Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On 2-Selmer ranks of quadratic twists of elliptic curves   We study the $2$-Selmer ranks of elliptic curves. We prove that for an
arbitrary elliptic curve $E$ over an arbitrary number field $K$, if the set
$A_E$ of 2-Selmer ranks of quadratic twists of $E$ contains an integer $c$, it
contains all integers larger than $c$ and having the same parity as $c$. We
also find sufficient conditions on $A_E$ such that $A_E$ is equal to $\Z_{\ge
t_E}$ for some number $t_E$. When all points in $E[2]$ are rational, we give an
upper bound for $t_E$.

19
The categories of this paper are: eess
23
The categories of this paper are: solv-int
13
The categories of this paper are: quant-ph
34
The categories of this paper are: chem-ph
7
The categories of this paper are: hep-th
33
The categories of this paper are: plasm-ph
27
The categories of this paper are: comp-gas
28
The categories of this paper are: alg-geom
29
The categories of this paper are: funct-an
32
The categories of this paper are: atom-ph
3
The categories of this paper are: physics
6
The categories of this paper are: astro-ph
25
The categories of this paper are: patt-sol
12
The categories of this paper are: nucl-th
0
The categories of this paper are: hep-ph
31
The categories of this paper are: ao-sci