Related papers: Heegner Point Kolyvagin System and Iwasawa Main Co…

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…

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…

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…

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 $\Lambda$ be the anticyclotomic…

Let $f$ be a newform of even weight at least $4$, level $N$ and trivial character. Let $p\nmid N$ be an odd prime number that is ordinary for $f$ and let $K$ be an imaginary quadratic field satisfying a generalized Heegner hypothesis…

We prove Mazur and Rubin's Iwasawa-theoretic Gross-Zagier conjecture (under some restrictive hypotheses), which relates Heegner points in towers of number fields to the 2-variable p-adic L-function. The result generalizes Perrin-Riou's…

Our first goal in this note is to explain that a weak form of Perrin-Riou's conjecture on the non-triviality of Beilinson-Kato classes follows as an easy consequence of the Iwasawa main conjectures, and deduce its refined versions in the…

We upgrade Howard's divisibility toward Perrin-Riou's Heegner point Main Conjecture to an equality under some mild conditions. We do this by exploiting Wei Zhang's proof of the Kolyvagin conjecture. The main ingredient is an improvement of…

Let $E/\mathbb{Q}$ be an elliptic curve, and $p$ a 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…

In 1987, B. Perrin-Riou formulated a Heegner point main conjecture for elliptic curves at primes of ordinary reduction. In this paper, we formulate an analogue of Perrin-Riou's main conjecture for supersingular primes. We then prove this…

Building on the construction of big Heegner points in the quaternionic setting, and their relation to special values of Rankin-Selberg $L$-functions, we obtain anticyclotomic analogues of the results of Emerton-Pollack-Weston on the…

Let $E/\mathbb{Q}$ an elliptic curve with good supersingular reduction at a prime $p\geq 5$, and $K$ an imaginary quadratic field such that the root number of $E$ over $K$ equals $-1$. When $p$ splits in $K$, Castella and Wan formulated the…

In this article, we study the Iwasawa theory for cuspidal automorphic representations of $\mathrm{GL}(n)\times\mathrm{GL}(n+1)$ over CM fields along anticyclotomic directions, in the framework of the Gan--Gross--Prasad conjecture for…

Let $E/\mathbb{Q}$ be an elliptic curve and let $K$ be an imaginary quadratic field. Under a certain Heegner hypothesis, Kolyvagin constructed cohomology classes for $E$ using $K$-CM points and conjectured they did not all vanish.…

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…

We formulate an Iwasawa main conjecture for a higher rank Euler system for a general motive. We prove "one half" of the main conjecture under mild hypotheses. We also formulate a conjecture on "Darmon-type derivatives" of Euler systems and…

The aim of this paper is to prove the two-variable anticyclotomic Iwasawa main conjecture for Hida families and a definite version of the horizontal non-vanishing conjecture, which are formulated in Longo-Vigni. Our approach is based on the…

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…

In this article, we study the Iwasawa theory for Hilbert modular forms over the anticyclotomic extension of a CM field. We prove a one sided divisibility result toward the Iwasawa main conjecture. The proof relies on the first and second…

Let $E$ be an elliptic curve over $\mathbb Q$ and let $p\geq5$ be a prime of good supersingular reduction for $E$. Let $K$ be an imaginary quadratic field satisfying a modified "Heegner hypothesis" in which $p$ splits, write $K_\infty$ for…