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- W4386869523 abstract "By a theorem of Clemens and Landman, see Griffiths (Bull Am Math Soc 76:228–296, 1970, Theorem. 3.1) in complex geometry and Grothendieck (Séminaire de Géométrie Algébrique: Groupes de monodromie en géométrie algébrique, XIV 1.1.10, 1973) in arithmetic geometry, geometric (complex or $$ell $$ -adic) local systems have quasi-unipotent monodromies at infinity. We explain in this section why this property is good for going from complex models to models over finite fields, and why in the Betti moduli space the complex local systems with quasi-unipotent monodromy at infinity are Zariski dense. Further, we report on Biswas et al. (Geom Topol 26(2):679–719, 2022), Landesman and Litt (Geometric local systems on very general curves and isomonodromy), Landesman and Litt (Canonical representations of surface groups) showing that the arithmetic local systems on geometric generic curves in low rank cannot be Zariski dense in their Betti moduli, contrary to what was expected in Esnault and Kerz (Camb J Math 8(3):453–478, 2020, Question 9.1 (1)) and Esnault and Kerz (Israel J Math, 9 pp., Conjecture 1.1, to appear). Finally we report on the concept of weakly arithmetic local systems defined in de Jong and Esnault (Trans AMS, 18 pp., Section 3, to appear), which in particular have quasi-unipotent monodromies at infinity, and show that they are Zariski dense in their Betti moduli. So for certain problems (to be defined) one would then wish to follow Drinfeld’s method (Drinfeld, Math Res Lett, 8(5–6):713–728, 2001) to conclude that it is enough to check them on weakly arithmetic local systems." @default.
- W4386869523 created "2023-09-20" @default.
- W4386869523 creator A5073683726 @default.
- W4386869523 date "2023-01-01" @default.
- W4386869523 modified "2023-09-27" @default.
- W4386869523 title "Lecture 6: Density of Special Loci" @default.
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- W4386869523 doi "https://doi.org/10.1007/978-3-031-40840-3_6" @default.
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