Display L-function factors

[edgarcosta]

How should we display the factorization of the L-function?

At the moment we display them on a gaga box on the side: http://cmfs.lmfdb.xyz/L/ModularForm/GL2/Q/holomorphic/13/12/a/a/ The example above looks fine, as we only have one instance of each factor.

However, whenever we have more than on instance things look weird: http://cmfs.lmfdb.xyz/L/EllipticCurve/2.0.8.1/2592.3/c/ http://cmfs.lmfdb.xyz/L/EllipticCurve/2.0.3.1/75.1/a/ http://cmfs.lmfdb.xyz/L/Genus2Curve/Q/196/a/

I'm open to suggestions.

[davidfarmer]

When it is an L-functions home page, the factors should be L-functions, not geometric objects. The names of those L-function may be difficult to fit into the Gaga box, so probably the full names can't be listed there.

Or maybe those could be written as L(Isogeny class 288.a, s)

A confusing thing is that when the sources are listed as factors, there can be repeats (modular forms and elliptic curves, for example), so it is not actually a list of factors. Is there a way to group them into equivalence classes of objects with the same L-function?

I like the way the factors are shown on some Dedekind zeta-function pages, such as:

http://beta.lmfdb.org/L/NumberField/2.0.3.1/ (scroll down below Euler product). Those factors are not repeated in the properties box.

That approach has the advantage of handling cases with repeated factors.

On Thu, 11 Oct 2018, Edgar Costa wrote:

How should we display the factorization of the L-function?

At the moment we display them on a gaga box on the side: http://cmfs.lmfdb.xyz/L/ModularForm/GL2/Q/holomorphic/13/12/a/a/ The example above looks fine, as we only have one instance of each factor.

However, whenever we have more than on instance things look weird: http://cmfs.lmfdb.xyz/L/EllipticCurve/2.0.8.1/2592.3/c/ http://cmfs.lmfdb.xyz/L/EllipticCurve/2.0.3.1/75.1/a/ http://cmfs.lmfdb.xyz/L/Genus2Curve/Q/196/a/

I'm open to suggestions.

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[edgarcosta]

I agree with you on both counts.

If we move towards the formats in Dedekind zeta-functions, we still need to pick an L-function to link. Which one? We could perhaps pick the generic one: http://cmfs.lmfdb.xyz/L/Lhash/3408724833556919645538325794380/

In terms of formating, then I would say we should list the factorization above the euler product, as that usually takes a lot of vertical space.

[davidfarmer]

If you can link to the generic L-function, and it knows about its sources, then I think that makes sense.

I agree that the factorization of non-primitive L-functions should be prominent. Would it be silly to move that to the top of the page?

On Thu, 11 Oct 2018, Edgar Costa wrote:

I agree with you on both counts.

If we move towards the formats in Dedekind zeta-functions, we still need to pick an L-function to link. Which one? We could perhaps pick the generic one: http://cmfs.lmfdb.xyz/L/Lhash/3408724833556919645538325794380/

In terms of formating, then I would say we should list the factorization above the euler product, as that usually takes a lot of vertical space.

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[edgarcosta]

It doesn't look bad at all:

It could also show it under "invariants".

A side question is what to do regarding a large number of factors? eg http://cmfs.lmfdb.xyz/L/ModularForm/GL2/Q/holomorphic/99/2/p/a (this page already looks horrible in many ways)

[davidfarmer]

I like it, and I like the fact that it immediately shows it is non-primitive. Once you know that, the factors contain the key information.

If there are a lot of factors we just need to use multiple rows, with maybe a half-dozen factors per row.

On Thu, 11 Oct 2018, Edgar Costa wrote:

It doesn't look bad at all: screen shot 2018-10-11 at 16 13 56

It could also show it under "invariants".

A side question is what to do regarding a large number of factors? eg http://cmfs.lmfdb.xyz/L/ModularForm/GL2/Q/holomorphic/99/2/p/a (this page already looks horrible in many ways)

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[edgarcosta]

I have now renamed the box from Factors to Origins of factors, and also removed duplicates, see:

I will later on work on displaying the factorization as in Dedekind case

[davidlowryduda]

I had also envisioned using the generic L-functions. In fact, I think that displaying how L-functions factor was precisely the original reason for which the lhash pages were originally constructed.

[davidfarmer]

The way I think of it is:

The lhash tells you when a new L-function happens to be an L-function you have seen previously. (For example, an elliptic curve over a real quadratic field having the same L-function as an elliptic curve over an imaginary quadratic field.)

The first zeros tell you how it factors.

What we still need is a mapping to a mathematically meaningful URL for the L-function -- independent of its source. This has been discussed several times but not implemented. I am in the midst of doing this for Maass-type L-functions.

On Fri, 12 Oct 2018, David Lowry-Duda wrote:

I had also envisioned using the generic L-functions. In fact, I think that displaying how L-functions factor was precisely the original reason for which the lhash pages were originally constructed.

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[davidlowryduda]

So that I understand, you don't consider the lhash to be a mathematically meaningful URL, right? It definitely carries some mathematical meaning, but it's not at all human-understandable. A better URL is desirable, but I don't know the solution (even though I think I've butted in at the tail end of a few discussions in the past).

What sort of mathematically meaningful URL structure are you making for Maass-type L-functions?

[AndrewVSutherland]

@davidlowryduda if you are referring to the hash of a rational L-function based on taking a linear combination of traces modulo a large prime (as is done for genus 2 curves, for example), this is mathematically meaningful but it is not a suitable URL because it is not necessarily unique and it only applies to L-functions with rational Dirichlet coefficients.

[davidfarmer]

This URL has some of the right ingredients:

http://beta.lmfdb.org/L/ModularForm/GL3/Q/Maass/4/1/9.632444_1.374060/0.15012282/

The problem is similar to every other URL decision: that object has invariants, and there is a sense in which some invariants are more basic and others less so. After listing invariants in decreasing order of importance, you may have to resort to something like lexicographic order or an arbitrary counter.

For L-functions, the most important invariants, in decreasing order of importance, are

degree conductor central character Gamma-factor shape

(The Gamma-factor shape determines the degree, but I think it is important o have both. I would consider putting Gamma-factor shape immediately after degree, but for now I leave it that way.)

For very small degree L-functions, those 4 things might reduce you to a finite set, but in general they do not.

The later invariants are less obvious. Invariants worth considering are

the imaginary parts of the Gamma-shifts sign of the functional equation is is self-dual? is it algebraic? does it have rational integer coefficients?

Once you specify the imaginary parts of the Gamma-shifts, then (conjecturally) you are down to a finite set. Algebraic L-functions are (conjecturally) characterized by all of the imaginary parts of the Gamma-shifts are 0. For transcendental L-functions, the Gamma-shifts are a discrete set, so once you specify those shifts to enough decimals, you have specified them completely and are down to a finite set.

The degree 3 L-function at the link above could live at

http://beta.lmfdb.org/L/3/4/1/r0r0r0/9.63_1.37_-11.00/

That URL does not reveal that the sign of the functional equation is e^{2 Pi i/3}. Note that there is redundant information: the sum of the imaginary parts of the shifts is 0.

The L-function of the conductor 11 elliptic curve could be

http://beta.lmfdb.org/L/2/11/1/c1/

The L-function of the conductor 35 elliptic curve could be

http://beta.lmfdb.org/L/2/35/1/c1/something

where "something" distinguishes that L-function from the other two with the same initial date (the space of newforms of weight 2 and level 35 is 3. If there were modular forms of weight 2 with levels 5 and 7, then there also would be non-primitive L-functions with that same functional equation).

I don't quite have this to the point where I am making a specific proposal. Some inconsistencies bother me: should the imaginary shifts be mentioned even when they are not strictly needed? Should the fact that they sum to 0 be used? Should that conductor 11 L-function have a "0" at the end of its so that it and the conductor 35 example have URLS of the same shape?

I am trying to talk myself into putting the Gamma-factor shape immediately after the degree. That makes sense in terms of how people would browse.

On Fri, 12 Oct 2018, David Lowry-Duda wrote:

So that I understand, you don't consider the lhash to be a mathematically meaningful URL, right? It definitely carries some mathematical meaning, but it's not at all human-understandable. A better URL is desirable, but I don't know the solution (even though I think I've butted in at the tail end of a few discussions in the past).

What sort of mathematically meaningful URL structure are you making for Maass-type L-functions?

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