Related papers: On the vanishing of Selmer groups for elliptic cur…

Let $K$ be an imaginary quadratic field and $E/\mathbb{Q}$ an elliptic curves with complex multiplication by $\mathcal{O}_K$. Let $K_\infty/K$ be the anticyclotomic $\mathbb{Z}_p$-extension of $K$ and $K_n$ the intermediate layers. Under…

Let $F$ be a global function field of characteristic $p>0$, $\mathcal F/F$ a Galois extension with $Gal(\tilde F/F)\simeq \mathbb{Z}_p^{\mathbb N}$ and $E/F$ a non-isotrivial elliptic curve. We study the behaviour of Selmer groups…

Let A be an abelian variety over a number field K. An identity between the L-functions L(A/K_i,s) for extensions K_i of K induces a conjectural relation between the Birch-Swinnerton-Dyer quotients. We prove these relations modulo finiteness…

Fix a prime number $p$. Let $\mathbb{F}_q$ be a finite field of characteristic coprime to 2, 3, and $p$, which also contains the primitive $p$-th root of unity $\mu_p$. Based on the works by Swinnerton-Dyer and Klagsbrun, Mazur, and Rubin,…

We study elliptic curves of the form $x^3+y^3=2p$ and $x^3+y^3=2p^2$ where $p$ is any odd prime satisfying $p\equiv 2\bmod 9$ or $p\equiv 5\bmod 9$. We first show that the $3$-part of the Birch-Swinnerton-Dyer conjecture holds for these…

We consider $L$-functions attached to representations of the Galois group of the function field of a curve over a finite field. Under mild tameness hypotheses, we prove non-vanishing results for twists of these $L$-functions by characters…

Let k be a global field, $\bar{k}$ a separable closure of k, and $G_k$ the absolute Galois group $\Gal(\bar{k}/k)$ of $\bar{k}$ over k. For every g in $G_k$, let $\bar{k}^g$ be the fixed subfield of $\bar{k}$ under g. Let E/k be an elliptic…

We study the elliptic curve E given by y^2=x(x+1)(x+t) over the rational function field k(t) and its extensions K_d=k(\mu_d,t^{1/d}). When k is finite of characteristic p and d=p^f+1, we write down explicit points on E and show by…

Let $E$ be an elliptic curve over a number field $K$ defined by a monic irreducible cubic polynomial $F(x)$. When $E$ is \textit{nice} at all finite primes of $K$, we bound its $2$-Selmer rank in terms of the $2$-rank of a modified ideal…

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.…

Let $p\ge 5$ be a prime number and $E/\mathbf{Q}$ an elliptic curve with good supersingular reduction at $p$. Under the generalized Heegner hypothesis, we investigate the $p$-primary subgroups of the Tate--Shafarevich groups of $E$ over…

We generalise the Siegel-Voloch theorem about S-integral points on elliptic curves as follows: let K/F denote a global function field over a finite field F of characteristic p>3, let S denote a finite set of places of K and let E/K denote a…

We show that if F is the rational numbers or a multiquadratic number field, p is 2,3, or 5, and K/F is a Galois extension of degree a power of p, then for elliptic curves E/Q ordered by height, the average dimension of the p-Selmer groups…

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…

For an elliptic curve $E$ over $K$, the Birch and Swinnerton-Dyer conjecture predicts that the rank of Mordell-Weil group $E(K)$ is equal to the order of the zero of $L(E_{/ K},s)$ at $s=1$. In this paper, we shall give a proof for elliptic…

Let $p$ be a fixed odd prime. Let $E$ be an elliptic curve defined over a number field with either good ordinary reduction or multiplicative reduction at each prime of $F$ above $p$. We shall study the characteristic element of the Selmer…

Let $q$ be any prime $\equiv 7 \mod 16$, $K = \mathbb{Q}(\sqrt{-q})$, and let $H$ be the Hilbert class field of $K$. Let $A/H$ be the Gross elliptic curve defined over $H$ with complex multiplication by the ring of integers of $K$. We prove…

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…

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…

Let $C$ be a smooth projective irreducible curve defined over a finite field $\mathbb{F}_q$ and $K=\mathbb{F}_q(C)$. Let $A\subset K$ be the ring of functions regular outside a fixed place $\infty$ of $K$. Let…