Related papers: Rank stability of elliptic curves in certain non-a…

We study the growth of the rank of elliptic curves and, more generally, Abelian varieties upon extensions of number fields. First, we show that if $L/K$ is a finite Galois extension of number fields such that $\Gal(L/K)$ does not have an…

We obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields. Suppose $K/k$ is a quadratic extension of number fields, $E$ is an elliptic curve defined over $k$, and $p$ is an odd prime. Let $F$…

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 $K$ be a field of characteristic $0$ and $E/K$ an elliptic curve over $K$. For a finite extension $L/K$ and a prime~$\ell$, we provide Galois-theoretic sufficient conditions on $L/K$ under which…

We study the behavior of Selmer groups of an elliptic curve $E/\mathbb{Q}$ in finite Galois extensions with prescribed Galois group. Fix a prime $\ell \geq 5$, a finite group $G$ with $\#G = \ell^n$, and an elliptic curve $E/\mathbb{Q}$…

Let p be an odd prime number, E an elliptic curve over a number field k, and F/k a Galois extension of degree twice a power of p. We study the Z_p-corank rk_p(E/F) of the p-power Selmer group of E over F. We obtain lower bounds for…

We investigate the rank gain of elliptic curves, and more generally, Jacobian varieties, over non-Galois extensions whose Galois closure has Galois group permutation-isomorphic to a prescribed group $G$ (in short, "$G$-extensions"). In…

We consider the problem of classifying quadruples $(K,E,m_1,m_2)$ where $K$ is a number field, $E$ is an elliptic curve defined over $K$ and $(m_1,m_2)$ is a pair of relatively prime positive integers for which the intersection $K(E[m_1])…

We investigate the rank growth of elliptic curves from $\mathbb{Q}$ to $S_4$ and $A_4$ quartic extensions $K/\mathbb{Q}$. In particular, we are interested in the quantity $\mathrm{rk}(E/K) - \mathrm{rk}(E/\mathbb{Q})$ for fixed $E$ and…

It is known, that for every elliptic curve over Q there exists a quadratic extension in which the rank does not go up. For a large class of elliptic curves, the same is known with the rank replaced by the 2-Selmer group. We show, however,…

We prove that if $f:X \rightarrow A$ is a morphism from a smooth projective variety $X$ to an abelian variety $A$ over a number field $K$, and $G$ is a subgroup of automorphisms of $X$ satisfying certain properties, and if a prime $p$…

Let $p \in \{3, 5\}$ and consider a cyclic $p$-extension $L/\mathbb{Q}$. We show that there exists an effective positive density of elliptic curves $ E $ defined over $ \mathbb{Q} $, ordered by height, that are diophantine stable in $ L $.

Let $\mathcal{X}$ be a Riemann surface of genus $g>0$ defined over a number field $K$ which is a degree $d$-covering of $\mathbb{P}^1_K$. In this paper we show the existence of infinitely many linearly disjoint degree $d$-extensions $L/K$…

For an elliptic curve $E/\mathbb{Q}$ we show that there are infinitely many cyclic sextic extensions $K/\mathbb{Q}$ such that the Mordell-Weil group $E(K)$ has rank greater than the subgroup of $E(K)$ generated by all the $E(F)$ for the…

Let $E$ be a nonisotrivial elliptic curve over $\mathbb{Q}(T)$ and denote the rank of the abelian group $E(\mathbb{Q}(T))$ by $r$. For all but finitely many $t\in \mathbb{Q}$, specialization will give an elliptic curve $E_t$ over…

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…

We prove that there exist infinitely many elliptic curves over \Q with given modular invariant, and rank >=2. Furthermore, there exist infinitely many elliptic curves over $\Q$ with invariant equal at 0 (resp. 1728) and rank >=6 (resp.…

Let E/Q be an elliptic curve with good supersingular reduction at p with a_p(E)=0. We give a conjecture on the existence of analytic plus and minus p-adic L-functions of E over the Zp-cyclotomic extension of a finite Galois extension of Q…

We prove that there exist infinitely many elliptic curves over $\mathbb{Q}(i)$ with $j$-invariant $1728$ and rank exactly $2$ which are not obtained by base change from $\mathbb{Q}$. The rank of each such curve is determined via 2-isogeny…

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…