Variation of local systems and parabolic cohomology

Abstract

Given a family of local systems on a punctured Riemann sphere, with moving singularities, its first parabolic cohomology is a local system on the base space. We study this situation from different points of view. For instance, we derive universal formulas for the monodromy of the resulting local system. We use a particular example of our construction to prove that the simple groups PSL2(p 2 ) admit regular realizations over the field Q(t) for primes p ≡ 1, 4, 16 mod 21. Finally, we compute the monodromy of the Euler-Picard equation, reproving a classical result of Picard.

Universal deformation rings and dihedral defect groups

“…≡ 1, 4, 16 mod 21), it was proved in [31] (resp. in [19]) that PSL 2 (F q ) occurs regularly over Q(t), implying that there are PSL 2 (F q ) extensions of any number field L for such q. On the other hand, it was shown in [33] that for any prime there are infinitely many positive integers r such that for q = r , PSL 2 (F q ) of 3-tubes and use the results from [9] about the ordinary irreducible characters belonging to B to determine the universal deformation rings of V (see Theorem 5.1).…”

The classification of orthogonally rigid $G_2$-local systems and related differential operators

“…, x r , y} by restriction). Under certain conditions (made explicit in [7]), the analytic continuation of the integral (2.3) near the singularities is in the image of the local monodromy and therefore contained in the parabolic cohomology group [9]. By Remark 2.1, for varying y, the integral C p a (f )(y) can hence be viewed as a section of MC χ (L).…”

“…, x r , y}, L ⊗ L χ(x−y) ), cf. [9]. By Remark 2.1, for varying y, the integral C p a (f )(y) can hence be viewed as a section of MC χ (L).…”

“…) (in the case that they contradict the Scott Formula (Lemma 4.1) this is noted in the last column by red. ): 13,9) red.…”

The Grothendieck-Teichmüller Group and Galois Theory of the Rational Numbers – European Network GTEM

“…(1) Realization of Galois groups by 'middle convolution' and 'parabolic cohomology' (Dettweiler-Reiter [DR00], Völklein [Vö01], Dettweiler-Wewers [DW03]).…”