Expand all Collapse all | Results 1 - 6 of 6 |
1. CJM 2010 (vol 62 pp. 1011)
Functoriality of the Canonical Fractional Galois Ideal
The fractional Galois ideal
is a conjectural improvement on the higher Stickelberger
ideals defined at negative integers, and is expected to provide
non-trivial annihilators for higher $K$-groups of rings of integers of
number fields. In this article, we extend the definition of the
fractional Galois ideal to arbitrary (possibly infinite and
non-abelian) Galois extensions of number fields under the assumption
of Stark's conjectures and prove naturality properties under
canonical changes of extension. We discuss applications of this to the
construction of ideals in non-commutative Iwasawa algebras.
Categories:11R42, 11R23, 11R70 |
2. CJM 2009 (vol 62 pp. 157)
Special Values of Class Group $L$-Functions for CM Fields Let $H$ be the Hilbert class field of a CM number field $K$ with
maximal totally real subfield $F$ of degree $n$ over $\mathbb{Q}$. We
evaluate the second term in the Taylor expansion at $s=0$ of the
Galois-equivariant $L$-function $\Theta_{S_{\infty}}(s)$ associated to
the unramified abelian characters of $\operatorname{Gal}(H/K)$. This is an identity
in the group ring $\mathbb{C}[\operatorname{Gal}(H/K)]$ expressing
$\Theta^{(n)}_{S_{\infty}}(0)$ as essentially a linear combination of
logarithms of special values $\{\Psi(z_{\sigma})\}$, where $\Psi\colon
\mathbb{H}^{n} \rightarrow \mathbb{R}$ is a Hilbert modular function for a congruence
subgroup of $SL_{2}(\mathcal{O}_{F})$ and $\{z_{\sigma}: \sigma \in
\operatorname{Gal}(H/K)\}$ are CM points on a universal Hilbert modular variety. We
apply this result to express the relative class number $h_{H}/h_{K}$
as a rational multiple of the determinant of an $(h_{K}-1) \times
(h_{K}-1)$ matrix of logarithms of ratios of special values
$\Psi(z_{\sigma})$, thus giving rise to candidates for higher analogs
of elliptic units. Finally, we obtain a product formula for
$\Psi(z_{\sigma})$ in terms of exponentials of special values of
$L$-functions.
Keywords:Artin $L$-function, CM point, Hilbert modular function, Rubin-Stark conjecture Categories:11R42, 11F30 |
3. CJM 2006 (vol 58 pp. 419)
Stark's Conjecture and New Stickelberger Phenomena We introduce a new conjecture concerning the construction
of elements in the annihilator ideal
associated to a Galois action on the higher-dimensional algebraic
$K$-groups of rings of integers in number fields. Our conjecture is
motivic in the sense that it involves the (transcendental) Borel
regulator as well as being related to $l$-adic \'{e}tale
cohomology. In addition, the conjecture generalises the well-known
Coates--Sinnott conjecture. For example, for a totally real
extension when $r = -2, -4, -6, \dotsc$ the Coates--Sinnott
conjecture merely predicts that zero annihilates $K_{-2r}$ of the
ring of $S$-integers while our conjecture predicts a non-trivial
annihilator. By way of supporting evidence, we prove the
corresponding (conjecturally equivalent) conjecture for the Galois
action on the \'{e}tale cohomology of the cyclotomic extensions of
the rationals.
Categories:11G55, 11R34, 11R42, 19F27 |
4. CJM 2004 (vol 56 pp. 71)
Euclidean Rings of Algebraic Integers Let $K$ be a finite Galois extension of the field of rational numbers
with unit rank greater than~3. We prove that the ring of integers of
$K$ is a Euclidean domain if and only if it is a principal ideal
domain. This was previously known under the assumption of the
generalized Riemann hypothesis for Dedekind zeta functions. We now
prove this unconditionally.
Categories:11R04, 11R27, 11R32, 11R42, 11N36 |
5. CJM 2001 (vol 53 pp. 1194)
Explicit Upper Bounds for Residues of Dedekind Zeta Functions and Values of $L$-Functions at $s=1$, and Explicit Lower Bounds for Relative Class Numbers of $\CM$-Fields |
Explicit Upper Bounds for Residues of Dedekind Zeta Functions and Values of $L$-Functions at $s=1$, and Explicit Lower Bounds for Relative Class Numbers of $\CM$-Fields We provide the reader with a uniform approach for obtaining various
useful explicit upper bounds on residues of Dedekind zeta functions of
numbers fields and on absolute values of values at $s=1$ of $L$-series
associated with primitive characters on ray class groups of number
fields. To make it quite clear to the reader how useful such bounds
are when dealing with class number problems for $\CM$-fields, we
deduce an upper bound for the root discriminants of the normal
$\CM$-fields with (relative) class number one.
Keywords:Dedekind zeta functions, $L$-functions, relative class numbers, $\CM$-fields Categories:11R42, 11R29 |
6. CJM 1998 (vol 50 pp. 794)
Upper bounds on $|L(1,\chi)|$ and applications We give upper bounds on the modulus of the values at $s=1$ of
Artin $L$-functions of abelian extensions unramified at all
the infinite places. We also explain how we can compute better
upper bounds and explain how useful such computed bounds are
when dealing with class number problems for $\CM$-fields. For
example, we will reduce the determination of all the
non-abelian normal $\CM$-fields of degree $24$ with Galois
group $\SL_2(F_3)$ (the special linear group over the finite
field with three elements) which have class number one to the
computation of the class numbers of $23$ such $\CM$-fields.
Keywords:Dedekind zeta function, Dirichlet series, $\CM$-field, relative class number Categories:11M20, 11R42, 11Y35, 11R29 |