 CoCalc Public Fileswww / papers / thesis-old / congruences.tex
Author: William A. Stein
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1% congruences.tex
2
3\section{Congruences}
4\label{sec:congruences}
5
6Lo\"\i{}c's paper Arithmetic of elliptic curves and
7diophantine equations'' is very relevant to this
8section, esp. his section THREE.  Do not forget
10
11Let $f, g$ be nonconjugate newforms and $H=H_1(X_0(N),\Z)$.
12
13\begin{proposition}
14$(\Adual_f\intersect A_g^{\vee})[p]\neq 0$
15if and only if the mod $p$ rank of $H[I_f]+H[I_g]$
16is strictly less than $\rank H[I_f] + \rank H[I_g]$.
17\end{proposition}
18\begin{proof}
19By (\ref{Af}) $\Lambda_f=H[I_f]$ (resp., $\Lambda_g=H[I_g]$) is the submodule of
20$H$ which defines $A_f$ (resp., $A_g$).  By reduction mod $p$ we mean
21the map $H\ra H\tensor \Fp$. Suppose
22  $$\rank (\Lambda_f + \Lambda_g)\md p < \rank\Lambda_f + \rank \Lambda_g.$$
23Since $\Lambda_f$ (resp., $\Lambda_g$) is a kernel, it is saturated, so
24  $\rank \Lambda_f \md p = \rank \Lambda_f$ (resp., for $\Lambda_g$).
25We conclude that the mod $p$ linear dependence must involve vectors
26from both $\Lambda_f$ and $\Lambda_g$; there is $v\in\Lambda_f$ and
27$w\in\Lambda_g$ so that $v, w\not\equiv 0\md p$ but $v+w\con 0 \md p$.
28Thus $\frac{v+w}{p}\in H$ is integral, i.e., in $J_0(N)(\C)$ we have
29$\frac{1}{p}v - (-\frac{1}{p} w)=0$.  But $\frac{1}{p}v \not \in \Lambda_f$
30and $\frac{1}{p} w\not\in\Lambda_g$ (otherwise $v$ and $w$ would
31be $0\md p$), so
32$\frac{1}{p}v$ and $-\frac{1}{p}w$ are both nontrivial $p$-torsion in
33$\Adual_f$, $A_g^{\vee}$, resp.  Conclusion:
34$0\neq \frac{1}{p}v = -\frac{1}{p} w \in (\Adual_f\intersect A_g^{\vee})[p]$.
35
36Conversely, suppose $0\neq x\in (\Adual_f\intersect A_g^{\vee})[p]$.  Choose
37lifts modulo $H$ to $x_f\in\frac{1}{p}\Lambda_f$ and
38$x_g\in\frac{1}{p}\Lambda_g$.
39Then $px_f\in\Lambda_f$  (resp., $px_g\in\Lambda_g$), but
40$px_f\not\in p H$ (resp., $px_g\not\in pH$)
41because $x\neq 0$.  Since $x_f-x_g\in H$,
42$px_f - px_g = p(x_f-x_g)\equiv 0\md p$.
43This is a nontrivial linear relation between
44$\Lambda_f$ and $\Lambda_g$.
45\end{proof}
46
47\begin{corollary}
48If $p>2$ and the sign of some Atkin-Lehner involution
49for $f$ is different than that for $g$ then
50$(\Adual_f\intersect A_g^{\vee})[p]=0$.
51\end{corollary}
52\begin{proof}
53Suppose $w_q(f) \neq w_q(g)$ and
54let $G=(\Adual_f\intersect A_g^{\vee})[p]$.
55Observe that $W_q$ acts as $w_q(f)\md p$ on $\Adual_f[p]$
56and as $w_q(g)\md p$ on $A_g^{\vee}[p]$.  Hence $W_q$ acts
57as both $w_q(f)\md p$ and $w_q(g)\md p$ on $G$.  Since
58$p>2$, this is not possible when $G\neq 0$.
59\end{proof}
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