Proving a congruence relation for elements of modular group.

Asked Dec 22 '19 at 19:36

Active Dec 22 '19 at 19:48

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I am trying exercises of Tom M Apostol modular functions and Dirichlet series in number theory and I am struck on this problem of Chapter 2 .

Problem is - If (m, n) =1 and A $\epsilon $ $\Gamma$ , then prove that there exists A' in $\Gamma$ such that A'$\equiv$ A ( mod n) and A' $\equiv $ I ( mod m) . NOTE I is identity matrix here.

This problem visit related to this problem whose link I am giving -->

Deducing congruence relations from given congruence relations

Using above mentioned problem it becomes clear that such a choice of A' is possible. In fact, it is same as given in the answer of Bill Dubuque. But how to prove that A' $\equiv $ I ( mod m) . For this, I must show that b= qm for some q from integers and other required conditions.

Can somebody please explain how to find such entries in A' such that A'$\equiv $I ( mod m) .

edited Dec 22 '19 at 19:48

asked Dec 22 '19 at 19:36

What is $\Gamma$? and is $I\equiv1\pmod m$? – J. W. Tanner Dec 22 '19 at 19:38
Do you know the Chinese remainder theorem? – J. W. Tanner Dec 22 '19 at 19:45
@J. W. Tanner yes I know Chinese reminder theorem – Dec 22 '19 at 19:46
@J. W. Tanner $\Gamma$ is modular group . I is Identity matrix here. – Dec 22 '19 at 19:47
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Once you know that $SL_2(\Bbb{Z})\to SL_2(\Bbb{Z}/N\Bbb{Z})$ is surjective then take $N=nm$ and $SL_2(\Bbb{Z}/N\Bbb{Z})\cong SL_2(\Bbb{Z}/n\Bbb{Z})\times SL_2(\Bbb{Z}/m\Bbb{Z})$ – reuns Dec 23 '19 at 00:04
@reuns thank you very much. – Dec 23 '19 at 10:12

Proving a congruence relation for elements of modular group.

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