User idoneal - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T11:08:33Z http://mathoverflow.net/feeds/user/2344 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/114145/a-sentence-in-shimuras-on-the-periods-of-modular-forms A sentence in Shimura's "On The Periods of Modular Forms" Idoneal 2012-11-22T08:45:01Z 2013-02-06T11:20:08Z <p>Let $K_f$ denote the number field generated by the Fourier coefficients $a_n$ of a normalized primitive holomorphic cusp form $f$. On page 2, line 6 of the paper mentioned in the title, Shimura writes that $K_f$ is generated by $a_p$ for almost all primes $p$. In the next sentence, he says that it follows trivially from the Multiplicity one theorem. </p> <p>I don't see how it follows. I shall appreciate any comment.</p> http://mathoverflow.net/questions/105971/how-should-an-analytic-number-theorist-look-at-bessel-functions/105986#105986 Answer by Idoneal for How should an analytic number theorist look at Bessel functions? Idoneal 2012-08-30T19:46:52Z 2012-08-30T19:53:37Z <p>One way is to think of Bessel functions as some sort of non-abelian analogues of the exponential function. In fact, you can form a complete orthogonal system of $L^2 (\mathbb{R^+}, x^{-1}d x)$ using the Bessel functions. See Chapter 16 of Iwaniec-Kowalski (page 411).</p> <p>For manipulations, the formulae given in Appendix B of Iwaniec's Spectral Methods book usually suffice.</p> <p>Yes, Dan Brown is more appropriate than Gradhsteyn and Rizhik for airport reading.</p> http://mathoverflow.net/questions/7988/riemann-surfaces-explicit-algebraic-equations Riemann surfaces: explicit algebraic equations Idoneal 2009-12-06T11:07:16Z 2010-11-11T05:45:05Z <p>Suppose $\Gamma$ is a nice discrete subgroup of $SL(2,\mathbb{R})$ such that the genus of the Riemann surface $\mathbb{H}/\Gamma$ is larger than 1. We know that this Riemann surface is also an algebraic curve over $\mathbb{C}$ defined by a bunch of polynomials. Is there any explicit/canonical way of going back and forth between the group $\Gamma$ and a set of polynomials defining the curve? For example, if the group is given in terms of generators and relations, is there any algorithm for obtaining a set of polynomials defining the associated curve?</p> <p>Somehow I couldn't add a comment so let me write it here.</p> <p>Thanks for the responses. It will take me some time to digest the suggestions and look up the references provided. Please ignore the last comment about generators and relation. Let me ask a more specific question just to clarify what I wanted. What I am wondering is whether something akin to what happens for genus one curve also happens for higher genus. Recall that if $L=\mathbb{Z}+\tau \mathbb{Z}$, then we can write an equation for the elliptic curve $\mathbb{C}/L$ with the Eisenstein series $G_4(\tau)$ and $G_6(\tau)$ as coefficients. Can we do (or hope to do) something similar if we replace $\mathbb{C}$ by $\mathbb{H}$ and the lattice by a discrete group ? </p> http://mathoverflow.net/questions/44326/most-memorable-titles/44418#44418 Answer by Idoneal for Most memorable titles Idoneal 2010-11-01T05:29:33Z 2010-11-01T11:56:41Z <p><a href="http://books.google.com/books?id=P82xkbGioL8C" rel="nofollow"><strong>Ideals and reality</strong></a></p> <p>projective modules and number of generators of ideals </p> <p>By Friedrich Ischebeck and Ravi A. Rao</p> http://mathoverflow.net/questions/43464/complex-analysis-applications-toward-number-theory/43493#43493 Answer by Idoneal for Complex Analysis applications toward Number Theory Idoneal 2010-10-25T08:41:59Z 2010-10-25T10:50:04Z <p>Complex analysis is often used in analytic number theory as a tool to evaluate or estimate sums $\sum a_n$ by studying the analytic behaviour (like existence of poles or how fast it grows) of the associated Dirichlet series $\sum a_n n^{-s}$. So for most interesting arithmetical functions (like a_n = number of divisors of n, say), one can study the corresponding Dirichlet series (possibly factor it as a product of various $L$-functions) to obtain information about the sum. One uses the Perron formula or Mellin inversion formula to pass from the sum to a contour integral. Davenport's book is the canonical reference.</p> http://mathoverflow.net/questions/42900/refinements-of-the-riemann-hypothesis/42906#42906 Answer by Idoneal for Refinements of the Riemann hypothesis Idoneal 2010-10-20T16:18:16Z 2010-10-20T16:18:16Z <p>For the Riemann zeta function, there are several conjectures that are stronger than the Riemann Hypothesis and these imply stronger results on distribution of prime numbers than what are known on RH alone. The most notable is probably the Pair Correlation Conjecture of Montgomery. This is a conjecture that assumes the Riemann hypothesis and predicts on finer distribution of the gaps between the imaginary parts of the zeros. There are similar conjectures due to Katz and Sarnak for many other $L$-functions. A good place to read about this is the article by Katz and Sarnak <a href="http://www.ams.org/journals/bull/1999-36-01/S0273-0979-99-00766-1/home.html" rel="nofollow">http://www.ams.org/journals/bull/1999-36-01/S0273-0979-99-00766-1/home.html</a></p> <p>About the conjecture you have mentioned, I don't know if any standard conjecture on the zeros of the Riemann zeta function implies it. Heath-Brown has a result on the application of PCC on gaps between consecutive primes but I don't think it comes anywhere near this one.</p> http://mathoverflow.net/questions/42533/poles-of-kloosterman-zeta-function/42613#42613 Answer by Idoneal for Poles of Kloosterman Zeta Function Idoneal 2010-10-18T10:57:14Z 2010-10-18T10:57:14Z <p>For either of them, the only pole at an integer point is at $s=1$. There is a formula for the residue in Iwaniec's book on spectral theory of automorphic forms which is too involved to write here. Note, however, that your normalization is slightly different from his ( 2$s$ instead of $s$ ). There are other poles on the line Re $s=1$ (which are related to eigenvalues of the hyperbolic Laplacian) and these are the only poles. All these come from the spectral decomposition of the zeta function. </p> http://mathoverflow.net/questions/40997/fast-computation-of-multiplicative-inverse-modulo-q Fast computation of multiplicative inverse modulo q Idoneal 2010-10-04T09:05:57Z 2010-10-10T06:43:35Z <p>Given a large number $q$ (say, a prime) and a number $a$ between 2 and $q-1$ what is the fastest algorithm known for computing the inverse of $a$ in the group of residue classes modulo $q$?</p> http://mathoverflow.net/questions/25461/modular-forms-with-prime-fourier-coefficients-zero Modular forms with prime Fourier coefficients zero Idoneal 2010-05-21T05:50:19Z 2010-08-12T01:46:17Z <p>Can you give a non-trivial example of an integer weight cusp form which does not lie in the old subspace and it has $a_p=0$ for all primes $p$?</p> <p>If such a form cannot exist then why?</p> http://mathoverflow.net/questions/2437/is-there-an-image-for-you-that-epitomizes-mathematics/26803#26803 Answer by Idoneal for Is there an image for you that epitomizes mathematics? Idoneal 2010-06-02T08:54:42Z 2010-06-02T08:59:53Z <p>Here is a more mathematical rendition of Richard Kent's answer:</p> <p><a href="http://math-art.net/2007/12/03/eternal-scream-a-droste-effect/" rel="nofollow">http://math-art.net/2007/12/03/eternal-scream-a-droste-effect/</a></p> http://mathoverflow.net/questions/11349/are-most-cubic-plane-curves-over-the-rationals-elliptic Are most cubic plane curves over the rationals elliptic? Idoneal 2010-01-10T16:59:58Z 2010-04-23T12:56:37Z <p>%This is a new version of the original question modified in the light of the answers and comments.</p> <p>The word 'most' in the title is ambiguous. The following is one way of making it precise.</p> <p>Question1: (This seems to be open. See Poonen's answer below)</p> <p>A cubic projective curve over $\mathbb{Q}$ is given by ten relatively prime integers (the coefficients of its equation after clearing the denominators). Suppose we take a ten dimensional box $[-N,N]^{10}$ and choose points with integer coordinates with respect to the uniform measure and form the equation of the associated cubic curve. Suppose the number of points which give rise to a curve with a rational point is $E(N)$. Then what can we say about $E(N)/(2N+1)^{10}$ as $N\rightarrow \infty$?</p> <p>Should the limit exist and if it does, should it be one, zero, or some other number?</p> <p>Another question of interest is:</p> <p>Question 2: (There is a satisfactory answer to this. See Voloch's response below.)</p> <p>Are either of the sets {cubics with no rational point} and {cubics with at least one rational point} Zariski dense?</p> http://mathoverflow.net/questions/10603/does-p-adic-l-function-determine-the-l-function/21105#21105 Answer by Idoneal for Does p-adic $L$- function determine the $L$ function Idoneal 2010-04-12T12:38:58Z 2010-04-12T12:38:58Z <p>Yes. See the Invent. Math. paper by Luo and Ramakrishnan titled "Determination of modular forms by twists of critical L-values". They mention this result on the third page.</p> http://mathoverflow.net/questions/17461/conductor-of-monomial-forms-with-trivial-nebentypus Conductor of monomial forms with trivial nebentypus Idoneal 2010-03-08T11:16:18Z 2010-03-09T10:26:00Z <p>Is it true that the conductor of a holomoprhic or a Maass cusp form <em>with trivial nebentypus</em> corresponding to a two-dimensional dihedral representation (over $\mathbb{Q}$ )is non-square-free?</p> http://mathoverflow.net/questions/11677/modular-eigenforms-with-integral-coefficients-maedas-conjecture/11720#11720 Answer by Idoneal for modular eigenforms with integral coefficients [Maeda's Conjecture] Idoneal 2010-01-14T05:22:07Z 2010-01-14T08:15:49Z <p>This is a (far too long) comment on Buzzard's comment about Hida's remark.</p> <p>I think I can guess what Hida was saying. He was probably talking about non-vanishing of L-functions of Hecke eigenforms of level one and weight $k \equiv 0$ (mod 4). This is a long-standing (folklore? ) conjecture in its own right, well-known among analytic number-theorists. </p> <p>Here is how such a thing can be proven using Maeda's conjecture. There is a result of Shimura that says that the Galois group acts nicely on the central values (in fact any critical value) L-function of eigenforms. In particular, if one of them is zero then all the Galois twists are also zero and hence their sum is also zero. Now, even though it may be difficult to show that an L-function doesn't vanish at the centre, it is often easy to show that the sum of the central values of L-functions in a family is non-zero (see, for example, the work of Rohrlich and Rodriguez-Villegas on non-vanishing of L-functions of Hecke characters). </p> <p>In the case in question, Maeda's conjecture will imply that if one central L-value is zero then the sum of all the central L-values over the whole basis must be zero and I think a contradiction will ensue after one uses the approximate functional equation to write the central value in terms of the Fourier coefficients and then using the Petersson formula ( I need to check this up). </p> <p>Note 1: There is an article by Conrey and Farmer titled "Hecke operators and nonvanishing of L-functions" (Ahlgreen et al. (eds.), Topics in Number Theory, 1999) where they prove the above mentioned result along a different line. </p> <p>Note 2: I think the following is easier. One can think of $f\rightarrow L(f,k/2)$ as a linear functional on the space of cusp forms $S_k(\Gamma(1))$ and indeed it is possible to explicitly write a function $G$ such that </p> <p>$L(f,k/2)=\langle f,G \rangle$ </p> <p>for all Hecke eigenform $f$ in $S_k(\Gamma(1))$. Now Maeda + Shimura's result will imply that $G$ is orthogonal to the whole space and therefore zero. So it is just a matter of checking that $G$ is not identically zero, which shouldn't be too hard. </p> http://mathoverflow.net/questions/11053/whats-the-relationship-between-gauss-sums-and-the-normal-distribution/11228#11228 Answer by Idoneal for What's the relationship between Gauss sums and the normal distribution? Idoneal 2010-01-09T11:27:33Z 2010-01-12T16:28:42Z <p>I don't think there is anything deep going on here. The Fourier analysis on finite abelian group is fairly straightforward.</p> <p>Gauss sums are the Fourier coefficients you get when you expand an additive character $k \rightarrow e^{\frac{2\pi iak}{p}}$ with respect to the basis of multiplicative characters (i.e. those that give rise to Dirichlet characters when our group is $\mathbb{Z}/n\mathbb{Z}$). A Gauss sum is a sum of the product of an additive and a multiplicative characters and as such can be thought of as a finite group analogue of the Gamma function. Recall that the Gamma function is the integral on $\mathbb{R}^{>0}$ of the product of $e^{-x}$ (additive character on the reals) and $x^s$ (a multiplicative character on $\mathbb{R}^{>0}$) with respect to the Haar measure $\frac{dx}{x}$ on $\mathbb{R}^{>0}$.</p> <p>You are probably thinking ${\zeta_p}^{ak^2}$ as the finite analogue of the Gaussian $e^{-\pi x^2}$, but as you have written yourself,</p> <p>$g_p(a)=\sum_{k=0}^{p-1} {\zeta_p}^{ak^2}$,</p> <p>a Gauss sum is a sum of `things' that look like the Gaussian and there is no reason why a Gauss sum itself should be something like the Gaussian. </p> http://mathoverflow.net/questions/10419/depressed-graduate-student/10486#10486 Answer by Idoneal for Depressed graduate student. Idoneal 2010-01-02T11:30:32Z 2010-01-02T12:54:50Z <p>A career in mathematical research is fraught with ups and downs. Instead of wavering from phases of elation and depression it is better to adopt a calm and philosophical attitude to do good work in the long run.<br /> Also, drinking beer once in a while helps.</p> http://mathoverflow.net/questions/9083/how-many-l-values-determine-a-modular-form How many L-values determine a modular form? Idoneal 2009-12-16T06:38:15Z 2009-12-30T16:30:21Z <p>Suppose $f$ and $g$ are two newforms of certain levels, weights etc. If we know that L(f,n)=L(g,n) for all sufficiently large $n$, can we conclude that $f=g$? </p> <p>Same question when the forms have the same weight and $n$ runs over critical points.</p> http://mathoverflow.net/questions/10033/everywhere-locally-isomorphic-abelian-varieties/10044#10044 Answer by Idoneal for Everywhere locally isomorphic abelian varieties Idoneal 2009-12-29T16:20:35Z 2009-12-29T16:40:47Z <p>Selmer's curve $3X^3+4y^3+5z^3=0$ is a non-example (see the comment below) but somwhat relevant. See Theorem 1 in Mazur's article titled <a href="http://www.ams.org/bull/1993-29-01/S0273-0979-1993-00414-2/S0273-0979-1993-00414-2.pdf" rel="nofollow">ON THE PASSAGE FROM LOCAL TO GLOBAL IN NUMBER THEORY</a>.</p> http://mathoverflow.net/questions/9885/the-convergence-of-eisenstein-series-of-weight-zero/9940#9940 Answer by Idoneal for The convergence of Eisenstein series of weight zero Idoneal 2009-12-28T06:26:02Z 2009-12-28T06:26:02Z <p>It seems you have started reading something from the middle.</p> <p>Hint for the first one: Do it for $SL_2(\mathbb{Z})$ first. Note that </p> <p>$Im \frac{az+b}{cz+d} = y/|cz+d|^2$.</p> http://mathoverflow.net/questions/46769/covering-the-primes-by-arithmetic-progressions/46784#46784 Comment by Idoneal Idoneal 2010-11-21T06:34:52Z 2010-11-21T06:34:52Z In such problems, it is interesting to ignore the small primes and ask what happens for all sufficiently large primes. Perhaps that is what Fedor had meant. http://mathoverflow.net/questions/42533/poles-of-kloosterman-zeta-function/44017#44017 Comment by Idoneal Idoneal 2010-10-29T11:19:36Z 2010-10-29T11:19:36Z I am not sure I understand. I am using the fact that the full modular group has no exceptional eigenvalue. Also the Gamma function should not create problem at $s=1$. http://mathoverflow.net/questions/43580/barban-davenport-halberstam-without-von-mangoldt-weights/43710#43710 Comment by Idoneal Idoneal 2010-10-27T04:25:48Z 2010-10-27T04:25:48Z It is the modern bible of analytic number theory. <a href="http://books.google.com/books?id=8i7wpzjSWrIC&amp;printsec=frontcover&amp;dq=Iwaniec-Kowalski&amp;source=bl&amp;ots=GOKNmL0Au-&amp;sig=3WgRL402_K33SpdhuE1dIj3NzqM&amp;hl=en&amp;ei=kqnHTJS0I8q3cNz79LEF&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=1&amp;ved=0CBIQ6AEwAA#v=onepage&amp;q&amp;f=false" rel="nofollow">books.google.com/&hellip;</a> http://mathoverflow.net/questions/43580/barban-davenport-halberstam-without-von-mangoldt-weights Comment by Idoneal Idoneal 2010-10-26T13:33:36Z 2010-10-26T13:33:36Z Have you seen Theorem 17.5 of Iwaniec-Kowalski? I think that is all you need. http://mathoverflow.net/questions/43580/barban-davenport-halberstam-without-von-mangoldt-weights Comment by Idoneal Idoneal 2010-10-26T10:36:38Z 2010-10-26T10:36:38Z Well said Mukherjee! http://mathoverflow.net/questions/43580/barban-davenport-halberstam-without-von-mangoldt-weights Comment by Idoneal Idoneal 2010-10-26T04:29:27Z 2010-10-26T04:29:27Z I don't know of any such cut and dried reference but I find dealing with von-Mangoldt easier than dealing with primes. http://mathoverflow.net/questions/42728/fourier-coefficient-of-a-modular-form/43016#43016 Comment by Idoneal Idoneal 2010-10-21T09:23:45Z 2010-10-21T09:23:45Z &quot;This sounds tricky. Maybe Serre even conjectured once that this never happened if $p$ was sufficiently large. Let's say he did. Are you going to contradict Serre?&quot; - I liked your style of persuasion. http://mathoverflow.net/questions/42900/refinements-of-the-riemann-hypothesis/42906#42906 Comment by Idoneal Idoneal 2010-10-21T09:07:58Z 2010-10-21T09:07:58Z Yes. <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa41/aa4118.pdf" rel="nofollow">matwbn.icm.edu.pl/ksiazki/aa/aa41/aa4118.pdf</a> http://mathoverflow.net/questions/40997/fast-computation-of-multiplicative-inverse-modulo-q/41001#41001 Comment by Idoneal Idoneal 2010-10-05T07:12:31Z 2010-10-05T07:12:31Z Such things happen. I don't like it either. http://mathoverflow.net/questions/40997/fast-computation-of-multiplicative-inverse-modulo-q/41004#41004 Comment by Idoneal Idoneal 2010-10-05T06:18:04Z 2010-10-05T06:18:04Z I see. I haven't understood the half-GCD thing yet. It seems some 2-adic computation is going on. http://mathoverflow.net/questions/40997/fast-computation-of-multiplicative-inverse-modulo-q/41004#41004 Comment by Idoneal Idoneal 2010-10-04T14:44:35Z 2010-10-04T14:44:35Z Thanks. The way I understand it, the main idea (due to Lehmer) is carrying out the usual Euclidean algorithm for computing GCD with a little modification coming from the following observation: For computing the partial quotients, we can forget the tails and look at the leading digits. So for computing the GCD of 1234 and 102, first we need to find q and r such that 1234 =102.q+r with 0&lt;r&lt;102. Now, to find q, we might as well divide 10^3 by10^2 which takes less time. http://mathoverflow.net/questions/40997/fast-computation-of-multiplicative-inverse-modulo-q/41004#41004 Comment by Idoneal Idoneal 2010-10-04T10:56:41Z 2010-10-04T10:56:41Z Well, I tried reading the reference in Google books but it seems to be somewhat complicated and moreover it is for polynomials. Is it possible explain the main idea behind the fast Euclidean algorithm in a few words or by some simple example? http://mathoverflow.net/questions/40997/fast-computation-of-multiplicative-inverse-modulo-q Comment by Idoneal Idoneal 2010-10-04T10:35:56Z 2010-10-04T10:35:56Z Ahan! very good. http://mathoverflow.net/questions/40997/fast-computation-of-multiplicative-inverse-modulo-q/41001#41001 Comment by Idoneal Idoneal 2010-10-04T09:55:41Z 2010-10-04T09:55:41Z Is it faster than Euclid? http://mathoverflow.net/questions/40997/fast-computation-of-multiplicative-inverse-modulo-q Comment by Idoneal Idoneal 2010-10-04T09:54:22Z 2010-10-04T09:54:22Z To Buzzard: I wanted to check if there is any smarter way to do it than Euclid. Is there any reason to think that nothing better is possible?