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Register a new userStatistics of $K$-groups modulo $p$ for the ring of integers of a varying quadratic number field
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For each odd prime $p$, we conjecture the distribution of the $p$-torsion subgroup of $K_{2n}(mathcal{O}_F)$ as $F$ ranges over real quadratic fields, or over imaginary quadratic fields. We then prove that the average size of the $3$-torsion subgroup of $K_{2n}(mathcal{O}_F)$ is as predicted by this conjecture.
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We compute the etale cohomology ring $H^*(text{Spec } mathcal{O}_K,mathbb{Z}/nmathbb{Z})$ where $mathcal{O}_K$ is the ring of integers of a number field $K.$ As an application, we give a non-vanishing formula for an invariant defined by Minhyong Kim.
We present a variation of the modular algorithm for computing the Hermite normal form of an $mathcal O_K$-module presented by Cohen, where $mathcal O_K$ is the ring of integers of a number field $K$. An approach presented in (Cohen 1996) based on reductions modulo ideals was conjectured to run in polynomial time by Cohen, but so far, no such proof was available in the literature. In this paper, we present a modification of the approach of Cohen to prevent the coefficient swell and we rigorously assess its complexity with respect to the size of the input and the invariants of the field $K$.
Each p-ring class field K(f) modulo a p-admissible conductor f over a quadratic base field K with p-ring class rank r(f) mod f is classified according to Galois cohomology and differential principal factorization type of all members of its associated heterogeneous multiplet M(K(f))=[(N(c,i))_{1<=i<=m(c)}]_{c|f} of dihedral fields N(c,i) with various conductors c|f having p-multiplicities m(c) over K such that sum_{c|f} m(c)=(p^r(f)-1)/(p-1). The advanced viewpoint of classifying the entire collection M(K(f)), instead of its individual members separately, admits considerably deeper insight into the class field theoretic structure of ring class fields, and the actual construction of the multiplet M(K(f)) is enabled by exploiting the routines for abelian extensions in the computational algebra system Magma.
One of the many number theoretic topics investigated by the ancient Greeks was perfect numbers, which are positive integers equal to the sum of their proper positive integral divisors. Mathematicians from Euclid to Euler investigated these mysterious numbers. We present results on perfect numbers in the ring of Eisenstein integers.
As an analogue of a link group, we consider the Galois group of the maximal pro-$p$-extension of a number field with restricted ramification which is cyclotomically ramified at $p$, i.e, tamely ramified over the intermediate cyclotomic $mathbb Z_p$-extension of the number field. In some basic cases, such a pro-$p$ Galois group also has a Koch type presentation described by linking numbers and mod $2$ Milnor numbers (Redei symbols) of primes. Then the pro-$2$ Fox derivative yields a calculation of Iwasawa polynomials analogous to Alexander polynomials.
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