Academic literature on the topic 'Average ranks of elliptic curve'

Most systematic tables of data associated to ranks of elliptic curves order the curves by conductor. Recent developments, led by work of Bhargava and Shankar studying the average sizes of $n$-Selmer groups, have given new upper bounds on the average algebraic rank in families of elliptic curves over $\mathbb{Q}$, ordered by height. We describe databases of elliptic curves over $\mathbb{Q}$, ordered by height, in which we compute ranks and $2$-Selmer group sizes, the distributions of which may also be compared to these theoretical results. A striking new phenomenon that we observe in our database is that the average rank eventually decreases as height increases.

We prove quantitative upper bounds for the number of quadratic twists of a given elliptic curve E/Fq(C) over a function field over a finite field that have rank ≥ 2, and for their average rank. The main tools are constructions and results of Katz and uniform versions of the Chebotarev density theorem for varieties over finite fields. Moreover, we conditionally derive a bound in some cases where the degree of the conductor is unbounded.

AbstractLet ψ be a generic Drinfeld module of rank r ≥ 2. We study the first elementary divisor d1,℘ (ψ) of the reduction of ψ modulo a prime ℘, as ℘ varies. In particular, we prove the existence of the density of the primes ℘ for which d1,℘ (ψ) is fixed. For r = 2, we also study the second elementary divisor (the exponent) of the reduction of ψ modulo ℘ and prove that, on average, it has a large norm. Our work is motivated by J.-P. Serre's study of an elliptic curve analogue of Artin's Primitive Root Conjecture, and, moreover, by refinements to Serre's study developed by the first author and M. R. Murty.

This Ph.D. thesis collects the author's works and interests in several parts of Number Theory, from algebraic problems related to relations between number fields which are based on the factorization of prime numbers in the rings of integers, up to the application of tools concerning the density of primes with given splitting type in number fields to the computation of the average rank of specific families of elliptic curves, concluding finally with the classification and estimate of the main invariants of a number field, like the discriminant and the regulator, pursued by means of analytic formulas and algorithmic methods developed on the previous tools and implemented on suitable computer algebra systems, like PARI/GP.

Abstract This paper deals with numerical predictions of the leakage flow rates, drag power and rotordynamic force coefficients for three types of helically-grooved liquid annular seals, which include a liquid annular seal with helically-grooved stator (GS/SR seal), one with helically-grooved rotor (SS/GR seal), and one with helical grooves on stator and rotor (GS/GR seal). These seals are frequently used for multiple-stage centrifugal pumps as they have the advantage of low leakage (even to zero) due to the “pumping effect” of the helical grooves. However, the static and rotordynamic characteristics of helically-grooved liquid annular seals still are not fully understood, and even more pronounced is the lack of effective numerical models in the literature. A novel transient CFD-based perturbation method was proposed for the predictions of the leakage flow rates, drag power and rotordynamic force coefficients of helically-grooved liquid annular seals. This method is based on the unsteady Reynolds-Averaged Navier–Stokes (RANS) solution with the mesh deformation technique and the multiple reference frame theory. The time-varying fluid-induced forces acting on the rotor/stator surface were obtained as a response to the time-dependent perturbation of the seal stator surface with the periodic motion, based on the multiple-frequency elliptical-orbit stator whirling model. The frequency-independent rotordynamic force coefficients were determined using curve fit and Fast Fourier Transform (FFT) in the frequency domain. The CFD-based method was adequately validated by comparisons to the published experiment data of leakage flow rates and fluid response forces for three types of helically-grooved liquid annular seals. Based on the transient CFD-based perturbation method, numerical results of the leakage flow rates, drag powers and rotordynamic force coefficients were presented and compared for three types of helically-grooved liquid annular seals at five rotational speeds (n = 0.5 krpm, 1.0 krpm, 2.0 krpm, 3.0 krpm and 4.0 krpm), paying special attention to the effective stiffness coefficient and effective damping coefficient. Results show that the GS/GR seal has the best sealing capability, followed by the GS/SR seal and then the SS/GR seal. The leakage flow rate of all three helically-grooved seals monotonically decreases with the increasing rotational speed. The GS/SR seal possesses the best stiffness and damping capability, followed by the SS/GR seal and then the GS/GR seal. Rotordynamic instability problems are more likely caused by the GS/GR seal in multi-stage centrifugal pumps. From a rotordynamic viewpoint, the GS/SR helically-grooved liquid annular seal is a better seal concept for multi-stage centrifugal pumps.