# northern spain

hey, there are more pictures! hooray!

With lovely company I spent a quite different holiday than I am used to – without any bikes or tents involved. First we spent a few days in San Sebastian where we had very pleasant local guides to nice places, lovely food and interesting drinks.
Then we took a car and went to my first mountain experience, the national park picos de europa for a few day trips. Nice enough I was the only who stayed with fully working walking ability :)

# chinesisch für fortgeschrittene

A Bianchi group is some subgroup $\Gamma \subset$ $SL_2(\mathcal{O}_d)$ for the ring of integers of the imaginary quadratic field $\mathbb{Q}(\sqrt{d})$. There exists a homomorphism from $SL_2(\mathcal{O}_d)$ to $SL_2(\mathbb{F}_q)$ for a prime power $q$. $SL_2(\mathbb{F}_q)$ acts on the polynomial ring $\mathbb{F}_q[x,y]$ by
$\begin{pmatrix} a & b \\ c & d \end{pmatrix} \cdot f(x,y) = f(ax+cy,bx+dy)$
for $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL_2( \mathbb{F}_q)$ and $f \in \mathbb{F}[x,y]$.

This makes $\mathbb{F}_q[x,y]$ an $SL_2(\mathcal{O}_d)$-module. It is also a free module of finite rank over its ring of invariants under the action. Thus the first cohomology $H^1$ of $\Gamma$ with coefficients in $\mathbb{F}_q[x,y]$ can be considered. It is a graded module and specific interest lay in determining the dimensions of its homogeneous components. This information is obtained by computing its Hilbert-Poincaré series.

Wie geil ist das denn, latex gelb auf schwarz auf wordpress. just three clicks away. rock’n’roll, baby !

# party after

if you happened to help causing the mess, thanks. We enjoyed …

# 2 cracks

one here :

and one in my heart …

das leiden geht weiter, wie versprochen :p. Gut auch, dass meine lunge gleich mal mitgeröntgt (wie schreibt man denn das?) wurde.

# where were you?

WE were just cycling and dancing.