extending pp-adic character of the local intertia to the absolute Galois group

Suppose I have a number field F, and a finite place v of F. Let E be finite extension of Fv. I start with a continuous morphism χ:O×Fv→E×. where OFv is the ring of integers of the completion Fv of F at v. My question is if χ can be extended to a continuous morphism … Read more

Real field of definition of an abelian variety of CM-type?

Question 0. Can a field of definitions (without automorphisms) of an (almost arbitrary) abelian variety of CM-type, originally defined over ${\mathbb{C}}$, be chosen to be a totally real number field or at least a real number field? I specify. Let $L$ be a CM-field of degree $[L:{\mathbb{Q}}]=2n$, i.e., a totally imaginary quadratic extension of a … Read more

Polynomial equations in many variables have solutions (Lang 1952 paper)

I am trying to understand the proof of the following result: Suppose F is a function field in k variables over an algebraically closed field. Let f1,…,fr∈F[x1,…,xn] be polynomials without constant term of degrees d1,…,dr, respectively. Assume that n>dk1+…+dkr. Then f1,…,fr have a non-trivial common zero. If F is instead a function field in k … Read more

Reference request: ramified and local geometric class field theory

There are lots of references on global unramified geometric class field theory (following Deligne’s ℓ-adic sheaves approach). There are also some notes talking about how to extend Deligne’s approach to the ramified case e.g. Bhatt’s Oberwolfach 2016 report titled “geometric class field theory”. In his article, Bhatt sketched the main descend step in the ramified … Read more

Are there “elementary” proofs of the openness of norm subgroups and of the norm limitation theorem?

Let K be a local field and L/K be a finite extension. Let Lab be the maximal abelian subextension of K in L. Write NL (resp. NLab) for the image of the norm map from L (resp. Lab) to K; this is a subgroup of K×. Two standard consequences of local class field theory are: … Read more

Kummer congruences for totally real number fields

There is a generalization of the Kummer congruences to totally real number fields with characters due to Deligne-Ribet. For example, see the exposition here, more precisely see Theorem 2.1. What is confusing me is that they don’t seem to use the L-functions with the Euler factor at p removed (which I will call the p-adic … Read more

How to calculate genus number of number field using sage?

I am looking to find real quadratic fields whose Hilbert class field is abelian over Q. Then I learned about genus numbers and genus field of the number field. It is enough to find a number field whose class number is equal to the genus number. In YOSHIOMI FURUTA article I found the formula, but … Read more

A Kummer exact sequence involving μ∞\mu_\infty

Let k be a number field. We have the well-known Kummer exact sequence of etale sheaves on Speck: 1→μn→Gm→Gm→1. Question 0. Applying the etale cohomology functor Hiet(k,−), I know that Hiet(k,Gm)=Hi(k,ˉk∗), where the latter is a Galois cohomology group. What is the Galois cohomological equivalent for Hiet(k,μn)? Let μ∞:=colimnμn, this group can be interpreted as … Read more

What are the roots of unity in abelian extensions of imaginary quadratic fields?

What roots of unity can be contained in the abelian extensions of an imaginary quadratic number field $K = \mathbb{Q}(\sqrt{-d})$? In particular, I would like to know: Is $K(\zeta_n)/K$ an abelian extension for every $n$? What are the roots of unity in the ray class field of $K$ with conductor $\mathfrak{c}$? What are the roots … Read more

class groups of unramified cyclic p-extensions of imaginary quadratic fields

Let K be an imaginary quadratic number field with p-Sylow-class group A(K) and L/K be an unramified cyclic extension of K of degree p (p prime). Then I am looking for heuristics on ker(NL/K:A(L)→A(K)), where NL/K is the usual norm map on ideal classes. Numerical data suggest that: |KerNL/K|≤p2 with high probability. But that is … Read more