Given an elliptic curve $E$ over a finite field $\mathbb{F}_q$ we study the finite extensions $\mathbb{F}_{q^n}$ of $\mathbb{F}_q$ such that the number of $\mathbb{F}_{q^n}$-rational points on $E$ attains the Hasse upper bound. We obtain an upper bound on the degree $n$ for $E$ ordinary using an estimate for linear forms in logarithms, which allows us to compute the pairs of isogeny classes of such curves and degree $n$ for small $q$. Using a consequence of Schmidt's Subspace Theorem, we improve the upper bound to $n\leq 11$ for sufficiently large $q$. We also show that there are infinitely many isogeny classes of ordinary elliptic curves with $n=3$.
Let $E/\mathbb{Q}$ be a non-CM elliptic curve. Assuming GRH, we prove that, for a set of primes $p$ of density $1$, the absolute discriminant of the $\mathbb{F}_p$-endomorphism ring of the reduction of $E$ modulo $p$ is close to maximal.
Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let $\mathbb{Q}^{ab}$ be the maximal abelian extension of $\mathbb{Q}$. In this article we classify the groups that can arise as $E(\mathbb{Q}^{ab})_{\text{tors}}$ up to isomorphism. The method illustrates techniques for finding explicit models of modular curves of mixed level structure. Moreover we provide an explicit algorithm to compute $E(\mathbb{Q}^{ab})_{\text{tors}}$ for any elliptic curve $E/\mathbb{Q}$.
We study the intersection of an algebraic variety with the maximal compact subgroup of a universal vectorial extension of a product of elliptic curves. For this intersection we show a Manin-Mumford type statement. This answers some questions posed by Corvaja-Masser-Zannier which arose in connection with their investigation of the intersection of a curve with real analytic subgroups of various algebraic groups. They prove finiteness in the situation of a single elliptic curve. Using Khovanskii's zero-estimates combined with a stratification result of Gabrielov-Vorobjov and recent work of the au…
The torsion anomalous conjecture states that for any variety V in an abelian variety there are only finitely many maximal V-torsion anomalous varieties. We prove this conjecture for V of codimension 2 in a product E^N of any elliptic curve E. This was known only when E has CM. We also give an effective upper bound for the normalized height of these maximal V-torsion anomalous varieties.
Chevyrev and Galbraith recently devised an algorithm which inputs a maximal order of the quaternion algebra ramified at one prime and infinity and constructs a supersingular elliptic curve whose endomorphism ring is precisely this maximal order. They proved that their algorithm is correct whenever it halts, but did not show that it always terminates. They did however prove that the algorithm halts under a reasonable assumption which they conjectured to be true. It is the purpose of this paper to verify their conjecture and in turn prove that their algorithm always halts. More precisely, Chevyr…
We pick up the study of 13-torsion in elliptic curves where Mazur and Tate left off 45 years ago. We consider various questions concerning elliptic curves defined over the maximal totally real subfield of the 13th cyclotomic field (where J_1(13) acquires everywhere good reduction), and over quadratic extensions.
In this paper, we examine certain maximal rank sublattices of the Mordell-Weil lattices of two families of elliptic curves over fields of characteristic $p > 0$. We compute explicit lower bounds on the densest sphere packings of these sublattices by finding lower bounds on the minimal norms of the sublattices and explicitly computing the volumes of their fundamental domains.
Computing endomorphism rings of supersingular elliptic curves is an important problem in computational number theory, and it is also closely connected to the security of some of the recently proposed isogeny-based cryptosystems. In this paper we give a new algorithm for computing the endomorphism ring of a supersingular elliptic curve $E$ that runs, under certain heuristics, in time $O((\log p)^2p^{1/2})$. The algorithm works by first finding two cycles of a certain form in the supersingular $\ell$-isogeny graph $G(p,\ell)$, generating an order $Λ\subseteq \operatorname{End}(E)$. Then all maxi…
We give equations for 13 genus-2 curves over $\overline{\mathbb{Q}}$, with models over $\mathbb{Q}$, whose unpolarized Jacobians are isomorphic to the square of an elliptic curve with complex multiplication by a maximal order. If the Generalized Riemann Hypothesis is true, there are no further examples of such curves. More generally, we prove under the Generalized Riemann Hypothesis that there exist exactly 46 genus-2 curves over $\overline{\mathbb{Q}}$ with field of moduli $\mathbb{Q}$ whose Jacobians are isomorphic to the square of an elliptic curve with complex multiplication by a maximal o…
Stark-Heegner points, also known as Darmon points, were introduced by H. Darmon as certain local points on rational elliptic curves, conjecturally defined over abelian extensions of real quadratic fields. The rationality conjecture for these points is only known in the unramified case, namely, when these points are specializations of global points defined over the strict Hilbert class field $H^+_F$ of the real quadratic field $F$ and twisted by (unramified) quadratic characters of $Gal(H_c^+/F)$. We extend these results to the situation of ramified quadratic characters; more precisely, we show…
We study elliptic surfaces over $\mathbb{Q}(T)$ with coefficients of a Weierstrass model being polynomials in $\mathbb{Q}[T]$ with degree at most 2. We derive an explicit expression for their rank over $\mathbb{Q}(T)$ depending on the factorization and other simple properties of certain polynomials. Finally, we give sharp estimates for the ranks of the considered families and we present several applications, among which there are lists of rational points, generic families with maximal rank and generalizations of former results.
Let $E/\mathbb{Q}$ be an elliptic curve. For a prime $p$ of good reduction, let $r(E,p)$ be the smallest non-negative integer that gives the $x$-coordinate of a point of maximal order in the group $E(\mathbb{F}_p)$. We prove unconditionally that $r(E,p)> 0.72\log\log p$ for infinitely many $p$, and $r(E,p) > 0.36 \log p$ under the assumption of the Generalized Riemann Hypothesis. This can be viewed as elliptic curve analogues of classical lower bounds on the least primitive root of a prime.
We prove several results on torsion points and Galois representations for complex multiplication (CM) elliptic curves over a number field containing the CM field. One result computes the degree in which such an elliptic curve has a rational point of order $N$, refining results of Silverberg. Another result bounds the size of the torsion subgroup of an elliptic curve with CM by a nonmaximal order in terms of the torsion subgroup of an elliptic curve with CM by the maximal order. Our techniques also yield a complete classification of both the possible torsion subgroups and the rational cyclic is…
We present a database of rational elliptic curves, up to Q-isomorphism, with good reduction outside {2,3,5,7,11,13}. We provide a heuristic involving the abc and BSD conjectures that the database is likely to be the complete set of such curves. Moreover, proving completeness likely needs only more computation time to conclude. We present data on the distribution of various quantities associated to curves in the set. We also discuss the connection to S-unit equations and the existence of rational elliptic curves with maximal conductor.
Let K denote the quadratic field $\mathbb{Q}(\sqrt{d})$ where d=$-1$ or $-3$. Let E be an elliptic curve defined over K. In this paper, we analyze the torsion subgroups of E in the maximal elementary abelian $2$-extension of $K$.
Given an elliptic curve $E/k$ and a Galois extension $k'/k$, we construct an exact functor from torsion-free modules over the endomorphism ring ${\rm End}(E_{k'})$ with a semilinear ${\rm Gal}(k'/k)$ action to abelian varieties over $k$ that are $k'$-isogenous to a power of $E$. As an application, we show that every elliptic curve with complex multiplication geometrically is isogenous over the ground field to one with complex multiplication by a maximal order.
We introduce and study a new way to catagorize supersingular abelian varieties defined over a finite field by classifying them as fully maximal, mixed, or fully minimal. The type of $A$ depends on the normalized Weil numbers of $A$ and its twists. We analyze these types for supersingular abelian varieties and curves under conditions on the automorphism group. In particular, we present a complete analysis of these properties for supersingular elliptic curves and supersingular abelian surfaces in arbitrary characteristic, and for a one-dimensional family of supersingular curves of genus $3$ in c…
