Profinite Braid Groups, Galois Representations and Complex Multiplications

Arithmetic Topology in Ihara Theory

“…Let g X,n , g Y,n be the automorphisms of C n := C n × Spec Q Spec Q over P Now, the main results in [Ih1] are arithmetic descriptions of • values of F g (u 1 , u 2 ) at l-powerth roots of unity in terms of the Jacobi sums which arise from the Galois action on T(Jac n ), and • coefficients of F g (u 1 , u 2 ) in terms of l-adic Soulé cocycles which are defined by the Galois action on higher cyclotomic l-units.…”

“…In Section 6, we consider the case that S = {0, 1, ∞}. We show that the Ihara power series F g (u 1 , u 2 ) (g ∈ Gal Q ) introduced in [Ih1] coincides with our pro-l reduced Gassner cocycle, and give a formula which expresses F g (u 1 , u 2 ) in terms of l-adic Milnor numbers. Accordingly, using our formula and Ihara's formula, we express the Jacobi sum in Q(ζ l n ) as a (ζ l n − 1)-adic expansion with coefficients l-adic Milnor numbers.…”

“…We note that σ f p ∈ Gal(Ω S /Q(ζ l n )). By using Weil's theorem, Ihara showed the following Theorem 6.2.1 ([Ih1;Theorem 7]). Let a, b ∈ Z/l n Z\{0} such that a+b = 0 and (a, b, a + b, l) = 1.…”

Universal Deformations, Rigidity,and Ihara's Cocycle

“…However, as a first step in this direction, for each γ ∈ G Q , and each i = 0, 1, ∞. In [Iha86b], Ihara shows that ψ is encoded by a cocycle…”

Periods of Mixed Tate Motives, Examples, l-adic Side

“…Ihara in [12] and Deligne in [4] studied the action of the Galois group G Q on π 1 (P 1 Q \ {0, 1, ∞}; It is not known, at least to the author of this article, if the last morphism is injective. (This question was studied very much by Ihara and his students.)…”