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.

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.

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.

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.

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.

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.