Related papers: Average rank of elliptic curves

Assuming the Hasse--Weil conjecture and the generalized Riemann hypothesis for the $L$-functions of the elliptic curve, we give an upper bound of the average analytic rank of elliptic curves over the number field with a level structure such…

We show that average analytic rank of elliptic curves with prescribed torsion $G$ is bounded for every torsion group $G$ under GRH for elliptic curve $L$-functions.

We show that the average size of the $2$-Selmer group of the family of Jacobians of non-hyperelliptic genus-$3$ curves with a marked rational hyperflex point, when ordered by a natural height, is bounded above by $3$. We achieve this by…

We show the existence of families of elliptic curves over Q whose generic rank is at least 2 for the torsion groups Z/8Z and Z/2Z x Z/6Z. Also in both cases we prove the existence of infinitely many elliptic curves, which are parameterized…

Let $E$ be an elliptic curve defined over a number field $K$ where $p$ splits completely. Suppose that $E$ has good reduction at all primes above $p$. Generalizing previous works of Kobayashi and Sprung, we define multiply signed Selmer…

We develop a graph-theoretic algorithm to compute the $\varphi$-Selmer group of the elliptic curve $E_b: y^2 = x^3 + bx$ over $\mathbb{Q}(i)$, where $b \in \mathbb{Z}[i]$ and $\varphi$ is a degree 2 isogeny of $E_b$. We associate to $E_b$ a…

Let $E/k$ be a non-isotrivial elliptic curve over a global function field $k$ of characteristic $p>3$, and $G\subset \mathrm{Gal}(k^{\mathrm{sep}}/k)$ be a topologically finitely generated subgroup. We prove that if $E/k$ has analytic rank…

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…

This paper gives a conjectural characterization of those elliptic curves over the field of complex numbers which "should" be covered by standard modular curves. The elliptic curves in question all have algebraic j-invariant, so they can be…

For an elliptic curve $E$ over $\ratq$ and an integer $r$ let $\pi_E^r(x)$ be the number of primes $p\le x$ of good reduction such that the trace of the Frobenius morphism of $E/\fie_p$ equals $r$. We consider the quantity $\pi_E^r(x)$ on…

We present a new algorithm for computing the endomorphism ring of an ordinary abelian surface over a finite field which is subexponential and generalizes an algorithm of Bisson and Sutherland for elliptic curves. The correctness of this…

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…

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $G=\langle\sigma_1, \dots, \sigma_n\rangle$ be a finitely generated subgroup of $\operatorname{Gal}(\overline{\mathbb{Q}}/ \mathbb{Q})$. Larsen's conjecture claims that the rank of the…

We show that for all odd primes $p$, there exist ordinary elliptic curves over $\bar{\mathbb{F}}_p(x)$ with arbitrarily high rank and constant $j$-invariant. This shows in particular that there are elliptic curves with arbitrarily high rank…

We present a heuristic that suggests that ranks of elliptic curves over the rationals are bounded. In fact, it suggests that there are only finitely many elliptic curves of rank greater than 21. Our heuristic is based on modeling the ranks…

We characterize the possible groups $E(\mathbb{Z}/N\mathbb{Z})$ arising from elliptic curves over $\mathbb{Z}/N\mathbb{Z}$ in terms of the groups $E(\mathbb{F}_p)$, with $p$ varying among the prime divisors of $N$. This classification is…

We develop geometry-of-numbers methods to count orbits in coregular vector spaces having bounded invariants over any global field. We apply these techniques to bound the average ranks and determine average Selmer group sizes of elliptic…

Fix an elliptic curve $E$ over a number field $F$ and an integer $n$ which is a power of $3$. We study the growth of the Mordell--Weil rank of $E$ after base change to the fields $K_d = F(\sqrt[2n]{d})$. If $E$ admits a $3$-isogeny, then we…

We investigate variations of Selmer ranks under quadratic twists satisfying the Heegner hypothesis. In particular, starting with an elliptic curve $E/\mathbb{Q}$ with partial $2$-torsion and a common relaxed Selmer group, we derive explicit…

We prove that a majority (in fact, $>66\%$) of all elliptic curves over $\mathbb Q$, when ordered by height, satisfy the Birch and Swinnerton-Dyer rank conjecture.