Horizontal non-vanishing of Heegner points and toric periods

Let $F/\mathbb{Q}$ be a totally real field and $A$ a modular \GL_2-type abelian variety over $F$. Let $K/F$ be a CM quadratic extension. Let $\chi$ be a class group character over $K$ such that the Rankin-Selberg convolution $L(s,A,\chi)$ is self-dual with root number $-1$. We show that the number of class group characters $\chi$ with bounded ramification such that $L'(1, A, \chi) \neq 0$ increases with the absolute value of the discriminant of $K$. We also consider a rather general rank zero situation. Let $\pi$ be a cuspidal cohomological automorphic representation over \GL_{2}(\BA_{F}). Let $\chi$ be a Hecke character over $K$ such that the Rankin-Selberg convolution $L(s,\pi,\chi)$ is self-dual with root number $1$. We show that the number of Hecke characters $\chi$ with fixed $\infty$-type and bounded ramification such that $L(1/2, \pi, \chi) \neq 0$ 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 \cite{Ts, YZ, AGHP} on the Andr\'e-Oort conjecture is accordingly fundamental to the approach.Comment: Adv. Math., to appear. arXiv admin note: text overlap with arXiv:1712.0214

On the non-triviality of the pp-adic Abel-Jacobi image of generalised Heegner cycles modulo pp, I: modular curves

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 $\Z_l$-anticylotomic extension of $K$.Comment: J. Alg. Geom., to appea

Horizontal variation of Tate--Shafarevich groups

Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $p$ be an odd prime and $\iota: \overline{\mathbb{Q}}\hookrightarrow \mathbb{C}_p$ an embedding. Let $K$ be an imaginary quadratic field and $H_{K}$ the corresponding Hilbert class field. For a class group character $\chi$ over $K$, let $\mathbb{Q}(\chi)$ be the field generated by the image of $\chi$ and $\mathfrak{p}_{\chi}$ the prime of $\mathbb{Q}(\chi)$ above $p$ determined via $\iota_p$. Under mild hypotheses, we show that the number of class group characters $\chi$ such that the $\chi$-isotypic Tate--Shafarevich group of $E$ over $H_{K}$ is finite with trivial $\mathfrak{p}_{\chi}$-part increases with the absolute value of the discriminant of $K$

On the non-triviality of arithmetic invariants modulo p

Arithmetic invariants are often naturally associated to motives over number fields. One of the basic questions is the non-triviality of the invariants. One typically expects generic non-triviality of the invariants as the motive varies in a family. For a prime $p$, the invariants can often be normalised to be $p$-integral. One can thus further ask for the generic non-triviality of the invariants modulo $p$. The invariants can often be expressed in terms of modular forms. Accordingly, one can try to recast the non-triviality as a modular phenomenon. If the phenomena can be proven, the non-triviality typically follows in turn. This principle can be found in the work of Hida and Vatsal among a few others.\\\\We have been trying to explore a strategy initiated by Hida in the case of central criticial Hecke L-values over the $\Z_p$-anticyclotomic extension of a CM-field. The strategy crucially relies on a linear indepedence of mod $p$ Hilbert modular forms. Several arithmetic invariants seem to admit modular expression analogous to the case of Hecke L-values. This includes the case of Katz $p$-adic L-function, its cyclotomic derivative and $p$-adic Abel-Jacobi image of generalised Heegner cycles.We approach the non-triviality of these invariants based on the independence.An analysis of the zero set of the invariants suggests finer versions of the independence. We approach the versions based on Chai's theory of Hecke stable subvarieties of a mod $p$ Shimura variety.We formulate a conjecture regarding the analogue of the independence for mod $p$ modular forms on other Shimura varieties. We prove the analogue in the case of quaternionic Shimura varieties over a totally real field

p-converse to a theorem of Gross–Zagier, Kolyvagin and Rubin

Let E be a CM elliptic curve over the rationals and p > 3 a good ordinary prime for E. We show that Corank_(Z_p)Sel_(p^∞)(E/_Q) = 1 ⟹ ord_(s=1)L(s,E/_Q) = 1 for the p^∞-Selmer group Sel_(p^∞)(E/_Q) and the complex L-function L(s,E/_Q). In particular, the Tate–Shafarevich group X(E/_Q) is finite whenever corank_(Z_p)Selp^∞(E/_Q) = 1. We also prove an analogous p-converse for CM abelian varieties arising from weight two elliptic CM modular forms with trivial central character. For non-CM elliptic curves over the rationals, first general results towards such a p-converse theorem are independently due to Skinner (A converse to a theorem of Gross, Zagier and Kolyvagin, arXiv:1405.7294, 2014) and Zhang (Camb J Math 2(2):191–253, 2014)