Proposed talk title and abstract of Ján Mináč for the conference:

ARITHMETIC GALOIS THEORY AND RELATED MODULI SPACES

“Galois Cohomology, Quotients of Absolute Galois Groups, and A Little Modular Representation Theory”

ABSTRACT. Absolute Galois groups of fields are mysterious. One can try to find manageable but still non-trivial quotients of absolute Galois groups. Let p be a prime number. In joint work with D. Benson, N. Lemire, and J. Swallow we consider groups T(E/F )= G_F/Φ(G_E ), where F is a field containing a primitive pth root of unity such that its absolute Galois group G_F is a pro-p group, E/F is a cyclic extension of degree p and Φ(G_E) is the Frattini subgroup of G_E . We determine all possible groups T(E/F ). Further assuming the Bloch-Kato conjecture we determine the F_p[G_F/G_E ]-module Hⁱ(G_E , F_p)for all i =1, 2,... which extends the previous work of Boreviˇc and Faddeev on the F_p[G_F /G_E ] structure H¹ (G_E, F_p)in the case of local fields.