Heegner Points Research Articles

Generalized Birch lemma and the 2-part of the Birch and Swinnerton-Dyer conjecture for certain elliptic curves

Abstract In the present paper, we generalize the celebrated classical lemma of Birch and Heegner on quadratic twists of elliptic curves over ℚ {{\mathbb{Q}}} . We prove the existence of explicit infinite families of quadratic twists with analytic ranks 0 and 1 for a large class of elliptic curves, and use Heegner points to explicitly construct rational points of infinite order on the twists of rank 1. In addition, we show that these families of quadratic twists satisfy the 2-part of the Birch and Swinnerton-Dyer conjecture when the original curve does. We also prove a new result in the direction of the Goldfeld conjecture.

Heegner Point Kolyvagin System and Iwasawa Main Conjecture

We prove an anticyclotomic Iwasawa main conjecture proposed by Perrin-Riou for Heegner points for semi-stable elliptic curves E over a quadratic imaginary field K satisfying a certain generalized Heegner hypothesis, at an ordinary prime p. It states that the square of the index of the anticyclotomic family of Heegner points in E equals the characteristic ideal of the torsion part of its Bloch-Kato Selmer group (see Theorem 1.3 for precise statement). As a byproduct we also prove the equality in the Greenberg-Iwasawa main conjecture for certain Rankin-Selberg product (Theorem 1.7) under some local conditions, and an improvement of Skinner’s result on a converse of Gross-Zagier and Kolyvagin theorem (Corollary 1.11).

Intersection of class fields

Using class field theory, we prove a restriction on the intersection of the maximal abelian extensions associated with different number fields. This restriction is then used to improve a result of Rosen and Silverman about the linear independence of Heegner points. In addition, it yields effective restrictions for the special points lying on an algebraic subvariety in a product of modular curves. The latter application is related to the Andre-Oort conjecture.

The exceptional zero phenomenon for elliptic units

The exceptional zero phenomenon has been widely studied in the realm of p -adic L -functions, where the starting point lies in the foundational works of Ferrero–Greenberg, Katz, Gross, or Mazur–Tate–Teitelbaum, among others. This phenomenon also appears in the study of Euler systems, and in this case one is led to study higher order derivatives of cohomology classes in order to extract the arithmetic information which is usually encoded in the explicit reciprocity laws which make the connection with p -adic L -funtions. In this work, we focus on the elliptic units of an imaginary quadratic field and study this exceptional zero phenomenon, proving an explicit formula relating the logarithm of a derived elliptic unit either to special values of Katz’s two variable p -adic L -function or to its derivatives. Further, we interpret this fact in terms of an \mathcal L -invariant and relate this result to other approaches to the exceptional zero phenomenon, most notably to the work of Bley, and we also compare this setting with other scenarios concerning Heegner points and Beilinson–Flach elements.

Hybrid subconvexity for class group 𝐿-functions and uniform sup norm bounds of Eisenstein series

Abstract In this paper, we study hybrid subconvexity bounds for class group L-functions associated to quadratic extensions K / ℚ {K/\mathbb{Q}} (real or imaginary). Our proof relies on relating the class group L-functions to Eisenstein series evaluated at Heegner points using formulas due to Hecke. The main technical contribution is the uniform sup norm bound for Eisenstein series E ⁢ ( z , 1 / 2 + i ⁢ t ) ≪ ε y 1 / 2 ⁢ ( | t | + 1 ) 1 / 3 + ε {E(z,1/2+it)\ll_{\varepsilon}y^{1/2}(\lvert t\rvert+1)^{1/3+\varepsilon}} , y ≫ 1 {y\gg 1} , extending work of Blomer and Titchmarsh. Finally, we propose a uniform version of the sup norm conjecture for Eisenstein series.

Diagonal restrictions of p-adic Eisenstein families

We compute the diagonal restriction of the first derivative with respect to the weight of a p-adic family of Hilbert modular Eisenstein series attached to a general (odd) character of the narrow class group of a real quadratic field, and express the Fourier coefficients of its ordinary projection in terms of the values of a distinguished rigid analytic cocycle in the sense of Darmon and Vonk (Duke Math J, to appear, 2020) at appropriate real quadratic points of Drinfeld’s p-adic upper half-plane. This can be viewed as the p-adic counterpart of a seminal calculation of Gross and Zagier (J Reine Angew Math 355:191–220, 1985, §7) which arose in their “analytic proof” of the factorisation of differences of singular moduli, and whose inspiration can be traced to Siegel’s proof of the rationality of the values at negative integers of the Dedekind zeta function of a totally real field. Our main identity enriches the dictionary between the classical theory of complex multiplication and its extension to real quadratic fields based on RM values of rigid meromorphic cocycles, and leads to an expression for the p-adic logarithms of Gross–Stark units and Stark–Heegner points in terms of the first derivatives of certain twisted Rankin triple product p-adic L-functions.

Heegner points in Coleman families

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.

On the p-adic Birch and Swinnerton-Dyer conjecture for elliptic curves over number fields

We formulate a multivariable p-adic Birch and Swinnerton-Dyer conjecture for p-ordinary elliptic curves A over number fields K. It generalizes the one-variable conjecture of Mazur, Tate, and Teitelbaum, who studied the case K=Q and the phenomenon of exceptional zeros. We discuss old and new theoretical evidence toward our conjecture and in particular we fully prove it, under mild conditions, in the following situation: K is imaginary quadratic, A=EK is the base change to K of an elliptic curve over the rationals, and the rank of A is either 0 or 1. The proof is naturally divided into a few cases. Some of them are deduced from the purely cyclotomic case of elliptic curves over Q, which we obtain from a refinement of recent work of Venerucci alongside the results of Greenberg, Stevens, Perrin-Riou, and the author. The only genuinely multivariable case (rank 1, two exceptional zeros, three partial derivatives) is newly established here. Its proof generalizes to show that the “almost-anticyclotomic” case of our conjecture is a consequence of conjectures of Bertolini and Darmon on families of Heegner points, and of (partly conjectural) p-adic Gross–Zagier and Waldspurger formulas in families.

A conjecture of Gross and Zagier: Case E(ℚ)tor≅ℤ/2ℤ ⊕ ℤ/2ℤ, ℤ/2ℤ ⊕ ℤ/4ℤ or ℤ/2ℤ ⊕ ℤ/6ℤ

Let [Formula: see text] be an elliptic curve defined over [Formula: see text] of conductor [Formula: see text], [Formula: see text] the Manin constant of [Formula: see text], and [Formula: see text] the product of Tamagawa numbers of [Formula: see text] at prime divisors of [Formula: see text]. Let [Formula: see text] be an imaginary quadratic field where all prime divisors of [Formula: see text] split in [Formula: see text], [Formula: see text] the Heegner point in [Formula: see text], and [Formula: see text] the Shafarevich–Tate group of [Formula: see text] over [Formula: see text]. Let [Formula: see text] be the number of roots of unity contained in [Formula: see text]. Gross and Zagier conjectured that if [Formula: see text] has infinite order in [Formula: see text], then the integer [Formula: see text] is divisible by [Formula: see text]. In this paper, we show that this conjecture is true if [Formula: see text], [Formula: see text] or [Formula: see text].

Towards Hilbert’s tenth problem for rings of integers through Iwasawa theory and Heegner points

For a positive proportion of primes p and q, we prove that $${\mathbb {Z}}$$ is Diophantine in the ring of integers of $${\mathbb {Q}}(\root 3 \of {p},\sqrt{-q})$$ . This provides a new and explicit infinite family of number fields K such that Hilbert’s tenth problem for $$O_K$$ is unsolvable. Our methods use Iwasawa theory and congruences of Heegner points in order to obtain suitable rank stability properties for elliptic curves.

A converse to a theorem of Gross, Zagier, and Kolyvagin

Let $E$ be a semistable elliptic curve over $\mathbb{Q}$. We prove that if $E$ has non-split multiplicative reduction at at least one odd prime or split multiplicative reduction at at least two odd primes, then \[ \mathrm{rank}_{\mathbb{Z}} E(\mathbb{Q}) =1\quad \mathrm{and}\quad \#\Sha(E) \lt \infty X(E)\implies \mathrm{ord}_{s=1}L(E,s) = 1. \] We also prove the corresponding result for the abelian variety associated with a weight 2 newform $f$ of trivial character. These, and other related results, are consequences of our main theorem, which establishes criteria for $f$ and $H^1_f(\mathbb{Q},V)$, where $V$ is the $p$-adic Galois representation associated with $f$, that ensure that $\mathrm{ord}_{s=1}L(f,s) = 1$. The main theorem is proved using the Iwasawa theory of $V$ over an imaginary quadratic field to show that the $p$-adic logarithm of a suitable Heegner point is non-zero.

Etudes of the resolvent

Based on the notion of the resolvent and on the Hilbert identities, this paper presents a number of classical results in the theory of differential operators and some of their applications to the theory of automorphic functions and number theory from a unified point of view. For instance, for the Sturm–Liouville operator there is a derivation of the Gelfand–Levitan trace formula, and for the one-dimensional Schrödinger operator a derivation of Faddeev’s formula for the characteristic determinant and the Zakharov– Faddeev trace identities. Recent results on the spectral theory of a certain functional-difference operator arising in conformal field theory are then presented. The last section of the survey is devoted to the Laplace operator on a fundamental domain of a Fuchsian group of the first kind on the Lobachevsky plane. An algebraic scheme is given for proving analytic continuation of the integral kernel of the resolvent of the Laplace operator and the Eisenstein–Maass series. In conclusion there is a discussion of the relationship between the values of the Eisenstein–Maass series at Heegner points and the Dedekind zeta-functions of imaginary quadratic fields, and it is explained why pseudo-cusp forms for the case of the modular group do not provide any information about the zeros of the Riemann zeta-function. Bibliography: 50 titles.

Horizontal non-vanishing of Heegner points and toric periods

Let F be a totally real number field and A a modular GL2-type abelian variety over F. Let K/F be a CM quadratic extension. Let χ be a class group character over K such that the Rankin-Selberg convolution L(s,A,χ) is self-dual with root number −1. We show that the number of class group characters χ with bounded ramification such that L′(1,A,χ)≠0 increases with the absolute value of the discriminant of K.We also consider a rather general rank zero situation. Let π be a cuspidal cohomological automorphic representation over GL2(AF). Let χ be a Hecke character over K such that the Rankin–Selberg convolution L(s,π,χ) is self-dual with root number 1. We show that the number of Hecke characters χ with fixed ∞-type and bounded ramification such that L(1/2,π,χ)≠0 increases with the absolute value of the discriminant of K.The Gross–Zagier formula and the Waldspurger formula relate the question to horizontal non-vanishing of Heegner points and toric periods, respectively. For both situations, the strategy is geometric relying on the Zariski density of CM points on self-products of a quaternionic Shimura variety. The recent result [26,31,1] on the André–Oort conjecture is accordingly fundamental to the approach.

CM-points and lattice counting on arithmetic compact Riemann surfaces

Let X(D,1)=Γ(D,1)\\H denote the Shimura curve of level N=1 arising from an indefinite quaternion algebra of fixed discriminant D. We study the discrete average of the error term in the hyperbolic circle problem over Heegner points of discriminant d<0 on X(D,1) as d→−∞. We prove that if |d| is sufficiently large compared to the radius r≈log⁡X of the circle, we can improve on the classical O(X2/3)-bound of Selberg. Our result extends the result of Petridis and Risager for the modular surface to arithmetic compact Riemann surfaces.

A conjecture of Gross and Zagier: Case E(ℚ)tor≅ℤ/3ℤ

Let [Formula: see text] be an elliptic curve defined over [Formula: see text] of conductor [Formula: see text], [Formula: see text] the Manin constant of [Formula: see text], and [Formula: see text] the product of Tamagawa numbers of [Formula: see text] at prime divisors of [Formula: see text]. Let [Formula: see text] be an imaginary quadratic field where all prime divisors of [Formula: see text] split in [Formula: see text], [Formula: see text] the Heegner point in [Formula: see text], and [Formula: see text] the Shafarevich–Tate group of [Formula: see text] over [Formula: see text]. Let [Formula: see text] be the number of roots of unity contained in [Formula: see text]. Gross and Zagier conjectured that if [Formula: see text] has infinite order in [Formula: see text], then the integer [Formula: see text] is divisible by [Formula: see text]. In this paper, we show that this conjecture is true if [Formula: see text].

Big Heegner points and generalized Heegner cycles

For big Galois representations attached to Hida families of elliptic cuspforms, Howard constructed the systems of big Heegner points. On the other hand, for Galois representations attached to cuspforms, there are Euler systems of generalized Heegner cycles, which are constructed by Bertolini-Darmon-Prasanna. Castella recently proved that the systems of big Heegner points interpolate the systems of generalized Heegner cycles. In this paper, by a different approach we slightly improve his work.

Kolyvagin systems and Iwasawa theory of generalized Heegner cycles

Iwasawa theory of Heegner points on abelian varieties of GL2 type has been studied by, among others, Mazur, Perrin-Riou, Bertolini, and Howard. The purpose of this article 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 greater than 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-theoretic techniques, by Fouquet in the context of Hida families of modular forms.

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

Abstract We relate p-adic families of Jacobi forms to big Heegner points constructed by B. Howard, in the spirit of the Gross–Kohnen–Zagier theorem. We view this as a GL ⁢ ( 2 ) {\mathrm{GL}(2)} instance of a p-adic Kudla program.

Modularity of Galois traces of Weber's resolvents

The values of the classical j-invariant at CM points are called singular moduli. Zagier [15] proved that the traces of singular moduli are Fourier coefficients of a weakly holomorphic modular form of weight 3/2 and Bruinier-Funke [1] generalized his result to the sums of the values at Heegner points of modular functions on modular curves of arbitrary genus. Extending the work of [9], we construct the real-valued class invariants by using the modular functions defined by Weber's resolvents and identify their Galois traces with Fourier coefficients of a weight 3/2 weakly holomorphic modular form by using Shimura's reciprocity law.

ON THE -ADIC VARIATION OF HEEGNER POINTS

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-Dyer conjecture.