Heegner Points Research Articles

On Euler systems for motives and Heegner points

On Euler systems for motives and Heegner points

Plectic Jacobians

ABSTRACT Looking for a geometric framework to study plectic Heegner points, we define a collection of abelian varieties – called plectic Jacobians—using the middle-degree cohomology of quaternionic Shimura varieties (QSVs). The construction is inspired by the definition of Griffiths’ intermediate Jacobians and rests on Nekovář–Scholl’s notion of plectic Hodge structures. Moreover, we construct exotic Abel–Jacobi maps sending certain zero cycles on QSVs to plectic Jacobians.

Theta cycles and the Beilinson–Bloch–Kato conjectures

We introduce ‘canonical’ classes in the Selmer groups of certain Galois representations with a conjugate-symplectic symmetry. They are images of special cycles in unitary Shimura varieties, and defined uniquely up to a scalar. The construction is a slight refinement of one of Y. Liu, based on the conjectural modularity of Kudla's theta series of special cycles. For 2-dimensional representations, Theta cycles are (the Selmer images of) Heegner points. In general, they conjecturally exhibit an analogous strong relation with the Beilinson–Bloch–Kato conjectures in rank 1, for which we gather the available evidence.

On the Iwasawa main conjecture for generalized Heegner classes in a quaternionic setting

Abstract We prove one divisibility relation of the anticyclotomic Iwasawa Main Conjecture for a higher weight ordinary modular form f and an imaginary quadratic field satisfying a “relaxed” Heegner hypothesis. Let Λ be the anticyclotomic Iwasawa algebra. Following the approach of Howard and Longo–Vigni, we construct the Λ-adic Kolyvagin system of generalized Heegner classes coming from Heegner points on a suitable Shimura curve. As its application, we also prove one divisibility relation in the Iwasawa–Greenberg main conjecture for the p-adic L-function defined by Magrone.

A higher Gross–Zagier formula and the structure of Selmer groups

We describe a Kolyvagin system-theoretic refinement of Gross–Zagier formula by comparing Heegner point Kolyvagin systems with Kurihara numbers when the root number of a rational elliptic curve E E over an imaginary quadratic field K K is − 1 -1 . When the root number of E E over K K is 1, we first establish the structure theorem of the p ∞ p^\infty -Selmer group of E E over K K . The description is given by the values of certain families of quaternionic automorphic forms, which is a part of bipartite Euler systems. By comparing bipartite Euler systems with Kurihara numbers, we also obtain an analogous refinement of Waldspurger formula. No low analytic rank assumption is imposed in both refinements. We also prove the equivalence between the non-triviality of various “Kolyvagin systems” and the corresponding main conjecture localized at the augmentation ideal. As consequences, we obtain new applications of (weaker versions of) the Heegner point main conjecture and the anticyclotomic main conjecture to the structure of p ∞ p^\infty -Selmer groups of elliptic curves of arbitrary rank. In particular, the Heegner point main conjecture localized at the augmentation ideal implies the strong rank one p p -converse to the theorem of Gross–Zagier and Kolyvagin.

On the Algebraicity of Polyquadratic Plectic Points

Abstract We establish direct evidence of the arithmetic significance of plectic Stark–Heegner points for elliptic curves of arbitrarily large rank. The main contribution is a proof of the algebraicity of plectic points associated to polyquadratic CM extensions of totally real number fields. Moreover, we relate the non-vanishing of plectic points to analytic and algebraic ranks of elliptic curves.

Plectic Stark–Heegner points

We propose a conjectural construction of determinants of global points on modular elliptic curves over arbitrary number fields, generalizing both the p-adic construction of Heegner points via Čerednik–Drinfeld uniformization and the definition of classical Stark–Heegner points. In alignment with Nekovář 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 plectic invariants.

On the anticyclotomic Iwasawa main conjecture for Hilbert modular forms of parallel weights

In this article, we study the Iwasawa theory for Hilbert modular forms over the anticyclotomic extension of a CM field. We prove an one-sided divisibility result toward the Iwasawa main conjecture in this setting. The proof relies on the first and second reciprocity laws relating theta elements to Heegner point Euler systems on Shimura curves. As a by-product we also prove a result towards the rank 0 case of certain Bloch–Kato conjecture and a parity conjecture.

Explicit realization of elements of the Tate–Shafarevich group constructed from Kolyvagin classes

We consider the Kolyvagin cohomology classes, associated to an elliptic curve E/{mathbb Q}, from a computational point of view. We explain how to go from a model of a class as an element of (E(L)/pE(L))^{mathrm {Gal}(L/{mathbb Q})}, where p is prime and L is a dihedral extension of {mathbb Q} of degree 2p, to a geometric model as a genus one curve embedded in mathbb { P} ^{p-1}. We adapt the existing methods to compute Heegner points to our situation, and explicitly compute them as elements of E(L). Finally, we compute explicit equations for several genus one curves that represent non-trivial elements of , for p le 11, and hence are counterexamples to the Hasse principle.

Optimal Small Scale Equidistribution of Lattice Points on the Sphere, Heegner Points, and Closed Geodesics

AbstractWe asymptotically estimate the variance of the number of lattice points in a thin, randomly rotated annulus lying on the surface of the sphere. This partially resolves a conjecture of Bourgain, Rudnick, and Sarnak. We also obtain estimates that are valid for all balls and annuli that are not too small. Our results have several consequences: for a conjecture of Linnik on sums of two squares and a “microsquare”, a conjecture of Bourgain and Rudnick on the number of lattice points lying in small balls on the surface of the sphere, the covering radius of the sphere, and the distribution of lattice points in almost all thin regions lying on the surface of the sphere. Finally, we show that for a density 1. subsequence of squarefree integers, the variance exhibits a different asymptotic behaviour for balls of volume with .We also obtain analogous results for Heegner points and closed geodesics. Interestingly, we are able to prove some slightly stronger results for closed geodesics than for Heegner points or lattice points on the surface of the sphere. A crucial observation that underpins our proof is the different behaviour of weighting functions for annuli and for balls. © 2022 Courant Institute of Mathematics and Wiley Periodicals LLC.

Heegner points, stark-Heegner points, and diagonal classes

Heegner points, stark-Heegner points, and diagonal classes

The Iwasawa Main Conjectures for GL2 and derivatives of p-adic L-functions

We prove under mild hypotheses the three-variable Iwasawa Main Conjecture for p-ordinary modular forms base changed to an imaginary quadratic field K in which p splits in the indefinite setting (in the definite setting this is a result due to Skinner–Urban). Being in a setting encompassing Heegner points and their variation in p-adic families, our main result has new applications to Greenberg's nonvanishing conjecture for central derivatives of p-adic L-functions of Hida families with root number −1.

THE p-ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES, II: NONSPLIT PRIMES

AbstractThe formula of the title relates p-adic heights of Heegner points and derivatives of p-adic L-functions. It was originally proved by Perrin-Riou for p-ordinary elliptic curves over the rationals, under the assumption that p splits in the relevant quadratic extension. We remove this assumption, in the more general setting of Hilbert-modular abelian varieties.

Heegner points and Beilinson–Kato elements: A conjecture of Perrin-Riou

A conjecture of Perrin-Riou relating Heegner cycles to Beilinson–Kato elements is proved, by relating both objects to p-adic families of Beilinson–Flach elements in the higher Chow groups of products of two modular curves.

Torsion properties of modified diagonal classes on triple products of modular curves

AbstractConsider three normalized cuspidal eigenforms of weight $2$ and prime level p. Under the assumption that the global root number of the associated triple product L-function is $+1$ , we prove that the complex Abel–Jacobi image of the modified diagonal cycle of Gross–Kudla–Schoen on the triple product of the modular curve $X_0(p)$ is torsion in the corresponding Hecke isotypic component of the Griffiths intermediate Jacobian. The same result holds with the complex Abel–Jacobi map replaced by its étale counterpart. As an application, we deduce torsion properties of Chow–Heegner points associated with modified diagonal cycles on elliptic curves of prime conductor with split multiplicative reduction. The approach also works in the case of composite square-free level.

Restriction of Eisenstein series and Stark–Heegner points

In a recent work of Darmon, Pozzi and Vonk, the authors consider a particular p-adic family of Hilbert–Eisenstein series E k (1,ϕ) associated with an odd character ϕ of the narrow ideal class group of a real quadratic field F and compute the first derivative of a certain one-variable twisted triple product p-adic L-series attached to E k (1,ϕ) and an elliptic newform f of weight 2 on Γ 0 (p). In this paper, we generalize their construction to include the cyclotomic variable and thus obtain a two-variable twisted triple product p-adic L-series. Moreover, when f is associated with an elliptic curve E over ℚ, we prove that the first derivative of this p-adic L-series along the weight direction is a product of the p-adic logarithm of a Stark–Heegner point of E over F introduced by Darmon and the cyclotomic p-adic L-function for E.

A proof of Perrin-Riou’s Heegner point main conjecture

Let $E/\mathbf{Q}$ be an elliptic curve of conductor $N$, let $p>3$ be a prime where $E$ has good ordinary reduction, and let $K$ be an imaginary quadratic field satisfying the Heegner hypothesis. In 1987, Perrin-Riou formulated an Iwasawa main conjecture for the Tate-Shafarevich group of $E$ over the anticyclotomic $\mathbf{Z}_p$-extension of $K$ in terms of Heegner points. In this paper, we give a proof of Perrin-Riou's conjecture under mild hypotheses. Our proof builds on Howard's theory of bipartite Euler systems and Wei Zhang's work on Kolyvagin's conjecture. In the case when $p$ splits in $K$, we also obtain a proof of the Iwasawa-Greenberg main conjecture for the $p$-adic $L$-functions of Bertolini-Darmon-Prasanna.

Construction of anti-cyclotomic Euler systems of modular abelian varieties, and the ranks of their Mordell-Weil groups

We generalize Birch's construction of the Heegner points on X0(N) to construct new points X1(N) (and therefore construct points on the abelian varieties associated to J1(N)). Then, we show that these points form an Euler system, and we improve Kolyvagin's Euler system techniques to show that for our point PτK/c and any ring class character χ of the extended ring class field of conductor c satisfying χ=χ‾, if PτK/cχ is non-torsion and GK→AutAf[π] is surjective, then the corank of Sel(Aχ/K) is 1, which implies the rank of Af(K)χ is 1.

On the anticyclotomic Iwasawa theory of rational elliptic curves at Eisenstein primes

Let \(E/{\mathbb {Q}}\) be an elliptic curve and p an odd prime where E has good reduction, and assume that E admits a rational p-isogeny. In this paper we study the anticyclotomic Iwasawa theory of E over an imaginary quadratic field in which p splits, which we relate to the anticyclotomic Iwasawa theory of characters by a variation of the method of Greenberg–Vatsal. As a result of our study we obtain proofs (under relatively mild hypotheses) of Perrin-Riou’s Heegner point main conjecture, a p-converse to the theorem of Gross–Zagier and Kolyvagin, and the p-part of the Birch–Swinnerton-Dyer formula in analytic rank 1, for Eisenstein primes p.

Tiling number problem and $\boldsymbol{\theta}$-congruent number problem

The tiling number problem is closely related to the arithmetic of elliptic curves. In this paper, we construct a family of tiling numbers with multiple prime factors by using Birch lemma and the induction method developed by one of the authors(Ye Tian) to prove the non-torsioness of Heegner points. In addition, we prove the $2$-part BSD conjecture of the related elliptic curves. We deal with the quadratic twist family of elliptic curves without complex multiplication and the distribution of $2$-Selmer groups of this family is not predicted by the general conjecture or the known results.