Related papers: Selmer groups and anticyclotomic $\mathbb{Z}_p$-ex…

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…

We prove a $p$-converse theorem for elliptic curves $E/\mathbb{Q}$ with complex multiplication by the ring of integers $\mathcal{O}_K$ of an imaginary quadratic field $K$ in which $p$ is ramified. Namely, letting $r_p =…

We study the rank of the $p$-Selmer group $Sel_p(A/k)$ of an abelian variety $A/k$, where $k$ is a function field. If $K/k$ is a quadratic extension and $F/k$ is a dihedral extension and the $\mathbb{Z}_p$-corank of $Sel_p (A/K)$ is odd, we…

Let $p\geq 5$ be a prime number. Let $\mathsf{E}/\mathbb{Q}$ be an elliptic curve with good ordinary reduction at $p$. Let $K$ be an imaginary quadratic field where $p$ splits, and such that the generalized Heegner hypothesis holds. Under…

Let $E$ be an elliptic curve over $\mathbb{Q}$. Greenberg has posed a question whether the structure of the fine Selmer group over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ can be described by cyclotomic polynomials in a…

We prove a $p$-converse to the theorem of Gross-Zagier and Kolyvagin for elliptic curves $E/\mathbf{Q}$ at primes $p>3$ of multiplicative reduction. Two key ingredients in the argument are an extension to this setting of a $p$-adic formula…

Let $A$ be an abelian variety defined over a number field $F$ with supersingular reduction at all primes of $F$ above $p$. We establish an equivalence between the weak Leopoldt conjecture and the expected value of the corank of the…

Let $E$ be an elliptic curve with $j$-invariant $0$ or $1728$ and let $\widetilde{E}$ be a $k^{th}$ twist of $E$. We show that for any prime $p$ of good reduction of $\widetilde{E}$, a degree $k$ relative $p$-class group and the root number…

For an elliptic curve, we study how many Selmer groups are cotorsion over the anti-cyclotomic $\mathbb{Z}_p$-extension as one varies the prime $p$ or the quadratic imaginary field in question.

Let $p\ge5$ be a prime number. Let $E/\mathbb{Q}$ be an elliptic curve with complex multiplication by an imaginary quadratic field $K$ such that $p$ is inert in $K$ and that $E$ has good reduction at $p$. Let $K_\infty$ be the…

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 $E$ over an imaginary quadratic field $K$ is…

Iwasawa theory of modular forms over anticyclotomic $\mathbb{Z}_p$-extensions of imaginary quadratic fields has been studied by several authors, starting from the works of Bertolini-Darmon and Iovita-Spiess, under the crucial assumption…

Let $E/\mathbb{Q}$ be an elliptic curve and let $p\ge 5$ be a prime of good supersingular reduction. We generalize results due to Meng Fai Lim proving Kida's formula and integrality results for characteristic elements of signed Selmer…

Let $k = \mathbb{Q}(\sqrt {-m})$ and $p \geq 3$ split in $k$. We prove new properties of the $\mathbb{Z}_p$-extensions $K/k$, distinct from the cyclotomic one; we do not assume $K/k$ totally ramified, nor the triviality of the $p$-class…

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…

Let $E$ be an elliptic curve over $\mathbb{Q}$ with good supersingular reduction at a prime $p\geq 3$ and $a_p=0$. We generalise the definition of Kobayashi's plus/minus Selmer groups over $\mathbb{Q}(\mu_{p^\infty})$ to $p$-adic Lie…

In this paper, we consider normalized newforms $f\in S_k(\Gamma_0(N),\varepsilon_f)$ whose non-constant term Fourier coefficients are congruent to those of an Eisenstein series modulo some prime ideal above a rational prime $p$. In this…

We review the main conjecture for an elliptic curve on $\Q$ having good supersingular reduction at $p$ and give some consequences of it. Then we define the notion of $\lambda$-invariant and of $\mu$- invariant in this situation,…

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.…

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…