## Zev Klagsbrun

Department of Mathematics
University of Wisconsin - Madison
480 Lincoln Dr.
Office: Van Vleck 313
Phone: (608) 263-4698
E-mail: klagsbru@math.wisc.edu

I am currently a Van Vleck Visiting Assistant Professor at the University of Wisconsin - Madison. My research area is algebraic number theory.

I received my Ph.D. from UC-Irvine in June 2011 under the direction of Karl Rubin. My thesis was titled Selmer Ranks of Quadratic Twists of Elliptic Curves.

My primary research interest at the moment is the distribution of Selmer ranks of within quadratic twist families of elliptic curves, but I am in the process of branching out into questions relating to the Cohen-Lenstra heuristics.

## Publications

Disparity in Selmer ranks of quadratic twists of elliptic curves (with Karl Rubin and Barry Mazur). To appear in Annals of Mathematics.

If E is an elliptic curve over Q, then Goldfeld's conjecture famously states that half of the twists of E have rank zero and half have rank one. Goldfeld's conjecture is based on the observation that half of the twists of E have root number 1 and half have root number -1. Combined with the parity conjecture and a minimalist philosophy, this suggests that half of the twists of E have rank zero and half have rank one.

It turns out that if E is defined over a totally complex number field, then the proportion of twists of E having even 2-Selmer rank is not neccessarily one-half. We show that there is a factor ρ coming from a computable local product such that the proportion of twists of E having even 2-Selmer rank is ρ. In general, ρ will be different from one-half; moreover, we show that as E and K are allowed to vary, ρ is actually be dense in [0,1]. Because the parity of the 2-Selmer rank is conjecturally the same as the parity of the rank of E, the same minimalist philosophy behind Goldfeld's conjecture suggests that the proportion of twists of E having rank zero is ρ and the proportion of twists of E having rank one is 1 - ρ.

A family of elliptic curves with a lower bound on 2-Selmer ranks of quadratic twists. To appear in Math Research Letters.

If E is an elliptic curve over a number field K, it is natural to ask which non-negatives integer appear as the 2-Selmer rank of some quadratic twist of E. Dokchitser and Dokchitser obtained negative results over totally complex fields, showing that certain special curves exhibited a phenomenom call constant 2-Selmer parity, in which the 2-Selmer ranks of all of the quadratic twists of E have the same parity. This paper exhibits another obstruction to certain integers appearing as 2-Selmer ranks within the quadratic twist family of E. In particular, we show that if K has a complex place, then there is an infinite class of elliptic curves for which the 2-Selmer rank of every quadratic twist of every curve in this family has 2-Selmer rank greater than the number of complex places of K.

This paper addresses the same general problem as the previous paper - which non-negatives integer appear as the 2-Selmer rank of some quadratic twist of an elliptic curve E. Previously existing results due to Mazur and Rubin showed that if Gal(K(E[2])/K) is the full symmetric group and either K has a real embedding or E has a place v of multiplicative reduction where the order of v in the discriminant of E is odd, then every non-negative integer does in fact occur. This paper achieves slightly strong results when E has a single rational point of order two. Specifically, we show that if E does not have a cyclic 4-isogeny defined over K(E[2]), then all non-negative integers do in fact occur. We also show that if E has a cyclic 4-isogeny defined over K(E[2]) but not over K, then every non-negative integer greater than or equal to the number of complex places of K appears as well.

In 1994, Heath-Brown showed that the 2-Selmer ranks in the quadratic twist family of the congruent number curve satisfied a particular distribution in which every non-negative integer appeared as a 2-Selmer rank within the quadratic twist family. This work was expanded by Swinnerton-Dyer and Kane to include all curves over Q with full two-torsion that do not have a cyclic 4-isogeny. This work gained particular significance recently when Poonen and Rains showed how Heath-Brown's distribution arose naturally in an elliptic curve context. In this paper, we show that if the curve E in question has a single rational point of order two and no cyclic 4-isogeny defined over Q(E[2]), then not only do the 2-Selmer ranks in the quadratic twist family of E not satisfy the Heath-Brown distribution, but in fact they do not satisfy ANY distribution. In particular, if r is some fixed positive integer, we show that at least half of the quadratic twists of E have 2-Selmer rank greater than r.

A Markov Model For Selmer Ranks in Families of Twists (with Karl Rubin and Barry Mazur). In preparation We show that if Gal(K(E[2])/K) is the full symmetric group on three letters, then the quadratic twists of E satisfy the Heath-Brown/ Poonen-Rains distribution described in the previous item.

## Teaching

Math 490, Spring 2013

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