Related papers: Non-vanishing theorems for central $L$-values of s…

The paper uses Iwasawa theory at the prime $p=2$ to prove non-vanishing theorems for the value at $s=1$ of the complex $L$-series of certain quadratic twists of the Gross family of elliptic curves with complex multiplication by the field $K…

We prove non-vanishing theorems for the central values of $L$-series of quadratic twists of the Gross elliptic curve with complex multiplication by the imaginary quadratic field $\mathbb{Q}(\sqrt{-q})$, where $q$ is any prime congruent to…

In this paper, we show that, by applying some results on modular symbols, for a family of certain elliptic curves defined over $\mathbb Q$, there is a large class of explicit quadratic twists whose complex $L$-series does not vanish at…

For primes $q \equiv 7 \mod 16$, the present manuscript shows that elementary methods enable one to prove surprisingly strong results about the Iwasawa theory of the Gross family of elliptic curves with complex multiplication by the ring of…

Let $K=\Bbb Q(\sqrt{-q})$, where $q$ is a prime congruent to $3$ modulo $4$. Let $A=A(q)$ denote the Gross curve. Let $E=A^{(-\beta)}$ denote its quadratic twist, with $\beta=\sqrt{-q}$. The curve $E$ is defined over the Hilbert class field…

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 paper, we give some non-vanishing results on the central values of prime twists of modular $L$-functions by imaginary quadratic fields for specific elliptic modular forms. In particular, we show that the central values of prime…

Let $E$ be a semistable elliptic curve over $\mathbb{Q}$. We prove that if $E$ has non-split multiplicative reduction at at least one odd prime or split multiplicative reduction at at least two odd primes and if the rank of $E(\mathbb{Q})$…

Let $K=\mathbb{Q}(\sqrt{-p})$ be a quadratic field for an odd prime $p$. We show that there exist infinitely many primes $p$ for which no elliptic curve $E/\mathbb{Q}$ has torsion subgroup $\mathbb{Z}/2\mathbb{Z}\times…

An abelian variety $A/K$ is heavenly at $\ell$ if the extension $K(A[\ell^\infty])/K(\mu_{\ell^{\infty}}\!)$ is both pro-$\ell$ and unramified away from $\ell$. It is known that for a fixed quadratic field $K$, the number of $K$-isomorphism…

In this paper, we investigate the coefficients of the Taylor expansion of the complex $L$-series of any elliptic curve over $\mathbb{Q}$. We prove that, in the family of quadratic twists by all the discriminants $d$, these coefficients are…

We use Iwasawa theory, at a prime $p$ inert in a quadratic imaginary field $K$, to study the arithmetic properties of mock plectic invariants for elliptic curves of rank two. More precisely, under some minor technical assumptions, we prove…

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…

In arXiv:math/0404297 a non-commutative Iwasawa Main Conjecture for elliptic curves over $\mathbb{Q}$ has been formulated. In this note we show that it holds for all CM-elliptic curves $E$ defined over $\mathbb{Q}$. This was claimed in…

In this paper, we study central values of the family of quadratic twists of modular $L$-functions of moduli $8p$, with $p$ ranging over odd primes. Assuming the truth of the generalized Riemann hypothesis, we establish a positive proportion…

In this paper, we show that Tian's induction method can be generalised to study the Birch-Swinnerton-Dyer conjecture for the quadratic twists, both with global root number $+1$ and with global root number $-1$, of certain elliptic curves…

For any quadratic extension $L/K$ of number fields, we prove that there are infinitely many elliptic curves $E$ over $K$ so that the abelian groups $E(K)$ and $E(L)$ both have rank $1$. In particular, there are infinitely many elliptic…

Let $q$ be a prime with $q \equiv 7 \mod 8$, and let $K=\mathbb{Q}(\sqrt{-q})$. Then $2$ splits in $K$, and we write $\mathfrak{p}$ for either of the primes $K$ above $2$. Let $K_\infty$ be the unique $\mathbb{Z}_2$-extension of $K$…

In this paper, we show the nonvanishing of some Hecke characters on cyclotomic fields. The main ingredient of this paper is a computation of eigenfunctions and the action of Weil representation at some primes including the primes above $2$.…

Let N = 1 mod 4 be the negative of a prime, K=Q(sqrt{N}) and O_K its ring of integers. Let D be a prime ideal in O_K of prime norm congruent to 3 modulo 4. Under these assumptions, there exists Hecke characters $\psi_{\D}$ of K with…