User idoneal - MathOverflow

A sentence in Shimura's "On The Periods of Modular Forms"

2012-11-22T08:45:01Z

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.

I don't see how it follows. I shall appreciate any comment.

Answer by Idoneal for How should an analytic number theorist look at Bessel functions?

2012-08-30T19:46:52Z

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).

For manipulations, the formulae given in Appendix B of Iwaniec's Spectral Methods book usually suffice.

Yes, Dan Brown is more appropriate than Gradhsteyn and Rizhik for airport reading.

Riemann surfaces: explicit algebraic equations

2009-12-06T11:07:16Z

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?

Somehow I couldn't add a comment so let me write it here.

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 ?

Answer by Idoneal for Most memorable titles

2010-11-01T05:29:33Z

Ideals and reality

projective modules and number of generators of ideals

By Friedrich Ischebeck and Ravi A. Rao

Answer by Idoneal for Complex Analysis applications toward Number Theory

2010-10-25T08:41:59Z

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.

Answer by Idoneal for Refinements of the Riemann hypothesis

2010-10-20T16:18:16Z

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 http://www.ams.org/journals/bull/1999-36-01/S0273-0979-99-00766-1/home.html

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.

Answer by Idoneal for Poles of Kloosterman Zeta Function

2010-10-18T10:57:14Z

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.

Fast computation of multiplicative inverse modulo q

2010-10-04T09:05:57Z

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 $q$?

Modular forms with prime Fourier coefficients zero

2010-05-21T05:50:19Z

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 $p$?

If such a form cannot exist then why?

Answer by Idoneal for Is there an image for you that epitomizes mathematics?

2010-06-02T08:54:42Z

Here is a more mathematical rendition of Richard Kent's answer:

http://math-art.net/2007/12/03/eternal-scream-a-droste-effect/

Are most cubic plane curves over the rationals elliptic?

2010-01-10T16:59:58Z

%This is a new version of the original question modified in the light of the answers and comments.

The word 'most' in the title is ambiguous. The following is one way of making it precise.

Question1: (This seems to be open. See Poonen's answer below)

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 $N\rightarrow \infty$?

Should the limit exist and if it does, should it be one, zero, or some other number?

Another question of interest is:

Question 2: (There is a satisfactory answer to this. See Voloch's response below.)

Are either of the sets {cubics with no rational point} and {cubics with at least one rational point} Zariski dense?

Answer by Idoneal for Does p-adic $L$- function determine the $L$ function

2010-04-12T12:38:58Z

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.

Conductor of monomial forms with trivial nebentypus

2010-03-08T11:16:18Z

Is it true that the conductor of a holomoprhic or a Maass cusp form with trivial nebentypus corresponding to a two-dimensional dihedral representation (over $\mathbb{Q}$ )is non-square-free?

Answer by Idoneal for modular eigenforms with integral coefficients [Maeda's Conjecture]

2010-01-14T05:22:07Z

This is a (far too long) comment on Buzzard's comment about Hida's remark.

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.

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).

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).

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.

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

$L(f,k/2)=\langle f,G \rangle$

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.

Answer by Idoneal for What's the relationship between Gauss sums and the normal distribution?

2010-01-09T11:27:33Z

I don't think there is anything deep going on here. The Fourier analysis on finite abelian group is fairly straightforward.

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 $\mathbb{R}^{>0}$.

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,

$g_p(a)=\sum_{k=0}^{p-1} {\zeta_p}^{ak^2}$,

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.

Answer by Idoneal for Depressed graduate student.

2010-01-02T11:30:32Z

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.
Also, drinking beer once in a while helps.

How many L-values determine a modular form?

2009-12-16T06:38:15Z

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 $f=g$?

Same question when the forms have the same weight and $n$ runs over critical points.

Answer by Idoneal for Everywhere locally isomorphic abelian varieties

2009-12-29T16:20:35Z

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 ON THE PASSAGE FROM LOCAL TO GLOBAL IN NUMBER THEORY.

Answer by Idoneal for The convergence of Eisenstein series of weight zero

2009-12-28T06:26:02Z

It seems you have started reading something from the middle.

Hint for the first one: Do it for $SL_2(\mathbb{Z})$ first. Note that

$Im \frac{az+b}{cz+d} = y/|cz+d|^2$.

Comment by Idoneal

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.

Comment by Idoneal

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 $s=1$.

Comment by Idoneal

2010-10-27T04:25:48Z

It is the modern bible of analytic number theory. books.google.com/…

Comment by Idoneal

2010-10-26T13:33:36Z

Have you seen Theorem 17.5 of Iwaniec-Kowalski? I think that is all you need.

Comment by Idoneal

2010-10-26T10:36:38Z

Well said Mukherjee!

Comment by Idoneal

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.

Comment by Idoneal

2010-10-21T09:23:45Z

"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?" - I liked your style of persuasion.

Comment by Idoneal

2010-10-21T09:07:58Z

Yes. matwbn.icm.edu.pl/ksiazki/aa/aa41/aa4118.pdf

Comment by Idoneal

2010-10-05T07:12:31Z

Such things happen. I don't like it either.

Comment by Idoneal

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.

Comment by Idoneal

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<r<102. Now, to find q, we might as well divide 10^3 by10^2 which takes less time.

Comment by Idoneal

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?

Comment by Idoneal

2010-10-04T10:35:56Z

Ahan! very good.

Comment by Idoneal

2010-10-04T09:55:41Z

Is it faster than Euclid?

Comment by Idoneal

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?