Timeline for Explanation for $\chi(\operatorname{SL}_2(\mathbb{Z})) = -1/12$ with zeta function

7 events

when toggle format	what		by	license	comment
Jan 9, 2022 at 16:48	history	edited	Carl-Fredrik Nyberg Brodda	CC BY-SA 4.0	Added link to review
Jan 9, 2022 at 13:17	comment	added	Carl-Fredrik Nyberg Brodda		@JHM Langlands' method extends to Chevalley groups; the Gauss-Bonnet measure $\mu$ on $G(\mathbb{R})/ K$ (as above) is explicitly calculated by Harder in the case of Chevalley groups $G$. Indeed, in this case $\mu = c\mu_a$ where $\mu_a$ is the arithmetic measure on $G(\mathbb{R})$, for some scalar $c$ (which is computed by Harder). Since $\mu_a(G(\mathbb{R})/G(\mathbb{Z}))$ is expressible using $\zeta$, this yields the connection in the cases considered by Langlands (and circumvents his to-me-seemingly more complicated methods). Perhaps the two papers are worth reading simultaneously?
Jan 9, 2022 at 12:31	history	edited	Carl-Fredrik Nyberg Brodda	CC BY-SA 4.0	Added more values
Jan 9, 2022 at 1:12	comment	added	JHM		I think zeta $\zeta$ specifically enters via Langlands computation of the integral of the Euler-Poincare form $\omega$ over the fundamental domain, but that is a paper i never could understand. "R. P. LANGLANDS, The volume of the fundamental domain for some arithmetical subgroups of Cheualley groups (Proc. of Symp. Math., Amer. Math. Soc., Providence, 1966. p. 143-148)."
Jan 8, 2022 at 22:22	vote	accept	Yotam Shomroni
Jan 8, 2022 at 22:22	vote	accept	Yotam Shomroni
Jan 8, 2022 at 22:22
Jan 8, 2022 at 22:03	history	answered	Carl-Fredrik Nyberg Brodda	CC BY-SA 4.0