Euclidean scissor congruence groups and mixed Tate motives over dual numbers

Euclidean scissor congruence groups and mixed Tate motives over dual numbers

We define Euclidean scissor congruence groups for an arbitrary algebraically closed field F and formulate a conjecture describing them. Using the Euclidean and NonEuclidean F–scissor congruence groups we construct a category which is conjecturally equivalent to a subcategory of the category MT (Fε) of mixed Tate motives over the dual numbers Fε := F [ε]/ε . 1. Euclidean scissor congruence group...

Mixed Artin–Tate motives over number rings

This paper studies Artin–Tate motives over bases S ⊂ Spec OF , for a number field F . As a subcategory of motives over S, the triangulated category of Artin–TatemotivesDATM(S) is generated by motives φ∗1(n), where φ is any finite map. After establishing the stability of these subcategories under pullback and pushforward along open and closed immersions, a motivic t-structure is constructed. Exa...

Congruence Relations Connecting Tate-Shafarevich Groups with Hurwitz Numbers

Multiple polylogarithms and mixed Tate motives

from analytic, Hodge-theoretic and motivic points of view. Let μN be the group of all N -th roots of unity. One of the reasons to develop such a theory was a mysterious correspondence (loc. cit.) between the structure of the motivic fundamental group π 1 (Gm − μN ) and the geometry and topology of modular varieties Y1(m;N) := Γ1(m;N)\GLm(R)/R ∗ + ·Om Here Γ1(m;N) ⊂ GLm(Z) is the subgroup fixing...

The slice filtration and mixed Tate motives

Using the ”slice filtration”, defined by effectivity conditions on Voevodsky’s triangulated motives, we define spectral sequences converging to their motivic cohomology and étale motivic cohomology. These spectral sequences are particularly interesting in the case of mixed Tate motives as their E2-terms then have a simple description. In particular this yields spectral sequences converging to t...