Vanishing cycles and non-classical parabolic cohomology
The paper deals with the formalism of vanishing cycles and some applications to non-classical parabolic cohomology. In particular, one constructs non-zero elements in motivic cohomology (i.e. algebraic \(K\)-groups) classifying extensions in the category of mixed motives over some number field \(K\), associated to subgroups of \(\text{SL}_2({\mathbb Z})\) which are not congruence subgroups. Let \(S\) be the spectrum of a strictly henselian discrete valuation ring with generic point \(\eta\) and closed point \(s\). Write \(I=\text{Gal}(\overline{\eta}/\eta)\), and let \(\ell\) be a prime different from the residue characteristic of \(S\). Consider a smooth curve \(f:X\rightarrow S\) with generic point \(\xi\). Assume \(f\) has two sections with images \(Z_1\), \(Z_2\) in \(X\) such that \(Z_1\), \(Z_2\) intersect in \(x\in X_s\). Write \(Z=Z_1\cup Z_2\) and \(q:Z\hookrightarrow X\) and \(g:U=X-Z\hookrightarrow X\) for the inclusions. Assume there is a \({\mathbb Q}_{\ell}\)-sheaf \({\mathcal F}\) on \(U\), lisse on the intersection of \(U\) with some neighbourhood of \(x\). Finally, assume the local monodromy representation of \(\pi_1(\overline{U},\overline{\xi})\) on \({\mathcal F}_{\overline{\xi}}\) is unipotent and admits a factorization \[ \pi_1(\overline{U},\overline{\xi})\longrightarrow\bigoplus_{r=1,2} \pi_1^{(\ell)}(\overline{X}_x-\overline{Z}_r,\overline{\xi}) {\buildrel\sim\over\longrightarrow}{\mathbb Z}_{\ell}(1)^2{\buildrel \chi\over\longrightarrow}\text{Aut}{\mathcal F}_{\overline{\xi}}, \] for some \(\chi\), where \(\overline{X}_x\) is the strict henselization of \(X\) at \(x\), \(\overline{U}=\overline{X}_x\times_XU\), and \(\overline{Z}_r=\overline{X}_x\times_XZ_r\). One has an action of \(I\) on \({\mathcal F}_{\overline{\xi}}\). For any \({\mathbb Q}_{\ell}\)-sheaf \({\mathcal G}\) on \(X\) denote the sheaves of vanishing cycles by \(R^i\Phi{\mathcal G}\). The main result of the paper now says that there is an \(I\)-equivariant isomorphism \((R^1\Phi g_!{\mathcal F})_x{\buildrel\sim\over\longrightarrow}{\mathcal F}_{\overline{\xi}}\). The proof is given in four steps: \((i)\) the characteristic zero case; \((ii)\) transition from characteristic zero to a simple mixed characteristic situation over a two-dimensional base; \((iii)\) the result in this particular case; \((iv)\) the existence of a morphism from \(S\) to the base as in \((ii)\) inducing a triple isomorphic to \((X,Z,{\mathcal F})\) for which the result holds by \((ii)\). The main result is applied in the setting of parabolic cohomology to construct, under suitable hypotheses, non-trivial extensions of the form \[ (*)\quad 0\longrightarrow H^1(\overline{X},g_*\text{Sym}^k{\mathcal F})\longrightarrow H^1(\overline{U},\text{Sym}^k{\mathcal F})\longrightarrow H^0(\overline{Z})(-k-1)\longrightarrow 0, \] \[ (**)\quad 0\longrightarrow H^0(\overline{Z})\longrightarrow H^1_c(\overline{U},\text{Sym}^k{\mathcal F})\longrightarrow H^1(\overline{X},g_*\text{Sym}^k{\mathcal F})\longrightarrow 0, \] where \(X\) is a smooth projective curve over some number field \(K\), \(\pi:E\rightarrow X\) denotes an elliptic surface with a section and satisfying suitable conditions, \(Z\subset X\) is a finite set of closed points over which \(\pi\) is not smooth, and \(g:U=X-Z\hookrightarrow X\) is the inclusion. Let \(\pi':E\times_XU\rightarrow U\) be the restriction of \(\pi\), then \({\mathcal F}\) is the rank two lisse \({\mathbb Q}_{\ell}\)-sheaf \(R^1\pi'_*{\mathbb Q}_{\ell}\) on \(U\). \(H^1(\overline{X},g_*\text{Sym}^k{\mathcal F})\simeq\text{Im}\left\{H^1_c(\overline{U},\text{Sym}^k{\mathcal F})\rightarrow H^1(\overline{U},\text{Sym}^k{\mathcal F})\right\}\) denotes parabolic cohomology. It is shown that, under the aforementioned conditions, \((*)\) and \((**)\) are non-trivial extensions when restricted to any open subgroup \(I_v\subset\text{Gal}(\overline{K}/K)\), where \(v\) is a finite prime of \(K\). As \((*)\) is just an avatar of the \(\ell\)-adic realization of an exact sequence in the category \(\mathcal{MM}_K\) of mixed motives (as proposed by U. Jannsen), it is classified by a non-zero element of the motivic cohomology group \(H^{k+2}_{\mathcal M}(E^{(k)},{\mathbb Q}(k+1))\) (under the aforementioned conditions). This element can be described more explicitly. The last section is concerned with three explicit examples of elliptic surfaces for general subgroups of \(\text{SL}_2({\mathbb Z})\) of finite type.
spectrum of a strictly henselian discrete valuation ring
scientific article; zbMATH DE number 866100