We provide a method to compute the dimension of the tangent space to the global infinitesimal deformation functor of a curve together with a subgroup of the group of automorphisms. The computational techniques we developed are applied to several examples including Fermat curves, $p$-cyclic covers of the affine line and to Lehr–Matignon curves.

We give an example of a projective smooth surface $X$ over a $p$-adic field $K$ such that for any prime $\ell$ different from $p$, the $\ell$-primary torsion subgroup of ${CH}_0\left(X\right)$, the Chow group of $0$-cycles on $X$, is infinite. A key step in the proof is disproving a variant of the Bloch–Kato conjecture which characterizes the image of an $\ell$-adic regulator map from a higher Chow group to a continuous étale cohomology of $X$ by using $p$-adic Hodge theory. With the aid of the theory of mixed Hodge modules, we reduce the problem to showing the exactness of the de Rham complex associated to a variation of Hodge structure, which is proved by the infinitesimal method in Hodge theory. Another key ingredient is the injectivity result on the cycle class map for Chow group of $1$-cycles on a proper smooth model of $X$ over the ring of integers in $K$, due to K. Sato and the second author.

We construct a singular homology theory on the category of schemes of finite type over a Dedekind domain and verify several basic properties. For arithmetic schemes we construct a reciprocity isomorphism between the integral singular homology in degree zero and the abelianized modified tame fundamental group.

In 1956, Brauer showed that there is a partitioning of the $p$-regular conjugacy classes of a group according to the $p$-blocks of its irreducible characters with close connections to the block theoretical invariants. But an explicit block splitting of regular classes has not been given so far for any family of finite groups. Here, this is now done for the 2-regular classes of the symmetric groups. To prove the result, a detour along the double covers of the symmetric groups is taken, and results on their 2-blocks and the 2-powers in the spin character values are exploited. Surprisingly, it also turns out that for the symmetric groups the 2-block splitting is unique.