User - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T11:25:06Z http://mathoverflow.net/feeds/user/4245 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103676/maximal-subfields-in-a-division-algebra-over-a-local-field Maximal subfields in a division algebra over a local field unknown (google) 2012-08-01T11:07:34Z 2012-10-05T01:06:20Z <p>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 </p> <ol> <li><p>Does $A$ contain a subfield isomorphic to $K$ ? Why?</p></li> <li><p>Could you descirbe the general picture about "maximal subfields in central simple algebras" over a (not necessarily local) field ? Or tell me some references.</p></li> </ol> <p>Thank you !</p> http://mathoverflow.net/questions/104397/base-change-and-automorphic-induction-for-gl-1 Base-Change and Automorphic-Induction for $GL_1$ unknown (google) 2012-08-10T10:26:44Z 2012-08-10T12:18:59Z <p>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</p> <p>Base-Change</p> <p>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: $$L(s,\chi_L)=\Pi _{\omega\in\Gamma }L(s,\chi\omega)$$</p> <p>Conversely, suppose $L/K$ is cyclic with $Gal(L/K)=&lt;\sigma >$. 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 $\mathbb A_K^{\times}$.</p> <p>Automorphic-Induction</p> <p>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 $$L(s,\omega)=\Pi_{i=1}^rL(s,\pi_i)$$ </p> <p>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.</p> <p>Thank you very much.</p> http://mathoverflow.net/questions/100606/what-is-an-automorphic-representation-of-cm-type What is an automorphic representation of CM type ? unknown (google) 2012-06-25T16:56:38Z 2012-06-25T17:35:02Z <p>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.</p> <p>My question is :</p> <ol> <li><p>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.</p> <p>I prefer a definition "purely" in terms of representation-theory, not of arithmetic-geometry. </p></li> <li><p>Why is the CM case excluded in that paper ?</p></li> </ol> <p>Any comments or references will be very welcome. Thanks </p> http://mathoverflow.net/questions/95261/questions-about-the-bernstein-center-of-a-p-adic-reductive-group Questions about the Bernstein center of a $p$-adic reductive group unknown (google) 2012-04-26T15:08:59Z 2012-05-01T10:17:24Z <p>Dear all, </p> <p>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 : </p> <p>"...in particular the centre of enveloping algebra acts on $\delta$ via a character at the infinite places and the <em>Bernstein centre</em> does so at the finite places... "</p> <p>So one may guess that it is some analogy of "the centre of enveloping algebra" at fintie places. </p> <p>My questions are:</p> <ol> <li>What is the definition of the Bernstein centre of a p-adic reductive group.</li> <li>What is the original motivation to introduce it ?</li> <li>What role does it play in the theory of automorphic forms ?</li> <li>Could you explain these in some concrete example ,say $GL_2$ ?</li> </ol> <p>Please feel free to choose part of the questions to reply. Any comments and references (in English) will also be very welcome !</p> <p>Thank you very much in advance!</p> http://mathoverflow.net/questions/86636/automorphic-forms-on-product-of-groups-g-times-h Automorphic Forms on product of groups $G\times H$ unknown (google) 2012-01-25T15:29:42Z 2012-04-26T20:24:35Z <p>Dear all, I have some difficulty in understanding the notion of automorphic forms on product of groups. </p> <p>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 $$L^{2}(G\times H)\simeq L^{2}(G)\hat{\otimes}L^{2}(H)$$</p> <p>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 $L^{2}(H)$ ?</p> <p>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?</p> <p>Note that the isomorphisms here are both for representations! So the question may be not so trivial.</p> <p>Thank you so much! Any comments, example or non-example will be welcomed!</p> http://mathoverflow.net/questions/87388/an-interesting-double-coset-in-the-theory-of-automorphic-forms An interesting double coset in the theory of automorphic forms unknown (google) 2012-02-03T00:53:07Z 2012-02-05T06:18:55Z <p>Dear all, </p> <p>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 $G$ .</p> <p>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. </p> <p>Let's start from a special case: say P is maximal parabolic.</p> <p>Any comments and references will be welcome. Thank you !</p> http://mathoverflow.net/questions/25630/major-mathematical-advances-past-age-fifty/55520#55520 Answer by unknown (google) for Major mathematical advances past age fifty unknown (google) 2011-02-15T14:40:25Z 2011-02-15T14:40:25Z <p>Andre Weil lay the modern foundation of "theta series" in Acta math. (1964/65) when he was almost 60 years old!</p> http://mathoverflow.net/questions/49811/measure-of-adeles-minus-ideles Measure of "adeles minus ideles" unknown (google) 2010-12-18T15:41:06Z 2010-12-20T08:54:41Z <p>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. </p> <p>Is it measurable, and what is its volume, with respect to the standard measure of adeles?<br> ("standard" means the same as in Tate's thesis) </p> <p>Thank you.</p> http://mathoverflow.net/questions/34595/eisenstein-series-as-sections-of-line-bundles-on-moduli-spaces Eisenstein series as sections of line bundles on moduli spaces unknown (google) 2010-08-05T08:55:09Z 2010-08-05T16:31:29Z <p>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). </p> <p>My question is </p> <p><em><strong>How to characterize Eisenstein series among such sections using geometric datas?</em></strong> </p> <p>For example, we know cusp forms are just sections of H^0(X,E^k(-cusps)).But how about Eisenstein series?</p> <p>Actually in his Introduction to "Abelian Varieties" 1970, Mumford writes:</p> <p>"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. ..."</p> <p>Could somebody explain the analytic-algebraic-representation aspects of Eisenstein series in some detail? </p> <p>Thank you!</p> http://mathoverflow.net/questions/26266/spectral-sequences-in-number-theory spectral sequences in number theory unknown (google) 2010-05-28T14:04:18Z 2010-05-29T13:59:07Z <p>What is your favorite examples of spectral sequences arising naturally in arithmetic geometry? Please explain it in some detail</p> http://mathoverflow.net/questions/16315/conceptual-understanding-of-the-gross-zagier-theorem/16380#16380 Answer by unknown (google) for Conceptual understanding of the Gross-Zagier theorem. unknown (google) 2010-02-25T09:41:18Z 2010-05-15T09:28:37Z <p>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.</p> http://mathoverflow.net/questions/24586/rallis-inner-product-formula-for-u2-2-and-u3/24589#24589 Answer by unknown (google) for Rallis inner product formula for U(2,2) and U(3) unknown (google) 2010-05-14T07:31:59Z 2010-05-14T07:31:59Z <p>See recent preprint of Harris-Li, which is base on Ichinos S-W formula for unitary groups</p> http://mathoverflow.net/questions/64419/explicit-formula-for-the-trace-of-an-unramified-principal-series-representation-o/64421#64421 Comment by 2012-08-28T12:08:12Z 2012-08-28T12:08:12Z Dear @Alexander Braverman : Have you finished running and been ready to &quot;write a more detailed proof &quot; ? http://mathoverflow.net/questions/47920/what-does-the-theta-divisor-of-a-number-field-know-about-its-arithmetic/104086#104086 Comment by 2012-08-06T16:29:39Z 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. http://mathoverflow.net/questions/14283/central-simple-algebras-approach-to-classfield-theory-merits-of/14289#14289 Comment by 2012-08-05T15:10:13Z 2012-08-05T15:10:13Z Weil's analytic approach will prepare one very well to attack the theory of &quot;automorphic representations&quot; e.g. the book &quot;Zeta functions of simple algebra&quot; by Godement-Jacquet is merely an continuation of Weil's book form $GL_1$ to $GL_n$. http://mathoverflow.net/questions/24719/suggestions-for-good-books-on-class-field-theory/24729#24729 Comment by 2012-08-04T21:13:42Z 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. http://mathoverflow.net/questions/103846/why-are-galois-representations-so-important-in-number-theory Comment by 2012-08-04T07:06:20Z 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) http://mathoverflow.net/questions/103846/why-are-galois-representations-so-important-in-number-theory Comment by 2012-08-04T07:03:20Z 2012-08-04T07:03:20Z I suggest you to read Richard Taylor's review article &quot;Galois representation&quot; which could be found on his website. http://mathoverflow.net/questions/103676/maximal-subfields-in-a-division-algebra-over-a-local-field/103740#103740 Comment by 2012-08-02T12:06:49Z 2012-08-02T12:06:49Z Note that Weil restricts to separable-extension case for simplicity. http://mathoverflow.net/questions/103676/maximal-subfields-in-a-division-algebra-over-a-local-field/103734#103734 Comment by 2012-08-02T07:18:15Z 2012-08-02T07:18:15Z Oh,I see! I just miss the condition &quot;of dimension $n^2$&quot; in your statement. Sorry and thanks again. http://mathoverflow.net/questions/103676/maximal-subfields-in-a-division-algebra-over-a-local-field/103734#103734 Comment by 2012-08-02T06:33:19Z 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 ? http://mathoverflow.net/questions/103676/maximal-subfields-in-a-division-algebra-over-a-local-field/103740#103740 Comment by 2012-08-02T06:26:03Z 2012-08-02T06:26:03Z Thank you so much! I just miss it. http://mathoverflow.net/questions/103676/maximal-subfields-in-a-division-algebra-over-a-local-field/103679#103679 Comment by 2012-08-01T16:27:35Z 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 &quot;compatible&quot; locally. http://mathoverflow.net/questions/103676/maximal-subfields-in-a-division-algebra-over-a-local-field/103679#103679 Comment by 2012-08-01T13:35:54Z 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 ? http://mathoverflow.net/questions/96352/lubin-tate-vs-cohomological-approach-to-local-cft Comment by 2012-06-30T09:43:18Z 2012-06-30T09:43:18Z @Theo Johnson-Freyd : The title is perfectly clear. Since &quot;Lubin-tate&quot; has nothing to do with conformal field theory. http://mathoverflow.net/questions/100276/can-one-prove-complex-multiplication-without-assuming-cft/100305#100305 Comment by 2012-06-28T04:11:21Z 2012-06-28T04:11:21Z Still, the senstence &quot;Complex Multiplication precedes Class Field Theory&quot; does not make sense at all. http://mathoverflow.net/questions/100667/why-should-i-believe-in-the-siegels-and-hasses-rationale Comment by 2012-06-26T10:54:58Z 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 &quot;Tamagawa measure one&quot; staement.