User robert k - MathOverflow

Hejhal's algorithm and computational methods for non-classical Maass wave forms

2013-04-14T18:27:35Z

Hejhal's algorithm [1] was a little gadget invented in the 90's for calculating the Hecke eigenvalues and Fourier coefficients of Maass wave forms. Later, Booker, Strombergsson, and Venkatesh (BSV) [2] took Hejhal's paper and made it more viable to perform high-precision calculations.

The basic idea is to assume the Fourier expansion takes a certain form, and write down a finite linear system (where the finitude is possible due to assumptions on the final order of magnitude of desired precision). Now take a horocycle on the complex upper half plane, and use these points in the linear system, and you can almost solve it. In BSV's words, "to solve for r [the desired eigenvalue] the above linear system is repeatedly solved for two different Y-values [horocycles], successively adjusting r to make the two solution vectors as nearly equal as possible."

My question is whether such an algorithm or the germ of an idea for such an algorithm exists for non-classical Maass wave forms. What would take the analogue of horocycles, and would we have enough information to solve an analogous linear system? (Remember, Fourier expansions are a good deal more complicated in multiple dimensions)

[1] Dennis A. Hejhal. On eigenfunctions of the Laplacian for Hecke triangle groups. In Emerging applications of number theory (Minneapolis, MN, 1996), volume 109 of IMA Vol. Math. Appl., pages 291–315. Springer, New York, 1999.

[2] http://math.stanford.edu/~akshay/research/bsv.pdf

Answer by Robert K for Removable base loci for non-projective varieties

2011-12-10T03:14:46Z

Short answer: No.

In his original paper, Fujita posed the question whether we can weaken the assumptions in the theorem. (Fujita 1983, 1.16) There has been one paper written since then with an attempt at improvement.

Let $R$ be a commutative Noetherian ring, $X$ a scheme proper over $R$, and $\mathcal{L}$ a line bundle on $X$. Define

$H^p_*(\mathcal{F},\mathcal{L}) = H^p(X, \mathcal{F}\otimes\text{Sym}\ \mathcal{L})=\bigoplus_{n\geq0} H^p(X,\mathcal{F}\otimes \mathcal{L}^{\otimes n})$, a module over the graded ring $\Gamma_*(\mathcal{L}) = H^0(X,\text{Sym} \ \mathcal{L})$.

Then there is

Theorem. (Schröer, 2001) Let $B$ be the stable base locus of $\mathcal{L}$. The following are equivalent:

$\mathcal{L}$ is semi-ample,
For each coherent sheaf $\mathcal{F}$ and $p \geq 0$, $H^p_*(\mathcal{F},\mathcal{L})$ is finitely generated over $\Gamma_*(\mathcal{L})$.
For all coherent ideal sheaves $\mathcal{I} \subset \mathcal{O}(B)$, the module $H^1_*(\mathcal{I}, \mathcal{L})$ is finitely generated over $\Gamma_*(\mathcal{L})$.

Schröer uses this generalization of Zariski-Fujita to characterize contractible curves in 1-dimensional families, where considers $X$ complete (i.e., proper over a base field) in a couple instances.

I think any other attempt will run into the same problem: the best you can do is a cohomological characterization.

Keeler (2003) worked in the setting of schemes with filters of line bundles, rather than a single individual line bundle. (This is again a cohomological condition.) He proved a generalization of Serre's vanishing theorem and some other results in Fujita's paper, but not a very promising lead on your question:

Proposition. (Keeler, 2003) Let $X$ be a complete variety, $\mathcal{L}$ a line bundle on $X$, and suppose the base locus is zero-dimensional or empty. Then $\mathcal{L}$ is nef.

There is also the paper of Ein (2000) where he finds a more elegant proof of Zariski-Fujita using Koszul complexes.

Ein, L., Linear systems with removable base loci, Comm. Algebra 28, n. 12 (2000), 5931–-5934
Fujita, Takao, Semipositive line bundles, J. Fac. Sci. Univ. Tokyo 30 (1983), 353--378
Keeler, Dennis S, Ample filters of invertible sheaves, J. Algebra 259 (2003), 243--283
Schroer, Stefan, A characterization of semiampleness and contractions of relative curves, Kodai Math. J. Volume 24, Number 2 (2001), 207--213.

Answer by Robert K for Mathematical Advice for Interested Highschool Students

2011-07-26T18:47:38Z

Any of the books in the Elementary section of Chris Jeris's undergraduate bibliography will be fun for a talented high schooler to look at.

Answer by Robert K for componentwise injective quasi coherent sheaves

2011-03-13T13:01:21Z

The condition you want is $X$ a locally noetherian scheme. Then by Hartshorne's "Residues & Dualities," Proposition 7.17, $\cal{F}$ is an injective ${\cal O}_X$-module if and only if for each $x \in X$, the stalks ${\cal F}_x$ are injective ${\cal O}_x$-modules. If the sections are injective ${\cal O}_X(U)$-modules, that should give injectivity on the stalks (abelian groups form a locally noetherian Grothendieck category, so use e.g., Henning Krause's "The Spectrum of a Module Category" Proposition A.11, which says direct limits of injective objects are injective). For the reverse question, I think you need $X$ to be noetherian.

Answer by Robert K for learning crystalline cohomology

2011-02-27T00:19:01Z

If you don't mind reading mimeographed notes, the exposition by Berthelot and Ogus is certainly worth taking a look at. It has the added advantage of only assuming a minimal background in algebra and algebraic geometry. If you find the DJVU version through Google, every other page is slightly cut off at the right, but this did not interfere with my reading of the document.

After you learn the subject, perhaps if you're feeling ambitious enough to learn some p-adic Hodge theory from Gabber and Ramero's long treatise (admittedly only lengthy because it starts from fundamentals), you could tell me if it treats the crystalline topoi--I have only skimmed it and it seems to only consider Zariski and etale sites.

Good luck. (Addendum to Pete Clark: Feeling a little jealous here ;)

Comment by Robert K

2013-04-18T19:49:23Z

Thank you! Do you know of a reference for the SL(3, Z) case?

Comment by Robert K

2013-04-15T18:32:26Z

You're right that the term I used is shaky, but let's say anything non-holomorphic. For example, in the case of GSp4 we could refer to modular forms associated to generic representations gsp4.org/cat-browser.php?categoryID=199 Thanks for the response so far!

Comment by Robert K

2013-04-14T20:07:33Z

Thanks, I got blindsighted by the fact it was hosted on his website.

Comment by Robert K

2012-11-23T20:32:06Z

Isn't the transpose its own inverse?

Comment by Robert K

2011-09-08T23:16:27Z

For a while, I thought Piatetski-Shapiro...

Comment by Robert K

2011-08-24T10:12:56Z

I assume you mean on the whole complex plane. The power series coefficients come from C, so there are very few meromorphic functions with integer coefficients. Certainly 1/z^k * f(z) for f(z) = e(z), sin(z), cos(z), but it's hard to say something specific. Can you expound on your question?

Comment by Robert K

2011-08-21T01:35:15Z

i.e. geometry without specifying a base

Comment by Robert K

2011-08-20T23:17:26Z

Seems like a homework problem. You may have more luck posting this on math.stackexchange

Comment by Robert K

2011-04-23T20:04:16Z

Is this exercise in a book somewhere, so I can refresh my memory on how to go about solving it?

Comment by Robert K

2011-04-23T19:54:31Z

No. Any explicit construction uses the trace map, which is taken over Q_p in char 0 and over the prime subfield F_p in char p. The former bears no significance as abstract or topological field like the prime subfield does. Hence, most authors are content to just state "fix an additive character"; the choice is usually of no practical importance.

Comment by Robert K

2011-04-23T19:13:43Z

Could you make your question more precise? The L ∞ norm for an arbitrary measure space is the essential supremum, inf({a >= 0 : mu({x : |f(x)| > a}) = 0}). For mu = P, the restriction P(Omega) = 1 does not change this definition.

Comment by Robert K

2011-03-13T15:24:26Z

@Pietro With the exception of Ehresmann and his school. :-)