27 citations · 86 across the 16 of their papers we have counts for
18 papers · 1 filter
Towards a classification of isolated -invariants
Abbey Bourdon, Sachi Hashimoto, Timo Keller +5
We develop an algorithm to test whether a non-CM elliptic curve gives rise to an isolated point of any degree on any modular curve of the form . This builds…
Computing nonsurjective primes associated to Galois representations of genus curves
Barinder S. Banwait, Armand Brumer, Hyun Jong Kim +4
For a genus curve over whose Jacobian admits only trivial geometric endomorphisms, Serre's open image theorem for abelian surfaces asserts that there are o…
New Rank Records For Elliptic Curves Having Rational Torsion
Noam D. Elkies, Zev Klagsbrun
We present rank-record breaking elliptic curves having torsion subgroups Z/2Z, Z/3Z, Z/4Z, Z/6Z, and Z/7Z.
Elements of given order in Tate-Shafarevich groups of abelian varieties in quadratic twist families
Manjul Bhargava, Zev Klagsbrun, Robert J. Lemke Oliver +1
Let be an abelian variety over a number field and let be a prime. Cohen-Lenstra-Delaunay-style heuristics predict that the Tate-Shafarevich group of should contai…
Three-isogeny Selmer groups and ranks of abelian varieties in quadratic twist families over a number field
Manjul Bhargava, Zev Klagsbrun, Robert J. Lemke Oliver +1
For an abelian variety over a number field , we prove that the average rank of the quadratic twists of is bounded, under the assumption that the multiplication-by-3 isog…
On a local invariant of elliptic curves with a p-isogeny
Matthew Gealy, Zev Klagsbrun
An elliptic curve defined over a -adic field with a -isogeny comes equipped with an invariant that measures the valuation of the l…
Statistics of -groups modulo for the ring of integers of a varying quadratic number field
Bruce W. Jordan, Zev Klagsbrun, Bjorn Poonen +2
For each odd prime , we conjecture the distribution of the -torsion subgroup of as ranges over real quadratic fields, or over imaginary quadratic…
On the Joint Distribution Of and in Quadratic Twist Families
Daniel Kane, Zev Klagsbrun
If is an elliptic curve with a point of order two, then work of Klagsbrun and Lemke Oliver shows that the distribution of $\dim_{\mathbb{F}_2}\mathrm{Sel}_ϕ(E^d/\mathbb{Q}) - \…
The average sizes of two-torsion subgroups in quotients of class groups of cubic fields
Zev Klagsbrun
We prove a generalization of a result of Bhargava regarding the average size as varies among cubic fields. For a fixed set of rational primes , we obtain…
Davenport-Heilbronn Theorems for Quotients of Class Groups
Zev Klagsbrun
We prove a generalization of the Davenport-Heilbronn theorem to quotients of ideal class groups of quadratic fields by the primes lying above a fixed set of rational primes . Ad…
The Elkies Curve has Rank 28 Subject only to GRH
Zev Klagsbrun, Travis Sherman, James Weigandt
In 2006, Elkies presented an elliptic curve with 28 independent rational points. We prove that subject to GRH, this curve has Mordell-Weil rank equal to 28 and analytic rank at mos…
The distribution of the Tamagawa ratio in the family of elliptic curves with a two-torsion point
Zev Klagsbrun, Robert J. Lemke Oliver
In recent work, Bhargava and Shankar have shown that the average size of the -Selmer group of an elliptic curve over is exactly , and Bhargava and Ho have shown…
The distribution of 2-Selmer ranks of quadratic twists of elliptic curves with partial two-torsion
Zev Klagsbrun, Robert J. Lemke Oliver
This paper presents a new result concerning the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve over an arbitrary number field K with a single poi…
A Markov model for Selmer ranks in families of twists
Zev Klagsbrun, Barry Mazur, Karl Rubin
We study the distribution of 2-Selmer ranks in the family of quadratic twists of an elliptic curve E over an arbitrary number field K. Under the assumption that Gal(K(E[2])/K) = S_…
On the Distribution of 2-Selmer Ranks within Quadratic Twist Families of Elliptic Curves with Partial Rational Two-Torsion
Zev Klagsbrun
This paper presents a new result concerning the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve with a single point of order two that does not hav…
Selmer Ranks of Quadratic Twists of Elliptic Curves with Partial Rational Two-Torsion
Zev Klagsbrun
This paper investigates which integers can appear as 2-Selmer ranks within the quadratic twist family of an elliptic curve E defined over a number field K with E(K)[2] = Z/2Z. We s…
A Family of Elliptic Curves with a Lower Bound on 2-Selmer Ranks of Quadratic Twists
Zev Klagsbrun
For any number field K with a complex place, we present an infinite family of elliptic curves defined over K such that $dim \mathbb{F}_2 Sel_2(E^F/K) \ge dim \mathbb{F}_2 E^F(K)[2]…
Disparity in Selmer ranks of quadratic twists of elliptic curves
Zev Klagsbrun, Barry Mazur, Karl Rubin
We study the parity of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. We prove that the fraction of twists (of…