The 22-Class Tower of Q(−5460)\mathbb{Q}(\sqrt{-5460})

The seminal papers in the field of root-discriminant bounds are those of Odlyzko and Martinet. Both papers include the question of whether the field $\mathbb{Q}(\sqrt{-5460})$ has finite or infinite $2$-class tower. This is a critical case that will either substantially lower the best known upper bound for lim inf of root-discriminants (if infinite) or else give a counter-example to what is often termed Martinet's conjecture or question (if finite). Using extensive computation and introducing some new techniques, we give strong evidence that the tower is in fact finite, establishing other properties of its Galois group en route

Towards Constructing Fully Homomorphic Encryption without Ciphertext Noise from Group Theory

In CRYPTO 2008, one year earlier than Gentry\u27s pioneering \lq\lq bootstrapping\u27\u27 technique on constructing the first fully homomorphic encryption (FHE) scheme, Ostrovsky and Skeith III had suggested a completely different approach towards achieving FHE. Namely, they showed that the $\mathsf{NAND}$ operator can be realized in some \emph{non-commutative} groups; consequently, in combination with the $\mathsf{NAND}$ operator realized in such a group, homomorphically encrypting the elements of the group will yield an FHE scheme. However, no observations on how to homomorphically encrypt the group elements were presented in their paper, and there have been no follow-up studies in the literature based on their approach. The aim of this paper is to exhibit more clearly what is sufficient and what seems to be effective for constructing FHE schemes based on their approach. First, we prove that it is sufficient to find a surjective homomorphism $\pi \colon \widetilde{G} \to G$ between finite groups for which bit operators are realized in $G$ and the elements of the kernel of $\pi$ are indistinguishable from the general elements of $\widetilde{G}$. Secondly, we propose new methodologies to realize bit operators in some groups, which enlarges the possibility of the group $G$ to be used in our framework. Thirdly, we give an observation that a naive approach using matrix groups would never yield secure FHE due to an attack utilizing the \lq\lq linearity\u27\u27 of the construction. Then we propose an idea to avoid such \lq\lq linearity\u27\u27 by using combinatorial group theory, and give a prototypical but still \emph{incomplete} construction in the sense that it is \lq\lq non-compact\u27\u27 FHE, i.e., the ciphertext size is unbounded (though the ciphertexts are noise-free as opposed to the existing FHE schemes). Completely realizing FHE schemes based on our proposed framework is left as a future research topic

Maximal subgroups and PST-groups

A subgroup H of a group G is said to permute with a subgroup K of G if HK is a subgroup of G. H is said to be permutable (resp. S-permutable) if it permutes with all the subgroups (resp. Sylow subgroups) of G. Finite groups in which permutability (resp. S-permutability) is a transitive relation are called PT-groups (resp. PST-groups). PT-, PST- and T-groups, or groups in which normality is transitive, have been extensively studied and characterised. Kaplan [Kaplan G., On T-groups, supersolvable groups, and maximal subgroups, Arch. Math. (Basel), 2011, 96(1), 19-25] presented some new characterisations of soluble T-groups. The main goal of this paper is to establish PT- and PST-versiosn of Kaplan's results, which enables a better understanding of the relationships between these classes

On the non-abelian Brumer-Stark conjecture and the equivariant Iwasawa main conjecture

We show that for an odd prime p, the p-primary parts of refinements of the (imprimitive) non-abelian Brumer and Brumer-Stark conjectures are implied by the equivariant Iwasawa main conjecture (EIMC) for totally real fields. Crucially, this result does not depend on the vanishing of the relevant Iwasawa mu-invariant. In combination with the authors' previous work on the EIMC, this leads to unconditional proofs of the non-abelian Brumer and Brumer-Stark conjectures in many new cases.Comment: 33 pages; to appear in Mathematische Zeitschrift; v3 many minor updates including new title; v2 some cohomological arguments simplified; v1 is a revised version of the second half of arXiv:1408.4934v

Groups with infinite homology

No abstract available

Polynomial Growth Harmonic Functions on Finitely Generated Abelian Groups

In the present paper, we develop geometric analytic techniques on Cayley graphs of finitely generated abelian groups to study the polynomial growth harmonic functions. We develop a geometric analytic proof of the classical Heilbronn theorem and the recent Nayar theorem on polynomial growth harmonic functions on lattices \mathds{Z}^n that does not use a representation formula for harmonic functions. We also calculate the precise dimension of the space of polynomial growth harmonic functions on finitely generated abelian groups. While the Cayley graph not only depends on the abelian group, but also on the choice of a generating set, we find that this dimension depends only on the group itself.Comment: 15 pages, to appear in Ann. Global Anal. Geo