Krdytkyu

4 $begingroup$ On several occasions I heard about the following result: For "certain" lattices $Lambda$ in $SL_2(mathbb{R})$ , and almost any prime $p$ there exists a lattice $Gamma$ in $SL_2(mathbb{R})times SL_2(mathbb{Q}_p)$ and a compact subgroup $K$ of $SL_2(mathbb{R})times SL_2(mathbb{Q}_p)$ such that there is an isomorphism between $$ Lambda backslash SL_2(mathbb{R}) $$ and $$ Gamma backslash SL_2(mathbb{R})times SL_2(mathbb{Q}_p)/K. $$ I know how to prove this for $Lambda = SL_2(mathbb{Z})$ . Then $Gamma = SL_2(mathbb{Z}[1/p])$ (diagonally in $SL_2(mathbb{R})times SL_2(mathbb{Q}_p)$ ) and $K={1}times SL_2(mathbb{Z}_p)$ and the isomorphism is a quite easy map. I would like to find a reference for more general $Lambda$ , preferably with an explicit statement of the isomorphism and

-1 $begingroup$ I am trying to show that given $$f(x) = xsin(1/x),quad x>0,quad f(0)=0, $$ the derivative is bounded. Namely, $ f'(x) < infty$ for $ x in (0,a)$ for some $ainmathbb{R}$ . My attempt at a solution lead me to taking the derivative using the Product Rule but I can't find a way to justify that $-frac{1}{x}cos(1/x) < infty$ as $x rightarrow 0^+$ despite Wolfram-Alpha telling me that the limit is indeed $< infty$ . real-analysis limits measure-theory share | cite | improve this question edited Dec 3 '18 at 1:35 raka