k is a totally real number field and E/k is parametrized by a Shimura curve. In both cases our…
We describe the specialization of Heegner points on Shimura curves at primes of bad reduction. Moreover, we give some reciprocity laws relating the Galois action on these points to natural actions on the set of singular points and the set of connected components of the fiber.
We explain how to find a rational point on a rational elliptic curve of rank 1 using Heegner points. We give some examples, and list new algorithms that are due to Cremona and Delaunay. These are notes from a short course given at the Institut Henri Poincare in December 2004.
This paper is complementary to the work Rosen-Silverman, which derives a criteria on the number fields for the independence of Heegner points associated to them on non-CM elliptic curves. We show that the same criteria holds for CM elliptic curves. A generalisation of the independence criteria for the Heegner points associated to an order of fixed conductor of quadratic imaginary fields can also be found in this paper.
Let $f$ be a primitive Hilbert modular form of parallel weight $2$ and level $N$ for the totally real field $F$, and let $p$ be a rational prime coprime to $2N$. If $f$ is ordinary at $p$ and $E$ is a CM extension of $F$ of relative discriminant $Δ$ prime to $Np$, we give an explicit construction of the $p$-adic Rankin-Selberg $L$-function $L_{p}(f_E,\cdot)$. When the sign of its functional equation is $-1$, we show, under the assumption that all primes $\wp\vert p$ are principal ideals of ${\mathcal O}_{F}$ which split in ${\mathcal O}_{E}$, that its central derivative is given by the $p$-adi…
We improve Kolyvagin's upper bound on the order of the $p$-primary part of the Shafarevich-Tate group of an elliptic curve of rank one over a quadratic imaginary field. In many cases, our bound is precisely the one predicted by the Birch and Swinnerton-Dyer conjectural formula.
Under a non-torsion assumption on Heegner points, results of Kolyvagin describe the structure of Shafarevich-Tate groups of elliptic curves. In this paper we prove analogous results for ($p$-primary) Shafarevich-Tate groups associated with higher weight modular forms over imaginary quadratic fields satisfying a "Heegner hypothesis". More precisely, we show that the structure of Shafarevich-Tate groups is controlled by cohomology classes built out of Nekovář's Heegner cycles on Kuga-Sato varieties. As an application of our main theorem, we improve on a result of Besser giving a bound on the ord…
In this paper, we prove an "explicit reciprocity law" relating Howard's system of big Heegner points to a two-variable $p$-adic $L$-function (constructed here) interpolating the $p$-adic Rankin $L$-series of Bertolini-Darmon-Prasanna in Hida families. As applications, we obtain a direct relation between classical Heegner cycles and the higher weight specializations of big Heegner points, refining earlier work of the author, and prove the vanishing of Selmer groups of CM elliptic curves twisted by 2-dimensional Artin representations in cases predicted by the equivariant Birch and Swinnerton-Dye…
Let $E$ be a rational elliptic curve, and $K$ be an imaginary quadratic field. In this article we give a method to construct Heegner points when $E$ has a prime bigger than $3$ of additive reduction ramifying in the field $K$. The ideas apply to more general contexts, like constructing Darmon points attached to real quadratic fields which is presented in the appendix.
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)) is -1 (respectively, +1), we prove a ``Jochnowitz congruence'' between the algebraic part of L'_K(E,1) (expressed in terms of Heegner points on Shimura curves) and the algebraic part of L_K(g,1). This establishes a relation between Zhang's formula of Gross-Zagier type for central derivatives of L-series and his formula of Gross type for special values. Our r…
Mohammed Ben Alhocain, in an Arab manuscript of the tenth century, stated that the principal object of the theory of rational right triangles is to find a square which when increased or diminished by a certain number $m$ becomes a square (see Dickson). In modern language, this object is to find a rational point of infinite order on the elliptic curve $my^2=x^3-x$. Heegner constructed (see also Monsky) such rational points in the case that $m$ are primes congruent to 5, 7 modulo 8 or twice primes congruent to 6 modulo 8. We extend Heegner's result to integers $m$ with many prime divisors.
Perrin-Riou has formulated a form of the Iwasawa main conjecture, which relates Heegner points to the Selmer group of an elliptic curve as one goes up the anticyclotomic Z_p extension of a quadratic imaginary field K. Building on the earlier work of Bertolini on this conjecture, and making use of the recent work of Mazur and Rubin on Kolyvagin's theory of Euler systems, we prove one divisibility of Perrin-Riou's conjectured equality. As a consequence, one obtains an upper bound on the rank of the Mordell-Weil group E(K) in terms of Heegner points.
We give a new proof of Howard's $Λ$-adic Gross-Zagier formula, which we extend to the context of indefinite Shimura curves over $\mathbf{Q}$ attached to nonsplit quaternion algebras. This formula relates the cyclotomic derivative of a two-variable $p$-adic $L$-function restricted to the anticyclotomic line to the cyclotomic $p$-adic heights of Heegner points over the anticyclotomic tower, and our proof, rather than inspired by the original approaches of Gross-Zagier and Perrin-Riou, is via Iwasawa theory, based on the connection between Heegner points, Beilinson-Flach elements, and their expli…
ATR points were introduced by Darmon as a conjectural construction of algebraic points on certain elliptic curves for which in general the Heegner point method is not available. So far the only numerical evidence, provided by Darmon--Logan and Gärtner, concerned curves arising as quotients of Shimura curves. In those special cases the ATR points can be obtained from the already existing Heegner points, thanks to results of Zhang and Darmon--Rotger--Zhao. In this paper we compute for the first time an algebraic ATR point on a curve which is not uniformizable by any Shimura curve, thus providing…
For certain elliptic curves $E$ over $\mathbb{Q}$ with multiplicative reduction at a prime $p\geq 5$, we prove the $p$-indivisibility of the derived Heegner classes defined with respect to an imaginary quadratic field $K$, as conjectured by Kolyvagin. The conditions on $E$ include that $E[p]$ be irreducible and not finite at $p$ and that $p$ split in the imaginary quadratic field $K$, along with certain $p$-indivisibility conditions on various Tamagawa factors. The proof extends the arguments of the second author for the case where $E$ has good ordinary reduction at~$p$.
In this paper we propose and prove an anticyclotomic Iwasawa main conjecture for Heegner points on supersingular elliptic curves with $a_p=0$. The result has a "$\pm$" nature in the sense of Kobayashi. The proof uses a recent work of the second author on one divisibility in the Iwasawa--Greenberg main conjecture for Rankin-Selberg $p$-adic $L$-functions, together with an argument of Howard (adapted to our "$\pm$"-situation). As a byproduct, we also obtain an improvement of Skinner's result on a converse to the Gross--Zagier--Kolyvagin theorem.
The main result of this text is a generalization of Perrin-Riou's p-adic Gross-Zagier formula to the case of Shimura curves over totally real fields. Let $F$ be a totally real field. Let $f$ be a Hilbert modular form over $F$ of parallel weight $2$, which is a new form and is ordinary at $p$. Let $E$ be a totally imaginary quadratic extension of $F$ of discriminant prime to $p$ and to the conductor of $f$. We may construct a $p$-adic $L$ function that interpolates special values of the complex $L$ functions associated to $f$, $E$ and finite order Hecke characters of $E$. The $p$-adic Gross-Zag…
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 Shimura curves, a formula of Gross and Zagier for the endomorphism ring of Heegner points over Artinian rings and the connection between Ribet's bimodules and the specialization of Heegner points. As an application, we provide a list of equations of Shimura curves and quotients of them obtained by our algorithm that had been conjectured by Kurihara.
The asymptotic behaviour of the Neron-Tate height of Heegner points on a rational elliptic curve attached to an arithmetically normalized new cusp form f of weight 2, level N and trivial character is studied in this paper. By Gross-Zagier formula, this height is related to the special value at the critical point for the derivative of the Rankin-Selberg convolution of f with a certain weight one theta series attached to some ideal class of some imaginary quadratic field. Asymptotic formula for the first moments asociated to these Dirichlet series are proved and experimental results are carefull…
Iwasawa theory of Heegner points on abelian varieties of GL_2 type has been studied by, among others, Mazur, Perrin-Riou, Bertolini and Howard. The purpose of this paper, the first in a series of two, is to describe extensions of some of their results in which abelian varieties are replaced by the Galois cohomology of Deligne's p-adic representation attached to a modular form f of even weight >2. In this more general setting, the role of Heegner points is played by higher-dimensional Heegner cycles in the sense of Nekovář. In particular, we prove that the Pontryagin dual of a certain Bloch-Kat…
