Databases of elliptic curves ordered by height and distributions of Selmer groups and ranks

Abstract

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 Q, ordered by height. We describe databases of elliptic curves over 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.

Ranks, 2-Selmer groups, and Tamagawa numbers of elliptic curves with ℤ∕2ℤ × ℤ∕8ℤ-torsion

“…The Tamagawa product of E is T (E) = p≤∞ c p (E). 1 The parameter height used here for an elliptic curve with a 2-torsion point E A,B : y 2 =…”

“…Tamagawa product. The average Tamagawa product in the (2, 8)-torsion family also behaves differently from the one in[1]…”

The Ubiquity of Elliptic Curves

“…Not a lot is known about the rank . Work of Manjul Bhargava [4] and others [1] suggest that the average value of is 1/2meaning roughly half of elliptic curves have rank = 0 while the other half have rank = 1. An example of Noam Elkies shows that the rank can be as large as = 28, but recent work of Bjorn Poonen et al [27] [28] suggests that there is a uniform upper bound on .…”

The Arithmetic Geometry of Resonant Rossby Wave Triads

“…However, a natural generalization of the widely believed Minimalist Conjecture predicts that, in the limit as N → ∞, 50% of the C(a, b) in [1, N ] 2 should have rank 1, and 50% rank 2. See [1] for background about this problem and extensive numerical computations.…”

“…Moreover, mesoscale waves observed in Jupiter's atmosphere have wavelengths of about 20 km, and would have wavenumbers of approximately 20000 if "extended to cover a significant part of Jupiter's surface" [3]. 1 The set of resonant triads may be described as the set of integer solutions to a Diophantine equation, as explained in the next section. In this paper, we give two descriptions of the solutions to this equation: a rational parametrization and an elliptic fibration.…”