Ordinary deformations are unobstructed in the cyclotomic limit

The deformation theory of ordinary representations of the absolute Galois groups of totally real number fields (over a finite field $k$) has been studied for a long time, starting with the work of Hida, Mazur and Tilouine, and continued by Wiles and others. Hida has studied the behaviour of these deformations when one considers the $p$-cyclotomic tower of extensions of the field. In the limit, one obtains a deformation ring classifying the ordinary deformations of the (Galois group of) the $p$-cyclotomic extension. We show that if this ring in Noetherian (a natural assumption considered by Hida) it is free over the ring of Witt vectors of $k$. This however imposes natural conditions on certain $\mu$-invariants

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

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 $K$

The conjecture of Birch and Swinnerton-Dyer for certain elliptic curves with complex multiplication

Let $E/F$ be an elliptic curve over a number field $F$ with complex multiplication by the ring of integers in an imaginary quadratic field $K$. We give a complete proof of the conjecture of Birch and Swinnerton-Dyer for $E/F$, as well as its equivariant refinement formulated by Gross, under the assumption that $L(E/F,1)\neq 0$ and that $F(E_{tors})/K$ is abelian. We also prove analogous results for CM abelian varieties $A/K$.Comment: To appear in Cambridge Journal of Mat

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

On the non-vanishing of pp-adic heights on CM abelian varieties, and the arithmetic of Katz pp-adic LL-functions

Let $B$ be a simple CM abelian variety over a CM field $E$, $p$ a rational prime. Suppose that $B$ has potentially ordinary reduction above $p$ and is self-dual with root number $-1$. Under some further conditions, we prove the generic non-vanishing of (cyclotomic) $p$-adic heights on $B$ along anticyclotomic $\Z_{p}$-extensions of $E$. This provides evidence towards Schneider's conjecture on the non-vanishing of $p$-adic heights. For CM elliptic curves over \Q, the result was previously known as a consequence of work of Bertrand, Gross--Zagier and Rohrlich in the 1980s. Our proof is based on non-vanishing results for Katz $p$-adic $L$-functions and a Gross--Zagier formula relating the latter to families of rational points on $B$.Comment: Ann. Inst. Fourier, to appea

Anticyclotomic Iwasawa theory of abelian varieties of GL2\mathrm{GL}_2-type at non-ordinary primes

Let $p\ge 5$ be a prime number, $E/\mathbb{Q}$ an elliptic curve with good supersingular reduction at $p$ and $K$ an imaginary quadratic field such that the root number of $E$ over $K$ is $+1$. When $p$ is split in $K$, Darmon and Iovita formulated the plus and minus Iwasawa main conjectures for $E$ over the anticyclotomic $\mathbb{Z}_p$-extension of $K$, and proved one-sided inclusion: an upper bound for plus and minus Selmer groups in terms of the associated $p$-adic $L$-functions. We generalize their results to two new settings: 1. Under the assumption that $p$ is split in $K$ but without assuming $a_p(E)=0$, we study Sprung-type Iwasawa main conjectures for abelian varieties of $\mathrm{GL}_2$-type, and prove an analogous inclusion. 2. We formulate, relying on the recent work of the first named author with Kobayashi and Ota, plus and minus Iwasawa main conjectures for elliptic curves when $p$ is inert in $K$, and prove an analogous inclusion.Comment: 43 page