The unit residue group

The unit residue group, to which the present thesis is devoted, is defined using the norm-residue symbol, which Hilbert introduced into algebraic number theory in 1897. By its definition, the unit residue group of a global field is a direct sum of local contributions. It has a subgroup of a global nature, called the virtual group.We give a precise description of the unit residue groups and their virtual subgroups for some classes of number fields, including all quadratic fields. In addition we point out connections to two classical theorems on ideal class groups, namely the theorem of Armitage and Froehlich on 2-ranks and Scholz’s theorem on 3-ranks.We also study certain subgroups of the multiplicative group of a local field that play an important role in an algorithm for computing norm-residue symbols, and group isomorphisms between the groups of quadratic characters of two number fields that preserve L-series.Number theory, Algebra and Geometr

Surpassing the Ratios Conjecture in the 1-level density of Dirichlet LL-functions

We study the $1$-level density of low-lying zeros of Dirichlet $L$-functions in the family of all characters modulo $q$, with $Q/2 < q\leq Q$. For test functions whose Fourier transform is supported in $(-3/2, 3/2)$, we calculate this quantity beyond the square-root cancellation expansion arising from the $L$-function Ratios Conjecture of Conrey, Farmer and Zirnbauer. We discover the existence of a new lower-order term which is not predicted by this powerful conjecture. This is the first family where the 1-level density is determined well enough to see a term which is not predicted by the Ratios Conjecture, and proves that the exponent of the error term $Q^{-\frac 12 +\epsilon}$ in the Ratios Conjecture is best possible. We also give more precise results when the support of the Fourier Transform of the test function is restricted to the interval $[-1,1]$. Finally we show how natural conjectures on the distribution of primes in arithmetic progressions allow one to extend the support. The most powerful conjecture is Montgomery's, which implies that the Ratios Conjecture's prediction holds for any finite support up to an error $Q^{-\frac 12 +\epsilon}$.Comment: Version 1.2, 30 page

Permuting operations on strings and the distribution of their prime numbers

Several ways of interleaving, as studied in theoretical computer science, and some subjects from mathematics can be modeled by length-preserving operations on strings, that only permute the symbol positions in strings. Each such operation X gives rise to a family {Xn}n≥2} of similar permutations. We call an integer n X-prime if Xn consists of a single cycle of length n(n≥2). For some instances of X - such as shuffle, twist, operations based on the Archimedes' spiral and on the Josephus problem - we investigate the distribution of X-primes and of the associated (ordinary) prime numbers, which leads to variations of some well-known conjectures in number theory

Two theorems about maximal Cohen--Macaulay modules

This paper contains two theorems concerning the theory of maximal Cohen--Macaulay modules. The first theorem proves that certain Ext groups between maximal Cohen--Macaulay modules $M$ and $N$ must have finite length, provided only finitely many isomorphism classes of maximal Cohen--Macaulay modules exist having ranks up to the sum of the ranks of $M$ and $N$. This has several corollaries. In particular it proves that a Cohen--Macaulay local ring of finite Cohen--Macaulay type has an isolated singularity. A well-known theorem of Auslander gives the same conclusion but requires that the ring be Henselian. Other corollaries of our result include statements concerning when a ring is Gorenstein or a complete intersection on the punctured spectrum, and the recent theorem of Leuschke and Wiegand that the completion of an excellent Cohen--Macaulay local ring of finite Cohen--Macaulay type is again of finite Cohen--Macaulay type. The second theorem proves that a complete local Gorenstein domain of positive characteristic $p$ and dimension $d$ is $F$-rational if and only if the number of copies of $R$ splitting out of $R^{1/p^e}$ divided by $p^{de}$ has a positive limit. This result generalizes work of Smith and Van den Bergh. We call this limit the $F$-signature of the ring and give some of its properties.Comment: 14 page