User 7-adic - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T14:13:07Z http://mathoverflow.net/feeds/user/2666 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/123706/maass-hecke-construction Maass-Hecke construction 7-adic 2013-03-06T05:28:03Z 2013-03-06T08:37:53Z <p>I heard this name that it can construct GL(2) automorphic forms or L-functions from GL(1)? I did not find it anywhere. Or does it have another name which we are familiar with?</p> http://mathoverflow.net/questions/123504/what-exactly-does-gg-and-ll-mean What exactly does \gg and \ll mean? 7-adic 2013-03-04T04:51:33Z 2013-03-04T14:22:46Z <p>For example, $f(T)\ll_T 1$ where $T$ is a positive number.</p> http://mathoverflow.net/questions/120713/decomposition-of-regular-representation-of-non-compact-lie-group Decomposition of Regular Representation of Non-compact Lie group 7-adic 2013-02-03T22:51:04Z 2013-02-04T01:36:43Z <p>Let G be a non-compact Lie group, such as SL(n,R), GL(n,C).</p> <p>How does the regular representation $L^2(G)$ decompose? Is there an analogue of Peter-Weyl theorem?</p> http://mathoverflow.net/questions/114579/reference-on-casselman-shalika-formula-for-gln-and-pgln Reference on Casselman-Shalika formula for GL(n) and PGL(n)? 7-adic 2012-11-26T20:14:20Z 2012-11-26T22:17:36Z <p>I am looking for reference on Casselman-Shalika formula for GL(n) and PGL(n) at finite place p.</p> http://mathoverflow.net/questions/113979/complex-finite-dimensional-representation-of-gln-c Complex Finite Dimensional Representation of GL(N,C) 7-adic 2012-11-20T19:49:31Z 2012-11-20T19:49:31Z <p>What are all the complex finite dimensional linear representation of $GL(N,\mathbb{C})$?</p> <p>We already know all the complex finite dimensional linear representation of SU(N).</p> http://mathoverflow.net/questions/112335/representation-ring-of-sun Representation ring of SU(n)? 7-adic 2012-11-14T01:44:34Z 2012-11-14T16:38:39Z <p>What's the structure of representation ring of SU(n)?</p> <p>What are the representations of generators?</p> http://mathoverflow.net/questions/109972/does-fe-of-selberg-zeta-function-imply-trace-formula Does FE of Selberg Zeta function imply Trace formula? 7-adic 2012-10-18T03:39:19Z 2012-10-18T09:56:20Z <p>Does the functional equation of the Selberg Zeta function imply the Selberg trace formula?</p> <p>BTW, the trace formula implies the functional equation.</p> http://mathoverflow.net/questions/101587/sato-tate-measure-for-gl3-automorphic-forms Sato-Tate measure for GL(3) Automorphic forms 7-adic 2012-07-07T15:14:34Z 2012-07-08T07:21:00Z <p>As we have known, the Sato-Tate measure for GL(2) turned out to be the half circle measure</p> <p>$\frac{1}{2\pi} \sqrt{4-x^2}dx$ on [-2,2],</p> <p>which appears in various versions of equi-distribution problems in GL(2).</p> <p>My question is what the corresponding measure for GL(3) should be.</p> <p>Firstly we shall note that Hecke eigenvalues in GL(3) can be imaginary. And we have to replace [-2,2] with {$z\in \mathbb{C}:|z|&lt;=3$} the ball of radius 3 in complex plane. I don't know anything further yet.</p> http://mathoverflow.net/questions/9235/a-problem-on-algebraic-number-theory-norm-of-ideals A problem on Algebraic Number Theory, Norm of Ideals 7-adic 2009-12-18T03:13:07Z 2012-05-23T04:50:24Z <p>A problem on Algebraic Number Theory</p> <p>K and L are number fields over Q.(Q is rational number filed) K is a subfield of L.</p> <p>O_K is the integers of K. and O_L is the integers of L.</p> <p>P is a prime ideal of O_L. p is a prime ideal of O_K. P is over p.</p> <p>The residue class degree f is defined to be f=[O_L/P:O_K/p]. The norm of P is Norm(P)=p^f</p> <p>This is the usual definition of Norm of an ideal.(See Serre's Local fields and Serge Lang's Algebraic Number Theory)</p> <p>Swinnerton-Dyer's A Brief Guide to Algebraic Number Theory has a different definition on Norm of an ideal.(Page 25)</p> <p>if A is an ideal of O_L, Norm(A)= ideal in O_K generated by elements Norm(a) where a is in A.</p> <p>I dont know why these two definitions are the same. Swinnerton-Dyer claims so in his book. Can anyone here give a hint, an explanation or anything else?</p> http://mathoverflow.net/questions/94037/any-reference-on-eisenstein-series-for-gamma-on-in-gl2 Any reference on Eisenstein Series for \Gamma_o(N) in GL(2) 7-adic 2012-04-14T15:40:17Z 2012-04-14T20:07:38Z <p>What's the best reference on Eisenstein Series for $\Gamma_o(N)$ in GL(2,R)?</p> <p>For fixed $\Gamma_o(N)$, should there be several Eisenstein series(corresponding to each cusp)?</p> http://mathoverflow.net/questions/93817/dual-maass-form-for-leveln-in-gl2 Dual Maass form for level=N in GL(2) 7-adic 2012-04-12T03:04:13Z 2012-04-12T04:50:12Z <p>Let $\Gamma=\Gamma_o(N)$ be the congruence subgroup. Let $f\in C^\infty(\Gamma\backslash GL(2,R)/SO(2,R)R^*)$ be a Maass form. How shall we define its dual(contragredient) Maass form $f'$?</p> <p>If $\Gamma=SL(2,Z)$, it is known that the dual Maass form should be $f'(z)=f(\omega (z^t)^{-1}\omega^{-1})$, where $$\omega=\begin{pmatrix}0&amp;-1\\ 1&amp;0\end{pmatrix}.$$</p> http://mathoverflow.net/questions/28857/how-to-associate-a-dirichlet-character-to-a-tate-character How to associate a Dirichlet character to a Tate character? 7-adic 2010-06-20T16:05:01Z 2012-04-07T16:57:52Z <p>A Dirichlet character is a multiplicative map from (<strong>Z</strong>/N)* to $S^1$. A Tate character is a continuous map from I/<strong>Q</strong> to $S^1$, where I is the Idele group of <strong>Q</strong>. It is always claimed that they are equivalent but I never see a convincing source.</p> <p>There is something involved called $F^N/P^N$ where $F^N$ is the group of fraction ideals not involving primes dividing $N$ and $P^N$ is the group of fraction ideals generated by $a$ such that a=1 mod N. </p> <p>I cannot see why a map on $F^N/P^N$ can be associated to a Dirichlet character, either.</p> http://mathoverflow.net/questions/75854/what-is-the-relationship-between-g-k-module-and-maass-forms What is the relationship between (g,K)-module and Maass forms? 7-adic 2011-09-19T14:57:17Z 2012-04-06T13:26:54Z <p>What is the relationship between (g,K)-module and Maass forms for GL(2)?</p> <p>(g,K)-modules are defined in chapter 2 of Bump, Automorphic forms and representations.</p> <p>There is a classification of (g,K)-modules.</p> <p>What is the relationship between (g,K)-module and Maass forms for GL(2)? and what does the classification of (g,K)-module imply for GL(2) Maass forms?</p> http://mathoverflow.net/questions/18928/union-of-closed-subschemes-with-the-structure-sheaf-over-it Union of closed subschemes with the structure sheaf over it 7-adic 2010-03-21T14:46:44Z 2011-11-22T02:57:48Z <p>Elementary commutative algebra fact: for two proper ideals I and J of a commutative ring R, we have $V(IJ)=V(I\cap J)=V(I)\cup V(J)$.</p> <p>Closed subschemes are related to sheaves of ideals. There is operation of intersection and product between sheaves of ideals, which is similar to the affine case.</p> <p>I see in many places that the structure sheaf over $V(I)\cup V(J)$ are defined to be $R/I\cap J$ rather than $R/IJ$. Why should the structure sheaf be define in that way?</p> http://mathoverflow.net/questions/77649/different-cuspidal-automorphic-representations-with-same-representations-at-infin Different cuspidal automorphic representations with same representations at infinity 7-adic 2011-10-10T02:30:43Z 2011-10-10T13:51:48Z <p>Let us fix a representation $\pi_\infty$ of GL(n,$\mathbb R$).</p> <p>Let us fix a character $\chi$ of K, where K is a compact subgroup of $GL(n,\mathbb A_{finite})$.</p> <p>$$K=\Pi_{v&lt;\infty}K_v$$</p> <p>$K_v$ is $GL(n,\mathbb Z_v)$ for almost all v.</p> <p>For automorphic forms of ($\chi$, K), we require that </p> <p>for $\phi: GL(n,\mathbb {A_Q})\to \mathbb C$ </p> <p>$\phi(x\gamma)=\chi(\gamma)\phi(x)$ for any $x\in GL(n,\mathbb{A_Q})$ and any $\gamma \in K$.</p> <p>How many cuspidal automorphic representations of GL(n,$\mathbb{A_Q}$) with character $\chi$ and with $\pi_\infty$ at infinity place are there?</p> <p>I am expecting answer to be "finitely many". Who and where is this proved?</p> http://mathoverflow.net/questions/77277/classification-of-irreducible-admissible-g-k-module-for-gl3-r classification of irreducible admissible (g,K)-module for GL(3,R) 7-adic 2011-10-05T19:22:06Z 2011-10-05T20:43:02Z <p>classification of irreducible admissible (g,K)-module for GL(3,R)</p> <p>Is there a classification of irreducible admissible (g,K)-module for GL(3,R)? </p> <p>For GL(2,R) we have principal series, discrete series and etc. Is there such a result for GL(3,R) or GL(n,R)?</p> http://mathoverflow.net/questions/58231/what-is-the-stirling-formula-for-xx1x2-xn-1 What is the Stirling formula for x(x+1)(x+2)...(x+n-1)? 7-adic 2011-03-12T04:54:35Z 2011-03-12T05:48:38Z <p>Let x be a complex number.</p> <p>What is the Stirling formula for x(x+1)(x+2)...(x+n-1) when n goes to infinity?</p> http://mathoverflow.net/questions/18665/what-does-being-analytically-isomorphic-imply-for-classification-of-singularities What does being Analytically Isomorphic imply for classification of singularities on curves? 7-adic 2010-03-18T21:55:06Z 2011-02-11T17:03:05Z <p>Hartshorne I.5 mentions the definition of being analytically isomorphic: P on X and Q on Y are analytically isomorphic iff the completion of O_P is isomorphic to the completion of O_Q where the completion is according to their maximal ideal in that local field.</p> <p>For any pair of regular points, they are always analytically isomorphic as long as they are of the same dimension according to Cohen Structure Theorem.</p> <p>But for singularities, they might be different. </p> <p>My question is: Is classification of the completion of O_P a first step to classify all singularities(on curves in particular)? What are known results about that classification? What does that imply if the classification of rings after completion was complete? WHAT IS THE CORRECT WAY TO CLASSIFY SINGULARITIES?</p> http://mathoverflow.net/questions/52744/what-is-the-relationship-between-modular-forms-and-maass-forms What is the relationship between modular forms and Maass forms? 7-adic 2011-01-21T08:08:46Z 2011-01-22T09:51:42Z <p>Modular forms are defined here: <a href="http://en.wikipedia.org/wiki/Modular_form#General_definitions" rel="nofollow">http://en.wikipedia.org/wiki/Modular_form#General_definitions</a></p> <p>Maass forms are defined here: <a href="http://en.wikipedia.org/wiki/Maass_wave_form" rel="nofollow">http://en.wikipedia.org/wiki/Maass_wave_form</a></p> <p>I wonder if modular forms can be transfered into Maass forms. Or they two are different categories of automorphic forms.</p> http://mathoverflow.net/questions/43240/what-is-the-l-function-version-of-quadratic-reciprocity What is the L-function version of quadratic reciprocity? 7-adic 2010-10-22T22:15:54Z 2010-11-10T00:58:09Z <p>Quadratic reciprocity theorems states that for two different odd prime p and q, we have (p/q)(q/p)=(-1)^(p-1)(q-1)/4.</p> <p>What is the statement of this theorem in L-function?</p> http://mathoverflow.net/questions/29114/relation-between-tates-thesis-and-class-field-theory Relation between Tate's thesis and Class Field Theory 7-adic 2010-06-22T16:37:13Z 2010-06-23T07:20:48Z <p>Class Field Theory states the correspondence between abelian extensions of k and congruence divisor class. In idelic language, there is a surjective map from $J_k/k^*$ to $Gal(k^{ab}/k)$ with its kernel unkonwn.</p> <p>Tate's Thesis proved some functional equations and analytic continuity(with a finite character of $J_k/k^*$). </p> <p>Question: Why Tate's thesis contributed to class field theory?</p> http://mathoverflow.net/questions/28113/cyclotomic-fields-over-q-and-prime-ideals Cyclotomic Fields over Q and prime ideals 7-adic 2010-06-14T12:36:40Z 2010-06-14T14:02:31Z <p><strong>Q</strong> is the rational number field. p is a prime number. q is a prime number other than p. $k_{p^r}$ is a cyclotomic field. $k_{p^r}$=<strong>Q</strong>(x) where x is exp(2$\pi$i/$p^r$). [$k_{p^r}$:<strong>Q</strong>]=$p^{r-1}(p-1)$.</p> <p>Question: Does q remain a prime in the integer ring of $k_{p^r}$?</p> http://mathoverflow.net/questions/20314/good-books-on-theory-of-distributions/20327#20327 Answer by 7-adic for Good books on theory of distributions 7-adic 2010-04-04T19:34:12Z 2010-04-04T19:34:12Z <p>I would say Fourier analysis, by Javier Duoandikoetxea, AMS.</p> http://mathoverflow.net/questions/20047/what-is-the-relationship-between-being-normal-and-being-regular What is the relationship between being normal and being regular? 7-adic 2010-04-01T07:45:51Z 2010-04-04T13:08:23Z <p>On a scheme, being normal means that each stalk of the structure sheaf is a integrally closed domain. Being regular means that each stalk of the structure sheaf is a regular local ring.</p> <p>As for a local ring, being regular or being integrally closed does not imply another.</p> <p>What is their connection with each other and classical/usual intuition of being smooth(being regular on stalk of each closed points)?</p> <p>Moreover, is there a smooth/regular variety which is not normal?</p> http://mathoverflow.net/questions/19388/degrees-of-subvarieties-of-projective-space/19391#19391 Answer by 7-adic for Degrees of subvarieties of projective space 7-adic 2010-03-26T04:37:49Z 2010-03-26T04:37:49Z <p>See the section about intersection of chapter I of hartshorne. Intuitively degree is the number of intersection of the dimension r subvariety and the GENERIC dimension n-r linear subspace if they are both in P^n. A rigorous definition was given in that section by the leading coefficient of the Hilbert polynomial.</p> http://mathoverflow.net/questions/19272/applications-of-brouwers-fixed-point-theorem/19346#19346 Answer by 7-adic for Applications of Brouwer's fixed point theorem 7-adic 2010-03-25T19:47:33Z 2010-03-25T19:47:33Z <p>By the fixed point theorem, one can prove that a polynomial with complex coefficients has at least a root in complex plane. It is not really cool but fits your audience. </p> http://mathoverflow.net/questions/3973/what-should-be-offered-in-undergraduate-mathematics-thats-currently-not-or-isn/18251#18251 Answer by 7-adic for What should be offered in undergraduate mathematics that's currently not (or isn't usually)? 7-adic 2010-03-15T07:21:05Z 2010-03-15T07:21:05Z <p>I am actually thinking about functional analysis and modern Fourier analysis.</p> http://mathoverflow.net/questions/18142/why-the-curve-t4-t3s-ts3-s4-is-not-projectively-normal-in-p3 Why the curve [t^4,t^3s,ts^3,s^4] is not projectively normal in P^3? 7-adic 2010-03-14T05:15:37Z 2010-03-14T06:15:55Z <p>Hartshorne EX I 3.18 b</p> <p>Define a curve by [t^4,t^3s,ts^3,s^4]. It is actually a P^1. Why the curve [t^4,t^3s,ts^3,s^4] is not projectively normal in P^3?</p> http://mathoverflow.net/questions/123706/maass-hecke-construction/123723#123723 Comment by 7-adic 7-adic 2013-03-06T13:30:49Z 2013-03-06T13:30:49Z thanks. so it is theta lifting. http://mathoverflow.net/questions/113979/complex-finite-dimensional-representation-of-gln-c Comment by 7-adic 7-adic 2012-11-20T20:26:59Z 2012-11-20T20:26:59Z does finite dimensional imply analytic and smooth? http://mathoverflow.net/questions/109972/does-fe-of-selberg-zeta-function-imply-trace-formula Comment by 7-adic 7-adic 2012-10-18T05:37:17Z 2012-10-18T05:37:17Z I am sure it is ture. By taking some special test function in the trace formula we get the functional equation of the Selberg Zeta function. http://mathoverflow.net/questions/101587/sato-tate-measure-for-gl3-automorphic-forms/101594#101594 Comment by 7-adic 7-adic 2012-09-01T18:55:21Z 2012-09-01T18:55:21Z @Qiaochu, not necessarily in number theory. Who firstly claimed that this measure should be the correct generalization of half-circle measure from U(2)? http://mathoverflow.net/questions/25929/u3-sato-tate-measure Comment by 7-adic 7-adic 2012-09-01T18:05:43Z 2012-09-01T18:05:43Z Did you know who firstly introduced this measure? Was that Katz-Sarnak? http://mathoverflow.net/questions/101587/sato-tate-measure-for-gl3-automorphic-forms/101594#101594 Comment by 7-adic 7-adic 2012-09-01T17:54:09Z 2012-09-01T17:54:09Z This is convincing. Do you know who was the first to introduce this measure for GL(3) or GL(n)? http://mathoverflow.net/questions/92207/atkinlehner-operator-for-gl3/92363#92363 Comment by 7-adic 7-adic 2012-03-29T18:06:42Z 2012-03-29T18:06:42Z why is atkin-lehner operator related to the decomposition of your induced representation in GL(2)? http://mathoverflow.net/questions/92207/atkinlehner-operator-for-gl3 Comment by 7-adic 7-adic 2012-03-26T05:39:37Z 2012-03-26T05:39:37Z I think you mean normalizer of the group $\Gamma_0(N)$, don't you? http://mathoverflow.net/questions/77649/different-cuspidal-automorphic-representations-with-same-representations-at-infin/77695#77695 Comment by 7-adic 7-adic 2011-10-10T19:50:11Z 2011-10-10T19:50:11Z to BR and Emerton, thank you so much! http://mathoverflow.net/questions/77649/different-cuspidal-automorphic-representations-with-same-representations-at-infin Comment by 7-adic 7-adic 2011-10-10T06:38:58Z 2011-10-10T06:38:58Z to Emerton, this character is not central character. http://mathoverflow.net/questions/77649/different-cuspidal-automorphic-representations-with-same-representations-at-infin Comment by 7-adic 7-adic 2011-10-10T06:18:27Z 2011-10-10T06:18:27Z modifying the original post and elaborating. K is a compact subgroup of $GL(n,\mathbb A_{finite})$. $$K=\Pi_{v&lt;\infty}K_v$$ $K_v$ is $GL(n,\mathbb Z_v)$ for almost all v. For automorphic forms of ($\chi$, K), we require that for $\phi: GL(n,\mathbb {A_Q})\to \mathbb C$ $\phi(x\gamma)=\chi(\gamma)\phi(x)$ for any $x\in GL(n,\mathbb{A_Q})$ and any $\gamma \in K$. http://mathoverflow.net/questions/75854/what-is-the-relationship-between-g-k-module-and-maass-forms Comment by 7-adic 7-adic 2011-09-19T16:09:21Z 2011-09-19T16:09:21Z David is correct. I have corrected them. http://mathoverflow.net/questions/28857/how-to-associate-a-dirichlet-character-to-a-tate-character Comment by 7-adic 7-adic 2010-06-20T16:09:31Z 2010-06-20T16:09:31Z For a map from I/Q to S^1, can one say something about the map on the infinite place R*? http://mathoverflow.net/questions/28113/cyclotomic-fields-over-q-and-prime-ideals/28117#28117 Comment by 7-adic 7-adic 2010-06-14T13:06:11Z 2010-06-14T13:06:11Z Could you elaborate on the congruence condition? The Galois group for $k_{p^r}$/**Q** is a cyclic group. Do you mean the condition that q does not divide p-1? http://mathoverflow.net/questions/17778/books-you-would-like-to-see-translated-into-english/17845#17845 Comment by 7-adic 7-adic 2010-04-09T10:53:08Z 2010-04-09T10:53:08Z It is at least translated to one other foreign language than Japanese.