Related papers: Averages of elliptic curve constants
An algorithm is given to compute a normal form for hyperelliptic curves. The elliptic case has been treated in a previous paper. In this paper the hyperelliptic case is treated.
We investigate the average rank in the family of quadratic twists of a given elliptic curve defined over $\mathbb{Q}$, when the curves are ordered using the canonical height of their lowest non-torsion rational point.
Consider elliptic curves $ E:\ y^{2} = x^{3} + D^{3} $ defined over the quadratic field $\ \Q(\sqrt{-3}) $. Hecke $ L-$series attached to $ E $ are studied, formulae for their values at $ s=1, $ and bound of 3-adic valuations of these…
We give some heuristics for counting elliptic curves with certain properties. In particular, we re-derive the Brumer-McGuinness heuristic for the number of curves with positive/negative discriminant up to $X$, which is an application of…
We prove $L_p$ estimates of solutions to a conormal derivative problem for divergence form complex-valued higher-order elliptic systems on a half space and on a Reifenberg flat domain. The leading coefficients are assumed to be merely…
We prove regularity, global existence, and convergence of Lagrangian mean curvature flows in the two-convex case. Such results were previously only known in the convex case, of which the current work represents a significant improvement.…
Let $D\neq 1$ be a fixed squarefree integer. For elliptic curves $E/\mathbb{Q}$, writing $E_D$ for the quadratic twist by $D$, we consider the question of how often $E(\mathbb{Q})$ and $E_D(\mathbb{Q})$ generate $E(\mathbb{Q}(\sqrt{D}))$.…
For stochastic perturbations of linear systems with non-zero pure imaginary spectrum we discuss the averaging theorems in terms of the slow-fast action-angle variables and in the sense of Krylov-Bogoliubov. Then we show that if the…
We observe that there are elliptic curves over number fields all of whose quadratic twists must have positive rank, assuming the Birch-Swinnerton-Dyer conjecture. We give a classification of such curves in terms of their local behaviour,…
We study the limiting behavior of multiple ergodic averages involving several not necessarily commuting measure preserving transformations. We work on two types of averages, one that uses iterates along combinatorial parallelepipeds, and…
We use an invariant-theoretic method to compute certain twists of the modular curves X(n) for n=7,9,11. Searching for rational points on these twists enables us to find non-trivial pairs of n-congruent elliptic curves over Q, i.e. pairs of…
In this short note, we give a method for computing a non-torsion point of smallest canonical height on a given elliptic curve $E/\mathbb{Q}$ over all number fields of a fixed degree. We then describe data collected using this method, and…
We prove that, when all elliptic curves over $\mathbb{Q}$ are ordered by naive height, a positive proportion have both algebraic and analytic rank one. It follows that the average rank and the average analytic rank of elliptic curves are…
The ancient unsolved problem of congruent numbers has been reduced to one of the major questions of contemporary arithmetic: the finiteness of the number of curves over $\bf Q$ which become isomorphic at every place to a given curve. We…
We show that if one can compute a little more than a particular moment for some family of L-functions, then one has upper bounds of the conjectured order of magnitude for all smaller (positive, real) moments and a one-sided central limit…
We show that the averaged equation for a one-frequency fast-oscillating Hamiltonian system is the result of symplectic reduction of a certain natural system on the corresponding $S^1$-bundle with respect to the circle action. Furthermore,…
We investigate the paramater of the average range of $M$-Lipschitz mapping of a given graph. We focus on well-known classes such as paths, complete graphs, complete bipartite graphs and cycles and show closed formulas for computing this…
We study double averages along orbits for measure preserving actions of $\mathbb{A}^\omega$, the direct sum of countably many copies of a finite abelian group $\mathbb{A}$. In this article we show an $L^p$ norm-variation estimate for these…
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 survey article, we summarise the known results towards the conjecture: elliptic curves over totally real number fields are modular. For understanding these recent results in the literature, we present some necessary background along…