Abstract
Abstract
For each prime p, we show that there exist geometrically simple abelian varieties A over
${\mathbb Q}$
with
. Specifically, for any prime
$N\equiv 1 \ \pmod p$
, let
$A_f$
be an optimal quotient of
$J_0(N)$
with a rational point P of order p, and let
$B = A_f/\langle P \rangle $
. Then the number of positive integers
$d \leq X$
with
is
$ \gg X/\log X$
, where
$\widehat B_d$
is the dual of the dth quadratic twist of B. We prove this more generally for abelian varieties of
$\operatorname {\mathrm {GL}}_2$
-type with a p-isogeny satisfying a mild technical condition. In the special case of elliptic curves, we give stronger results, including many examples where
for an explicit positive proportion of integers d.
Publisher
Cambridge University Press (CUP)
Subject
Computational Mathematics,Discrete Mathematics and Combinatorics,Geometry and Topology,Mathematical Physics,Statistics and Probability,Algebra and Number Theory,Theoretical Computer Science,Analysis
Reference34 articles.
1. Class Groups and Selmer Groups
2. Some examples of 5 and 7 descent for elliptic curves over Q
3. [Smi20] Smith, A. , ‘ ${\ell}^{\infty }$ -Selmer groups in degree $\ell$ twist families’, PhD thesis, Harvard University, Graduate School of Arts & Sciences (2020).
4. [BFS21] Bruin, N. , Flynn, E. V. , and Shnidman, A. , Genus $2$ curves with full $\sqrt{3}$ -level structure and Tate-Shafarevich groups. Preprint, 2021, arxiv.org/abs/2102.04319.
5. Quadratic Twists of Abelian Varieties With Real Multiplication