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Maximal subfields in a division algebra over a local field

2012-08-01T11:07:34Z

Hi, Let $A$ be a division algebra over a local field $F$ of dimension $n^2$ and $K$ be an extension of $F$ of degree $n$. Then if follows from COROLLARY 2 in page 225 of Weil's Basic Number Theory that $A$ is split over $K$. Now my question is

Does $A$ contain a subfield isomorphic to $K$ ? Why?
Could you descirbe the general picture about "maximal subfields in central simple algebras" over a (not necessarily local) field ? Or tell me some references.

Thank you !

Base-Change and Automorphic-Induction for $GL_1$

2012-08-10T10:26:44Z

Dear all, I try to understand the base-change and automorphic-induction in the theory of automorphic forms, for the simplest case: $GL_1$. Both are implied by Langlands conjectures

Base-Change

Let $L/K$ be an abelian extension of number fields of degree $n$ and let $\Gamma$ be the set of distinct Hecke characters of $\mathbb A_{K}^{\times} $ associated to this extension by class-field-theory. Now given a Hecke charactr $\chi$ of $\mathbb A_{K}^{\times}$, we define its base-change to $\mathbb A_{L}^{\times}$ by $\chi_L:=\chi \cdot \mathbb N_{L/K}$. Then one should check that: $$ L(s,\chi_L)=\Pi _{\omega\in\Gamma }L(s,\chi\omega)$$

Conversely, suppose $L/K$ is cyclic with $Gal(L/K)=<\sigma >$. Let $\chi^\prime$ be a Hecke character of $\mathbb A_L^\times$ with $\chi^\prime=\chi^\prime\cdot\sigma$, then one might expect that $$ \chi^\prime=\chi_L$$ for some Hecke character of $\mathbb A_K^{\times}$.

Automorphic-Induction

Let $L/K$ be an extension of number fields of degree $n$ and $\omega$ a Hecke character of $\mathbb A_L^\times$. Then there exists a partition $n=n_1+...+n_r$ and cuspidal automorphic representations $\pi_i$ of $GL_{n_i}$ such that $$L(s,\omega)=\Pi_{i=1}^rL(s,\pi_i)$$

My question is: How to prove these results ? Or tell me some reference. Please feel free to choose any one of questions to answer, not necessarily all.

Thank you very much.

What is an automorphic representation of CM type ?

2012-06-25T16:56:38Z

In a recent paper of BL-Gee-Geraghty: "Sato-Tate for Hilbert modular forms" (JAMS 2011), a theorem is proved for regular algebrai cuspidal automorphic representation of $GL_2(\mathbb A_F)$ with $F$ a totally real field, which is not of CM type. I could not find any definition or reference for "CM type" in that paper. But I expect it should correspond to CM elliptic curve in the classical modular case.

My question is :

What is the precise definition for "an automorphic representation of CM type", both in the $GL_2$ case here and for general reductive group over number fields.

I prefer a definition "purely" in terms of representation-theory, not of arithmetic-geometry.
Why is the CM case excluded in that paper ?

Any comments or references will be very welcome. Thanks

Questions about the Bernstein center of a $p$-adic reductive group

2012-04-26T15:08:59Z

Dear all,

The "Bernstein center" of a $p$-adic reductive group appears frequently in the literature of automorphic forms, often without a precise definition. For example, in page 233 of Moeglin-Waldspurger's classic "Spectral decomposition and Eisenstein series" , the couple tell us :

"...in particular the centre of enveloping algebra acts on $\delta$ via a character at the infinite places and the Bernstein centre does so at the finite places... "

So one may guess that it is some analogy of "the centre of enveloping algebra" at fintie places.

My questions are:

What is the definition of the Bernstein centre of a p-adic reductive group.
What is the original motivation to introduce it ?
What role does it play in the theory of automorphic forms ?
Could you explain these in some concrete example ,say $GL_2$ ?

Please feel free to choose part of the questions to reply. Any comments and references (in English) will also be very welcome !

Thank you very much in advance!

Automorphic Forms on product of groups $G\times H$

2012-01-25T15:29:42Z

Dear all, I have some difficulty in understanding the notion of automorphic forms on product of groups.

Let $G$, $H$ be two reductive groups defined over a number field $F$. Let $\mathcal{A}(G)$ be the space of automorphic forms on $G(F)\backslash G(\mathbb{A})$ and $L^{2}(G):=L^{2}(G(F)\backslash G(\mathbb{A})^{1})$ as in e.g. J. Arthur's papers . When are the following isomorphisms true (as representations)? $$ \mathcal{A}(G\times H)\cong\mathcal{A}(G)\otimes\mathcal{A}(H)$$ and $$ L^{2}(G\times H)\simeq L^{2}(G)\hat{\otimes}L^{2}(H)$$

How to obtain the spectral decomposition of $L^{2}(G\times H)$ (as direct integral of irreducibles) from the spectral decompositon of $L^{2}(G)$ and $L^{2}(H)$ ?

The examples I have in mind is $G=GL_{n}$ and $H=GL_{m}$ or $G=SO_{n}$ and $H=SO_{m}$. Are they true in these case?

Note that the isomorphisms here are both for representations! So the question may be not so trivial.

Thank you so much! Any comments, example or non-example will be welcomed!

An interesting double coset in the theory of automorphic forms

2012-02-03T00:53:07Z

Dear all,

Does anyone have some idea to describe the double coset $P(F)\backslash G(F)/H(F)$ , say using Weyl group elements ? Here $G=GL_n\times GL_{n-1}$ is defined over a number field $F$ , $H=GL_{n-1}$ diagnoal embedded into $G$ as a subgroup and $P$ is some standard parabolic of $G$ .

The interesting point is that $H$ is not the fixed point set of some involution on $G$ so the quotient is not a symmetric space. Such example appers e.g. in the theory of Rankin-Selberg convolutions.

Let's start from a special case: say P is maximal parabolic.

Any comments and references will be welcome. Thank you !

Answer by unknown (google) for Major mathematical advances past age fifty

2011-02-15T14:40:25Z

Andre Weil lay the modern foundation of "theta series" in Acta math. (1964/65) when he was almost 60 years old!

Measure of "adeles minus ideles"

2010-12-18T15:41:06Z

Hi, I am interested in the set $\mathbb A-\mathbb A^\times$ i.e. the complement of ideles in the adele ring of a number field.

Is it measurable, and what is its volume, with respect to the standard measure of adeles?
("standard" means the same as in Tate's thesis)

Thank you.

Eisenstein series as sections of line bundles on moduli spaces

2010-08-05T08:55:09Z

It is well known that a modular form of weight k and level \Gamma is a global section of k-power of a Hodge line bundle over some modular curve. e.g. H^0(X,E^k).

My question is

How to characterize Eisenstein series among such sections using geometric datas?

For example, we know cusp forms are just sections of H^0(X,E^k(-cusps)).But how about Eisenstein series?

Actually in his Introduction to "Abelian Varieties" 1970, Mumford writes:

"It is interesting to ask whether further ties between the analytic and algebraic theories exist: e.g. an algebraic defintion of the Eisenstein series as a section of a line bundle on the moduli space. ..."

Could somebody explain the analytic-algebraic-representation aspects of Eisenstein series in some detail?

Thank you!

spectral sequences in number theory

2010-05-28T14:04:18Z

What is your favorite examples of spectral sequences arising naturally in arithmetic geometry? Please explain it in some detail

Answer by unknown (google) for Conceptual understanding of the Gross-Zagier theorem.

2010-02-25T09:41:18Z

Indeed, there is a conceptual understanding of this via "incoherent Siegel-Weil Formula",cft S.Kudla `s papers.See also the last section of recent preprint of Gan-Gross-Prasad.

Answer by unknown (google) for Rallis inner product formula for U(2,2) and U(3)

2010-05-14T07:31:59Z

See recent preprint of Harris-Li, which is base on Ichino`s S-W formula for unitary groups

Comment by

2012-08-28T12:08:12Z

Dear @Alexander Braverman : Have you finished running and been ready to "write a more detailed proof " ?

Comment by

2012-08-06T16:29:39Z

To be honest, one could see nothing from what you wrote about your own understanding about the original question and the answer in your linked paper.

Comment by

2012-08-05T15:10:13Z

Weil's analytic approach will prepare one very well to attack the theory of "automorphic representations" e.g. the book "Zeta functions of simple algebra" by Godement-Jacquet is merely an continuation of Weil's book form $GL_1$ to $GL_n$.

Comment by

2012-08-04T21:13:42Z

Weil's book determines the kernel of reciprocity map in section 8 of the last chapter, which appears neither in Cassels-Frohlich nor in Milne's notes.

Comment by

2012-08-04T07:06:20Z

As for the connection with L-functions, the Skinner-Urban story use the constant term of Eisenstein series (Langlands-Shahidi) while the Siegel-Weil story use the integral-representation (Rankin-Selberg-PS-Rallis)

Comment by

2012-08-04T07:03:20Z

I suggest you to read Richard Taylor's review article "Galois representation" which could be found on his website.

Comment by

2012-08-02T12:06:49Z

Note that Weil restricts to separable-extension case for simplicity.

Comment by

2012-08-02T07:18:15Z

Oh,I see! I just miss the condition "of dimension $n^2$" in your statement. Sorry and thanks again.

Comment by

2012-08-02T06:33:19Z

Thanks very much for this nice reference for central simple algebra. I am a bit confused: don't you need to pass to an suitable $A^\prime$ similar to $A$ in (i) of the theorem ?

Comment by

2012-08-02T06:26:03Z

Thank you so much! I just miss it.

Comment by

2012-08-01T16:27:35Z

And in the case of global fields, when could the field $K$ be embedded into the division algebra $A$ over $F$ ? One expect $K$ and $A$ should be "compatible" locally.

Comment by

2012-08-01T13:35:54Z

Thanks. So any finite extension $K$ of a local field $F$ of degree $n$ could be embedded into the division algebra $A$ over $F$ of degree $n^2$. Is it right ?

Comment by

2012-06-30T09:43:18Z

@Theo Johnson-Freyd : The title is perfectly clear. Since "Lubin-tate" has nothing to do with conformal field theory.

Comment by

2012-06-28T04:11:21Z

Still, the senstence "Complex Multiplication precedes Class Field Theory" does not make sense at all.

Comment by

2012-06-26T10:54:58Z

That is the original paper of Siegel and hard to read. The referene I provided is more readable. Siegel's work could be translated using adele to the "Tamagawa measure one" staement.