In the early 19th century a young French mathematician E. Galois laid the foundations of abstract algebra by using the symmetries of a polynomial equation to describe the properties of its roots. One of his discoveries was a new type of structure, formed by these symmetries. This structure, now called a “group”, is central to much of modern mathematics. The groups that arise in the context of c...

We report on two classes of autoequivalences of the category of Yetter-Drinfeld modules over a finite group, or, equivalently the Drinfeld center of the category of representations of a finite group. Both operations are related to the r-th power operation, with r relatively prime to the exponent of the group. One is defined more generally for the group-theoretical fusion category defined by a f...

The main result of this paper is a characterization of the abelian varieties B/K defined over Galois number fields with the property that the zeta function L(B/K; s) is equivalent to the product of zeta functions of non-CM newforms for congruence subgroups Γ1(N). The characterization involves the structure of End(B), isogenies between the Galois conjugates of B, and a Galois cohomology class at...

This paper concerns the description of holomorphic extensions of algebraic number fields. After expanding the notion of adele class group to number fields of infinite degree over Q, a hyperbolized adele class group ŜK is assigned to every number field K/Q. The projectivization of the Hardy space PH•[K] of graded-holomorphic functions on ŜK possesses two operations ⊕ and ⊗ giving it the structur...

Proof. The discriminant of x + 1 is D = 256 = 2. We have x + 1 ≡ (x + 1) (mod 2). Let p be an odd prime (so p D), and suppose the irreducible factors of x + 1 have degrees n1, n2, . . . , nk. By Corollary 41, the Galois group of x + 1 contains an element with cycle structure (n1, n2, . . . , nk). Since the Galois group of x +1 over Q is the Klein 4-group, in which every element has order dividi...

Relations between the following classes of Galois extensions are given: (1) centrally projective Galois extensions (CP-Galois extensions), (2) faithfully Galois extensions, and (3) H-separable Galois extensions. Moreover, it is shown that the intersection of the class of CP-Galois extensions and the class of faithfully Galois extensions is the class of Azumaya Galois extensions.

Differential Galois theory, the theory of strongly normal extensions, has unfortunately languished. This may be due to its reliance on Kolchin’s elegant, but not widely adopted, axiomatization of the theory of algebraic groups. This paper attempts to revive the theory using a differential scheme in place of those axioms. We also avoid using a universal differential field, instead relying on a c...

The Chern-Galois theory is developed for corings or coalgebras over non-commutative rings. As the first step the notion of an entwined extension as an extension of algebras within a bijective entwining structure over a non-commutative ring is introduced. A strong connection for an entwined extension is defined and it is shown to be closely related to the Galois property and to the equivariant p...

Putting emphasis on the relation between rational conformal field theory (RCFT) and algebraic number theory, we consider a brane configuration in which the D-brane intersection is an elliptic curve corresponding to RCFT. A new approach to the generation structure of fermions is proposed in which the flavor symmetry including the R-parity has its origin in the Galois group on elliptic curves wit...