All Questions
8 questions with no upvoted or accepted answers
16
votes
0
answers
275
views
Why should an abelian variety with few places of bad reduction and a lot of endomorphisms not have many points?
In the paper "Points of Order 13 on Elliptic Curves" by Mazur-Tate, they say in the introduction:
It seemed ... that if such an abelian variety $J$, which has bad reduction at only one ...
- 7,746
10
votes
0
answers
217
views
Are the nonnegative rationals diophantine with only two quantifiers?
Definition: A subset $D\subseteq \mathbb{Q}$ is diophantine if it is the projection of the zero set of a polynomial, i.e. there exists a polynomial $f\in\mathbb{Q}[X,Y_1,\dots,Y_n]$ for some $n$ such ...
- 2,051
6
votes
0
answers
438
views
Brauer-Manin obstruction to surfaces of Kodaira dimension 1
Roughly speaking, the Kodaira dimension is an invariant of a variety that corresponds to curvature. One can show that curves of genus $\geq 2$ have Kodaira dimension 1 using Riemann-Roch. In Corollary ...
- 998
5
votes
0
answers
303
views
2-descent on elliptic curves, and units modulo squares of units
Setup: Let $p$ be a prime, let $f(x) \in \mathbb{Q}_p[x]$ be a separable monic cubic polynomial cutting out the maximal order $\mathcal{O}_{K_f}$ in the etale algebra $K_f := \mathbb{Q}_p[x]/(f(x))$, ...
5
votes
0
answers
184
views
Given a point $P$ on a genus-$1$ curve over $\mathbb{Q}_p$, is there an $R$ such that $2 \mid [P - R]$ and $x(\overline{P}) \neq x(\overline{R})$?
Let $p \in \mathbb{Z}$ be prime, and let $f \in \mathbb{Z}_p[x]$ be a quartic polynomial with nonzero discriminant. Let $C/\mathbb{Q}_p$ be the genus-$1$ curve with affine equation $y^2 = f(x)$. Let $\...
- 53
3
votes
0
answers
170
views
Smoothness of height in Manin conjecture
Set up: Let $K$ be a number field. Let $M_K$ be the places of $K$, and define the standard height on $\mathbb{P}^n(K)$ as
$$H([x_0, \cdots, x_n]) = \prod_{v \in M_K} \max\{|x_0|_v, \cdots, |x_n|_v\}$$
...
- 267
2
votes
0
answers
198
views
Finding rational points via birational map
Let $C$ be an affine curve given by $p_C(x,y)=0$ where $$ p_C=2x^3y + 2xy^3 +x^3 + y^3 + 5x^2y + 5xy^2 + 2x^2 + 2y^2 + 2x^2y^2 + 2xy $$
and let $\overline{C}$ denote the projective closure of $C$. For ...
- 21
2
votes
0
answers
107
views
the least point on a variety over a finite field
Let $p$ be a large prime parameter and $V\subseteq \mathbb{P}^n_{\mathbb{F}_p}$ a variety defined over the finite field $\mathbb{F}_p$ with bounded degree and dimension (w.r.t. $p$). Assume that $V$ ...
- 3,362