Distribution of Selmer groups of quadratic twists of a family of elliptic curves

Abstract. Using maximal isotropic submodules in a quadratic module over Z p , we prove the existence of a natural discrete probability distribution on the set of isomorphism classes of short exact sequences of co-finite type Z p -modules, and then conjecture that as E varies over elliptic curves over a fixed global field k, the distribution of→ 0 is that one. We show that this single conjecture would explain many of the known theorems and conjectures on ranks, Selmer groups, and Shafarevich-Tate groups of elliptic curves. We also prove the existence of a discrete probability distribution of the set of isomorphism classes of finite abelian p-groups equipped with a nondegenerate alternating pairing, defined in terms of the cokernel of a random alternating matrix over Z p , and we prove that the two probability distributions are compatible with each other and with Delaunay's predicted distribution for X. Finally, we prove new theorems on the fppf cohomology of elliptic curves in order to give further evidence for our conjecture.

show abstract

“…The average size of Sel 2 E can even be infinite in certain families. See [Yu05], [XZ09], [XZ08], and [FX12] for work in this direction.…”

mentioning

confidence: 99%

Modeling the distribution of ranks, Selmer groups, and Shafarevich–Tate groups of elliptic curves

Bhargava

Kane

Lenstra

et al. 2015

Cambridge Journal of Mathematics

View full text Add to dashboard Cite

show abstract

“…is of rank 0 for a positive proportion of squarefree integers D. We also note that the bound (37) plays an important role in studying the distribution of Selmer groups of similar families of curves, see [208].…”

Section: Ranks and Selmer Groups Of Elliptic Curvesmentioning

confidence: 92%

Modular hyperbolas

Shparlinski

2012

Jpn. J. Math.

View full text Add to dashboard Cite

show abstract

“…For example, the number of distinct prime factors of an integer n [4], of φ(n) [6], of the sum a + b when a and b are given in some dense set [5], of the number of points on an elliptic curve [11], of the characteristic polynomial of the Frobenius acting on Drinfeld modules [10], and of polynomials of several variables [16] are all with Gaussian distribution. Another example that falls into this type is the 2-rank of the Selmer groups of certain 2-isogenies of some families of elliptic curves [17,18]. A well-known unpublished result of Selberg on the distribution of values of Riemann-Zeta function ζ(s) on the critical line offers another type of Gaussian distribution result.…”

Section: Introductionmentioning

confidence: 99%

The Distribution of Values of Short Hybrid Exponential Sums on Curves over Finite Fields

Mak

Zaharescu²

2011

Mathematical Research Letters

Self Cite

View full text Add to dashboard Cite

Abstract. Let p be a prime number, X be an absolutely irreducible affine plane curve over Fp, and g, f ∈ Fp(x, y). We study the distribution of the values of the hybrid exponential sums Sn = Xon n ∈ I for some short interval I. We show that under some natural conditions the limiting distribution of the projections of the sum Sn, n ∈ I on any straight line through the origin is Gaussian as p tends to infinity.

show abstract

Distribution of Selmer groups of quadratic twists of a family of elliptic curves

Cited by 15 publications

References 31 publications

Modeling the distribution of ranks, Selmer groups, and Shafarevich–Tate groups of elliptic curves

Modeling the distribution of ranks, Selmer groups, and Shafarevich–Tate groups of elliptic curves

Modular hyperbolas

The Distribution of Values of Short Hybrid Exponential Sums on Curves over Finite Fields

Contact Info

Product

Resources

About