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 :)

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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 !

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