-
Notifications
You must be signed in to change notification settings - Fork 1
/
AIM_4torsion.tex
185 lines (155 loc) · 6.9 KB
/
AIM_4torsion.tex
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
\documentclass{article}
\usepackage{amsmath, amsthm, amssymb}
\usepackage[all]{xy}
\usepackage[pagebackref,colorlinks]{hyperref}
\usepackage{mathrsfs}
% Color comments!
\usepackage[usenames,dvipsnames]{color}
% Color comments
\newcommand{\jenn}[1]{{\color{magenta} \sf $\clubsuit\clubsuit\clubsuit$ Jenn: [#1]}}
\newcommand{\shahed}[1]{{\color{Purple} \sf $\clubsuit\clubsuit\clubsuit$ Shahed: [#1]}}
\newcommand{\sce}{\mathscr{C}^{\textsf{ex}}}
\newcommand{\scd}{\mathscr{C}}
\newcommand{\caff}{C_{\textsf{aff}}}
\theoremstyle{plain}
\newtheorem*{reftheorem}{Theorem}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{problem}{Problem}
\newtheorem{question}{Question}
\newtheorem*{question*}{Question}
\newtheorem{claim}{Claim}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
% General
\renewcommand{\emptyset}{\varnothing}
\newcommand{\hra}{\hookrightarrow}
\newcommand{\righthookarrow}{\hookrightarrow}
\newcommand{\isom}{\cong}
\newcommand{\too}{\longrightarrow}
\newcommand{\isomto}{\overset{\sim}{\longrightarrow}}
\newcommand{\nto}[1]{\overset{#1}{\longrightarrow}}
\newcommand{\nsubset}{\not\subset}
\renewcommand{\phi}{\varphi}
\newcommand{\To}{\Rightarrow}
\newcommand{\ilim}{\displaystyle\lim_{\leftarrow}}
\newcommand{\dirlim}{\displaystyle\lim_{\rightarrow}}
\newcommand{\eps}{\varepsilon}
\renewcommand{\bar}[1]{\overline{#1}}
\renewcommand{\tilde}[1]{\widetilde{#1}}
\DeclareMathOperator{\car}{char}
\DeclareMathOperator{\rk}{rk}
\DeclareMathOperator{\coker}{coker}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\Aut}{Aut}
\DeclareMathOperator{\End}{End}
\DeclareMathOperator{\im}{im}
\DeclareMathOperator{\pgl}{PGL}
\DeclareMathOperator{\Gl}{GL}
\DeclareMathOperator{\Sl}{SL}
% Number theory
\newcommand{\Qbar}{\ensuremath{\overline{\Q}}}
\newcommand{\Kb}{\overline{K}}
\newcommand{\Fb}{\overline{F}}
\newcommand{\kb}{\overline{k}}
\newcommand{\Xbar}{\overline{X}}
\newcommand{\Cbar}{\overline{C}}
\newcommand{\R}{\ensuremath{\mathbb{R}}}
\newcommand{\C}{\ensuremath{\mathbb{C}}}
\newcommand{\F}{\ensuremath{\mathbb{F}}}
\newcommand{\fp}{\ensuremath{\mathbb{F}_p}}
\newcommand{\sm}{\ensuremath{\mathfrak{m}}}
\newcommand{\Q}{\ensuremath{\mathbb{Q}}}
\newcommand{\Z}{\ensuremath{\mathbb{Z}}}
\newcommand{\ok}{\mathscr{O}_K}
\DeclareMathOperator{\Gal}{Gal}
\DeclareMathOperator{\inv}{inv}
\DeclareMathOperator{\Nm}{Nm}
\DeclareMathOperator{\tr}{Tr}
% Algebraic geometry
\newcommand{\sA}{\ensuremath{\mathscr{A}}}
\newcommand{\sO}{\ensuremath{\mathscr{O}}}
\newcommand{\sL}{\ensuremath{\mathscr{L}}}
\newcommand{\sK}{\ensuremath{\mathscr{K}}}
\newcommand{\sF}{\ensuremath{\mathscr{F}}}
\newcommand{\A}{\ensuremath{\mathbb{A}}}
\newcommand{\Pro}{\ensuremath{\mathbb{P}}}
\newcommand{\G}{\ensuremath{\mathbb{G}}}
\newcommand{\sG}{\mathscr{G}}
\newcommand{\sX}{\mathscr{X}}
\DeclareMathOperator{\Supp}{Supp}
\DeclareMathOperator{\Div}{Div}
\DeclareMathOperator{\dv}{div}
\DeclareMathOperator{\Pic}{Pic}
\DeclareMathOperator{\P0}{Pic^0}
\DeclareMathOperator{\Spec}{Spec}
\begin{document}
\title{Not the 4-torsion point you're looking for}
\author{Shahed Sharif}
\maketitle
Let $k$ be a finite field of odd characteristic, $g \geq 3$ an odd integer, $a_i$ for $1 \leq i \leq g$ distinct elements of $k^\times$, and $K = k(t)$. Let $C$ be the hyperelliptic curve with affine piece
\[
y^2 = x \prod^g (x + a_i)(a_i x + t).
\]
Let $u$ be a fixed square root of $t$ in an algebraic closure of $K$. Let $P \in C(K(u))$ be the point $(u, u^{\frac{g+1}{2}} \prod (u + a_i))$. The purpose of this note is to show that the divisor $2P - 2\infty$ is \emph{not} linearly equivalent to $(0,0) - \infty$. Note that the class of the latter divisor is 2-torsion. We will in fact prove a stronger claim:
\begin{proposition}
There is no $T \in C(\Kb)$ for which $T - \infty$ is linearly equivalent to $(0,0) - \infty$.
\end{proposition}
\section{Quotient by $\phi$}
\label{sec:quotient-phi}
Let $\phi \in \Aut C$ be given by
\[
\phi(x,y) = \bigg(\frac{t}{x}, \frac{yt^{\frac{g+1}{2}}}{x^{g+1}}\bigg).
\]
One checks that $\phi^2$ is the identity. As in Ren\'e's notes, let $C_1$ be the quotient $C/\phi$. Then $C_1$ is given by\footnote{The below corrects a minor error in Ren\'e's definition of $j$.}
\[
w^2 = j(v)
\]
where $v = x + \dfrac{t}{x}$, $w = \dfrac{y}{x^{\frac{g+1}{2}}}$, and
\[
j(v) = \prod^g a_i\bigg(v + a_i + \frac{t}{a_i}\bigg).
\]
Observe that $C_1$ is hyperelliptic of genus $(g-1)/2$. Since $g$ is odd, there is a single point at infinity. Let $p_1$ be the quotient map
\[
p_1: C \to C_1.
\]
\begin{remark}
The automorphism $\phi$ is defined even when $g$ is even (over the field $K(u)$), so one should be able to construct some model of $C_1$ in this case as well.
\end{remark}
\section{The divisor $P - \infty$}
\label{sec:divisor-p-infty}
One sees that $p_1(0,0) = p_1(\infty) = \infty$. Therefore $p_{1*}((0,0) - \infty)$ is the trivial divisor on $C_1$. To show that $2P - 2\infty$ is not linearly equivalent to $(0,0) - \infty$, then, it suffices to show that $p_{1*}(P - \infty)$ does not represent a 2-torsion class on $C_1$. But as $C_1$ is hyperelliptic, this is the same as showing that $p_1(P)$ is not a Weierstrass point. We calculate
\begin{align*}
p_1(P) &= p_1(u, u^{\frac{g+1}{2}} \prod (u + a_i)) \\
&= \bigg(u + \frac{t}{u}, \frac{t^{\frac{g+1}{2}}u^{\frac{g+1}{2}} \prod (u + a_i)}{u^{g+1}}\bigg) \\
&= \bigg(2u, u^{\frac{g+1}{2}} \prod (u + a_i)\bigg)
\end{align*}
The $w$-coordinate is nonzero, which concludes the proof.
\section{No other possibilities}
\label{sec:no-other-poss}
We now prove the proposition.
\begin{proof}
Suppose there is $T \in C(\Kb)$ such that $2T - 2\infty$ is linearly equivalent to $(0,0) - \infty$. Then $p_{1*}(T-\infty)$ is a 2-torsion class over $C_1$, or equivalently $T$ is a Weierstrass point of $C_1$. These points are precisely those for which
\[
p_1(T) = (a_i + t/a_i,0)
\]
for some $i$. One verifies that such $T$ are of the form $T = (a_i, 0)$ or $T = (t/a_i, 0)$. But such $T$ are Weierstrass points of $C$, so that $2T - 2\infty$ in fact lies in the trivial divisor class.
\end{proof}
\section{Miscellaneous}
\label{sec:miscellaneous}
Let $C_2 = C/(-\phi)$ as in Ren\'e's notes, and $p_2: C \to C_2$ the quotient map. Then $p_2(P)$ is a Weierstrass point of $C_2$. When $g = 1$, $C_1 \isom \Pro^1$, so these results verify Ulmer's calculations in the case of the Legendre curve. Finally, $p_2$ is in fact an \'etale, Galois cover, so there is a diagram
\[
\xymatrix{
C \ar[r] \ar[d] & A \ar[d] \\
C_2 \ar[r] & J_2
}
\]
where $J_2$ is the Jacobian of $C_2$, the map $C_2 \to J$ is given by $T \mapsto [T - \infty]$, the map $A \to J_2$ is an isogeny, and $C$ is the fiber product $C_2 \times_{J_2} A$.
\bibliographystyle{halpha}
\end{document}