User ramin - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T10:12:49Z http://mathoverflow.net/feeds/user/10458 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/130089/orders-in-number-fields Orders in number fields Ramin 2013-05-08T15:50:58Z 2013-05-21T01:09:08Z <p>Let $K$ be a degree $n$ extension of ${\mathbb Q}$ with ring of integers $R$. An order in $K$ is a subring with identity of $R$ which is a ${\mathbb Z}$-module of rank $n$. </p> <p>Question: Let $p$ be an unramified prime in $K$. Is it true that the number of orders in $R$ of index equal to $p^r$, for some natural number $r$, is less than or equal to the number of subrings with identity of ${\mathbb Z}^n$ of index equal to $p^r$?</p> <p>Nathan Kaplan and I need this fact for $n=5$ in a project where we are trying to find asymptotic formulae for the number of orders of bounded index in a given quintic field. </p> <p>We've been staring at Jos Brakenhoff's <a href="https://openaccess.leidenuniv.nl/bitstream/handle/1887/14539/proefschrift-brakenhoff.pdf?sequence=2" rel="nofollow">thesis</a> for a while, but I haven't gotten anywhere. Any advice will be greatly appreciated. Thanks. </p> <p>Added in edit: Here is an elementary reformulation of this problem. Let $r(x)$ be a polynomial with integer coefficients. Then show that for any natural number $a$ in order to maximize the number of subrings of $({\mathbb Z}/p^a {\mathbb Z})[x]/(r(x))$ of a given index, the polynomial $r(x)$ has to be a product of linear factors modulo $p$. </p> http://mathoverflow.net/questions/58507/how-was-the-importance-of-the-zeta-function-discovered/58537#58537 Answer by Ramin for How was the importance of the zeta function discovered? Ramin 2011-03-15T14:39:54Z 2013-05-08T17:05:34Z <p>Andre Weil has an article called "Prehistory of the zeta function" (reviewed by Jutila on mathscinet). I read this article many years ago, but this is basically what I remember of its content. Apparently the divergence of the harmonic series was known in 1650. Euler computed the special values at even integers and derived some kind of a functional equation. He also proved the Euler product formula and gave a proof of the infinitude of prime numbers using the Euler product. Dirichlet defined general L functions that now bear his name but only for real s>1. Riemann extended the definition of the zeta function to all complex values and proved the functional equation. According to Weil there were other people who had proved functional equations for functions that were closely related to the zeta function (namely, Malmstén, Schlömilch and Clausen from the review), but perhaps Riemann's contribution is the singular paper that established the importance of the zeta function as an important object to study. Weil believes that Riemann was influenced by his discussion with Eisenstein. </p> http://mathoverflow.net/questions/55953/torsion-points-in-abelian-varieties-over-number-fields Torsion points in Abelian varieties over number fields Ramin 2011-02-19T03:50:45Z 2013-01-07T08:56:58Z <p>Hello, Suppose $A$ is an Abelian variety of dimension $g$ over a number field $k$. Then using height functions one can show that there are non-torsion points in $A(\bar k)$. This looks like an overkill. Is there an easy, elementary way to see this? Thanks! Ramin</p> http://mathoverflow.net/questions/95205/a-question-about-partial-euler-products A question about partial Euler products Ramin 2012-04-25T22:08:39Z 2012-05-30T18:53:33Z <p>Let $K/{\mathbb Q}$ be an extension of degree $d$. Let $S$ be the set of primes $p$ which split completely in $K$. What can one say about the analytic properties of $$\zeta_{K, S}(s) : = \prod_{p \in S} \frac{1}{1-p^{-s}}.$$ More generally, one can define a similar partial Euler product for any splitting type, and ask the same question about the analytic properties of the resulting function. Any advice would be greatly appreciated. </p> http://mathoverflow.net/questions/95205/a-question-about-partial-euler-products/98407#98407 Answer by Ramin for A question about partial Euler products Ramin 2012-05-30T18:53:33Z 2012-05-30T18:53:33Z <p>This question turned out to be not too difficult. Please see <a href="http://www.math.uic.edu/~rtakloo/euler-product.pdf" rel="nofollow">http://www.math.uic.edu/~rtakloo/euler-product.pdf</a> for a (casual) writeup of an answer. Thanks for your comments and hints. </p> http://mathoverflow.net/questions/58569/question-related-to-diophantine-approximations-and-roths-theorem Question related to Diophantine approximations and Roth's theorem Ramin 2011-03-15T20:17:10Z 2012-02-09T00:22:19Z <p>The following question came up in my arithmetic geometry course yesterday. Suppose $\alpha$ is an irrational real algebraic integer, and suppose $\epsilon >0$ is given. Then by Roth's theorem there are at most finitely many rational numbers $\frac{h}{q}$ with $\gcd(h,q)=1$, $q>1$, such that $$\left| \alpha - \frac{h}{q}\right| &lt; \frac{1}{q^{2+\epsilon}}.$$ Are there any results on how large such $q$ can be? Thanks. </p> http://mathoverflow.net/questions/61466/non-vanishing-of-twists-of-l-functions-for-gl4 Non-vanishing of twists of L functions for GL(4) Ramin 2011-04-12T20:33:02Z 2011-04-12T20:33:02Z <p>Hello, </p> <p>This is a question in the spirit of <a href="http://mathoverflow.net/questions/56542/nonvanishing-of-central-l-values-of-quadratic-twists" rel="nofollow">http://mathoverflow.net/questions/56542/nonvanishing-of-central-l-values-of-quadratic-twists</a> and the application I have in mind is to p-adic L-functions a la Ash-Ginzburg. </p> <p>The question is this: </p> <p>Suppose $\pi$ is an irreducible unitary automorphic cuspidal representation of $GL(4)$, say, over ${\mathbb Q}$. Under what conditions on $\pi$, do we know the existence of a finite order $\chi$ such that $L(\frac{1}{2}, \pi \otimes \chi) \ne 0$? Here $L$ is the completed $L$-function. </p> <p>I know that by a result of Luo from 2005 if $\Re (s ) \ne 1/2$, then there is $\chi$ such that $L(s, \pi \otimes \chi) \ne 0$. </p> <p>Any help would be greatly appreciated. </p> http://mathoverflow.net/questions/60039/double-coset-spaces-of-reductive-groups-and-integral-representations-of-l-functio/60077#60077 Answer by Ramin for Double coset spaces of reductive groups and integral representations of L-functions Ramin 2011-03-30T14:59:11Z 2011-03-30T14:59:11Z <p>This is a very nice question! </p> <p>All of your examples are spherical quotients. Please take a look at <a href="http://andromeda.rutgers.edu/~sakellar/rs.pdf" rel="nofollow">http://andromeda.rutgers.edu/~sakellar/rs.pdf</a> and Yiannis' other papers for connections between spherical quotients and integral representations for L functions. </p> http://mathoverflow.net/questions/56807/how-to-be-updated-with-current-advances-in-mathematics/57207#57207 Answer by Ramin for How to be updated with current advances in mathematics Ramin 2011-03-03T04:03:36Z 2011-03-03T04:03:36Z <p>The same as everyone else: arxiv, conferences, in my case "Number theory web", talking to friends and colleagues, etc. But I've come to realize that all of this keeps me updated about a very small portion of mathematics. To remedy this, once every couple of months I walk down to the math periodicals section of our library and spend an hour or so browsing the most recent issues of good journals: read abstracts, scan keywords, etc. I somehow find this very helpful; granted, by the time papers appear in print, they are for the most part outdated, but this still gives me a sense, though with some delay, as to what's going on in the mathematical world. </p> http://mathoverflow.net/questions/52470/a-subring-question-revised A subring question (revised) Ramin 2011-01-19T04:06:03Z 2011-01-22T02:38:27Z <p>Hello, Let $K/{\mathbb Q}$ be a finite extension which is not necessarily Galois, and ${\mathcal O}$ be the ring of integers of $K$. Let $p$ be a prime in ${\mathbb Q}$ and let $p {\mathcal O}={\mathfrak p}_1^{k_1} \cdots {\mathfrak p}_r^{k_r}$ be its prime factorization in $K$. Next let ${\mathcal O}_i$ be the ring of integers of the completion of $K$ at ${\mathfrak p}_i$. Let ${\mathcal O}_p={\mathcal O}_1^{k_1} \times \cdots \times {\mathcal O}_r^{k_r}$. Is there a description of subrings $R$ of ${\mathcal O}_p$ such that $[{\mathcal O}_p:R ] &lt; \infty$ (additive subgroup index) ? Is it for example true that "most" of such $R$'s are of the form $R_1 \times \cdots \times R_r$? Here "most" should mean the following: Let ${\mathcal n}_t$ be the number of subrings $R$ such that $[{\mathcal O}_p: R] \leq t$ and let ${\mathcal n}'_t$ be the number of subrings $R$ of index at most $t$ which are expressible as a direct product $R_1 \times \cdots \times R_r$. Then is it true that $n'_t / n_t \to 1$ as $t \to \infty$? It would of course be most desirable if $n_t = n'_t$. My sincerest apologies ahead of time if this turns out to be a stupid question. </p> <p>Added in revision: Thank you for the answer and the comment. I would like to think of Laurent's example as ${\mathbb Z}_p^2$ and not ${\mathbb Z}_p \times {\mathbb Z}_p$. So it seems that I posed the question incorrectly; what I really need is to group together those ${\mathcal O}_i$'s that are isomorphic to each other, even if the the corresponding ${\mathfrak p}_i$'s are different. </p> <p>A word about the genesis of this problem: Nathan Kaplan and I have recently proved a theorem about counting the number of subrings of ${\mathbb Z}^n$ for small $n$ using $p$-adic integration techniques (we treat $n \leq 5$, the case $n=5$ seems to be new). Now I'm trying to see if we can use what we have proved to count orders in quintic fields (for cubic fields this is due to Davenport and Heilbronn, and for quartic fields this is a consequence of Nakagawa's work). To illustrate the idea, for a moment imagine that ${\mathbb Q}$ has a quintic Galois extension $K$. Except for finitely many exceptions, a given prime of $p$ either remains prime or splits as a product ${\mathfrak p}_1 \cdots {\mathfrak p}_5$. The situation for split primes is the same as counting the number of subrings of ${\mathbb Z}_p^5$ which we know how to do. The inert primes require separate treatment. </p> <p>It turns out that the question is quite a bit more interesting than I had originally thought (what else is new!). </p> http://mathoverflow.net/questions/19586/definition-of-l-function-attached-to-automorphic-representation/52575#52575 Answer by Ramin for Definition of L-function attached to automorphic representation Ramin 2011-01-20T03:49:58Z 2011-01-20T03:49:58Z <p>This question was posted a while back but I just saw it. Here are some thoughts. In practice there are a couple of methods to construct L functions for local ramified representations. The first one is the Langlands-Shahidi method which works for "generic representations" of quasi-split groups and the second one is the method of integral representations (Rankin-Selberg method, Shimura's integral and the doubling method, etc). It is probably a bit painful to give a meaningful description of these two methodologies in such a limited space, so instead let me refer you to a couple of places where you can see accessible accounts of the two approaches. A good reference for the basics of the Langlands-Shahidi method is the beautiful monograph "Analytic properties of automorphic L functions" by Gelbart and Shahidi. The same reference has a nice introduction to the method of integral representations. Dan Bump has written two very informative survey papers on the Rankin-Selberg method. Cogdell's ICTP lectures on the Rankin-Selberg method are lovely. Another good book to look at is the AMS book by Cogdell, Kim, Murty. </p> <p>As it stands there is no Langlands-Shahidi method for non-generic representations. What is missing from the picture is a good supply of easy to use unique models, like the Whittaker model in the generic setting. For orthogonal groups, however, recent progress by Waldspurger and others on the Gross-Prasad conjectures gives one the hope that maybe one can now develop a Langlands-Shahidi method, although there are serious obstacles to deal with. </p> <p>Most of the integral representations known to mankind too are closely linked with unique models (Whittaker, Bessel, etc). Sakellaridis has a theory that "explains" (some) integral representations in terms of spherical subgroups of reductive groups. </p> http://mathoverflow.net/questions/51386/prerequisites-for-p-adic-representations/51537#51537 Answer by Ramin for Prerequisites for P-adic Representations Ramin 2011-01-09T13:21:22Z 2011-01-13T01:40:39Z <p>Hello, I think a good first step is to learn the theory of admissible representations of p-adic groups and for this Godement's notes on Jacquet-Langlands theory and then Casselman's unpublished book on p-adic groups (available from his website) are good starting points. A good way to read Casselman's notes is to rewrite the proof of every theorem explicitly for a small non-GL(2) split group, say GL(3) or Sp(4). If you want to see the big picture shape of the theory and how it is connected to Galois representations you can look at the Trieste notes by Prasad and Raghuram (available from Dipendra Prasad's homepage at Tata). And in your first attempt to learn the theory don't worry too much about supercuspidal representations; just treat them like black boxes or elementary particles. One very nice thing about Godement's notes is that the theory is immediately followed by applications to Hecke theory and L functions. </p> <p>Let me add a couple of more points to address a question by Alex. You don't need much for Godement's notes; you do, however, need to be comfortable with Tate's thesis (p-adic integration, Haar measure, Poisson summation in the adelic setting, etc). I suppose that's the first thing you need to do if you haven't done already: </p> <p>"READ TATE'S THESIS." </p> <p>Tate's original writeup is amazing and much recommended. There is also the lovely book by Ramakrishnan and Valenza, as well as, of course, Bump's book. If you are already familiar with modular forms (and if not, why aren't you? :-) ) then Gelbart's classical book in the Princeton series is a good place to see the connections between the classical theory and the automorphic theory. When I was just starting to learn automorphic forms, I found Gelbart's treatment very nicely therapeutic. </p> http://mathoverflow.net/questions/43962/local-to-global-principle-for-reductive-groups/44300#44300 Answer by Ramin for Local to Global principle for reductive groups Ramin 2010-10-31T03:09:30Z 2010-11-23T03:35:44Z <p>There is also a theorem of I think Hakim (my apologies if this is not Jeff's theorem), generalized by Prasad and Schulze-Pillot, that allows you to globalize representations distinguished with respect to a subgroup. This one uses a simple relative trace formula. </p> http://mathoverflow.net/questions/130089/orders-in-number-fields Comment by Ramin Ramin 2013-05-21T18:58:03Z 2013-05-21T18:58:03Z In our work we only need this for unramified primes. http://mathoverflow.net/questions/72229/quotients-of-number-rings Comment by Ramin Ramin 2013-05-09T04:24:46Z 2013-05-09T04:24:46Z @KConrad. yes, it does, for example in Evanston, at 11:24 PM, sitting at a bar with Nathan Kaplan. http://mathoverflow.net/questions/95205/a-question-about-partial-euler-products Comment by Ramin Ramin 2012-04-25T23:20:58Z 2012-04-25T23:20:58Z Thanks! The quadratic case is Galois and Abelian. I need this sort of information for a quintic extension K/Q. http://mathoverflow.net/questions/69116/discrete-series-representations-for-sl-2-over-p-adic-field/69121#69121 Comment by Ramin Ramin 2011-07-01T16:04:45Z 2011-07-01T16:04:45Z I think the genesis of this construction is in Joseph Shalika's thesis, republished in his birthday volume in 2004. http://mathoverflow.net/questions/61466/non-vanishing-of-twists-of-l-functions-for-gl4 Comment by Ramin Ramin 2011-04-13T14:59:19Z 2011-04-13T14:59:19Z Hi Rob, This is very helpful! Thank you. http://mathoverflow.net/questions/61466/non-vanishing-of-twists-of-l-functions-for-gl4 Comment by Ramin Ramin 2011-04-13T02:17:13Z 2011-04-13T02:17:13Z David, I agree that the approximate functional equation method has some serious limitations, but let's keep in mind that Luo's result is better than what the naive method gives (he considers some second moments). And that makes me wonder if an $\epsilon$-improvement of the method, with $\epsilon$ possibly very big, might help deal with $\Re(s)=1/2$. Also, there are extremely non-trivial results obtained by Bump, Chinta, Friedberg, and Hoffstein (I might be missing some names) that give non-vanishing theorems using completely different methods. Could these results given something for GL(4)? http://mathoverflow.net/questions/58569/question-related-to-diophantine-approximations-and-roths-theorem/58571#58571 Comment by Ramin Ramin 2011-03-16T17:02:45Z 2011-03-16T17:02:45Z For some odd reason I kept missing the point. Sorry. I just changed the statement. http://mathoverflow.net/questions/58569/question-related-to-diophantine-approximations-and-roths-theorem Comment by Ramin Ramin 2011-03-16T17:01:51Z 2011-03-16T17:01:51Z @GH: You are right. Thank you. http://mathoverflow.net/questions/58569/question-related-to-diophantine-approximations-and-roths-theorem/58571#58571 Comment by Ramin Ramin 2011-03-16T02:39:08Z 2011-03-16T02:39:08Z I added $q&gt;1$ to the statement of the problem. As Antoine points out this came out of the discussion to figure out if it was possible to make the proof of Roth's theorem effective (or improve the Davenport-Roth bound). http://mathoverflow.net/questions/58569/question-related-to-diophantine-approximations-and-roths-theorem/58571#58571 Comment by Ramin Ramin 2011-03-15T20:48:31Z 2011-03-15T20:48:31Z Davenport and Roth's paper deals with bounding the number of $h/q$ such that the inequality is satisfied. They do say though that they do not know how to answer the question I'm asking. I was wondering if there has been any progress in this direction. I added the restriction that $\alpha$ is an algebraic integer. http://mathoverflow.net/questions/3359/has-anyone-read-this-paper-algebraic-real-analysis Comment by Ramin Ramin 2011-02-27T19:05:34Z 2011-02-27T19:05:34Z Actually Steven is asking a specific question. For those who did not bother to read his question before writing, let me quote his question: has anyone really looked into Freyd's claim that these ideas may be of use for defining functional integrals? Also if you bothered to look at the paper, it's 93 pages and densely written. I think Steven's question is a perfectly reasonable question for mathoverflow. http://mathoverflow.net/questions/56814/cohomology-of-real-algebraic-varieities Comment by Ramin Ramin 2011-02-27T18:55:50Z 2011-02-27T18:55:50Z and look for his papers with the word cohomology in the title. I do not know if there is a GAGA for o-minimal cohomology, but at least in stupid cases (e.g. nonstandard expansions of the real numbers) if done correctly coholomogy with constant coefficients is independent of the choice of the expansion (even though topologically these nonstandard fields are pretty disconnected). http://mathoverflow.net/questions/56814/cohomology-of-real-algebraic-varieities Comment by Ramin Ramin 2011-02-27T18:55:18Z 2011-02-27T18:55:18Z @Colin and Kevin: There is certainly a cohomology theory for semialgebraic varieties -- in fact, there is at least the beginning of a sheaf cohomology theory for any o-minimal structure. If you do a google search &quot;cohomology semialgebraic varieties&quot; you see get some interesting links. For cohomology theory in the o-minimal setting see the homepage of Mario Edmundo <a href="http://www.ciul.ul.pt/~edmundo/" rel="nofollow">ciul.ul.pt/~edmundo</a> http://mathoverflow.net/questions/55953/torsion-points-in-abelian-varieties-over-number-fields/55970#55970 Comment by Ramin Ramin 2011-02-20T22:34:03Z 2011-02-20T22:34:03Z Too bad mathoverflow doesn't let me accept more than one answer. http://mathoverflow.net/questions/55953/torsion-points-in-abelian-varieties-over-number-fields Comment by Ramin Ramin 2011-02-20T03:24:15Z 2011-02-20T03:24:15Z Maybe there is some kind of weak approximation lurking in the background?