We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
To save content items to your account,
please confirm that you agree to abide by our usage policies.
If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.
Find out more about saving content to .
To save content items to your Kindle, first ensure [email protected]
is added to your Approved Personal Document E-mail List under your Personal Document Settings
on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part
of your Kindle email address below.
Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations.
‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi.
‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
This chapter returns to more elementary mathematics, introducing Dirichlet, Hecke, and Artin L-functions. A proof of Dirichlet’s theorem on arithmetic progressions is given, by the method expounded by Serre; it would however be a shame to omit Dirichlet’s original method, which gave additional information and anticipated the analytic class number formulae. The two main generalisations of Dirichlet’s L-functions are then introduced: those of Hecke and Artin. Hecke’s main theorem is stated without proof: existence of an analytic continuation and a functional equation, and it is then explained how Artin and Brauer derived the same results for non-abelian L-functions.
Let $n$ be a positive even integer, and let $F$ be a totally real number field and $L$ be an abelian Galois extension which is totally real or $\text{CM}$. Fix a finite set $S$ of primes of $F$ containing the infinite primes and all those which ramify in $L$, and let ${{S}_{L}}$ denote the primes of $L$ lying above those in $S$. Then $\mathcal{O}_{L}^{S}$ denotes the ring of ${{S}_{L}}$-integers of $L$. Suppose that $\psi$ is a quadratic character of the Galois group of $L$ over $F$. Under the assumption of the motivic Lichtenbaum conjecture, we obtain a non-trivial annihilator of the motivic cohomology group $H_{\mathcal{M}}^{2}\left( \mathcal{O}_{L}^{S},\mathbb{Z}\left( n \right) \right)$ from the lead term of the Taylor series for the S-modified Artin $L$-function $L_{L/F}^{S}\left( s,\psi\right)$ at $s=1-n$.
We formulate an explicit conjecture for the leading term at s=1 of the equivariant Dedekind zeta-function that is associated to a Galois extension of number fields. We show that this conjecture refines well-known conjectures of Stark and Chinburg, and we use the functional equation of the zeta-function to compare it to a natural conjecture for the leading term at s=0.
Recommend this
Email your librarian or administrator to recommend adding this to your organisation's collection.