Timeline for Intuitive reasons for the existence of modular parametrizations
Current License: CC BY-SA 3.0
27 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 13, 2015 at 18:44 | comment | added | user40276 | ...See this answer too mathoverflow.net/questions/55288/… of course, the true reason lies behind the complicated proof that I don't know. | |
| Nov 13, 2015 at 18:44 | comment | added | user40276 | The fact that the modular curve is a curve is simply because the upper half plane parametrizes lattices together with a choice of a basis. Now if you mean why would someone expect that all elliptic curves can be produced as a direct summand of the jacobian of a modular curve, then you should look at the relationship between the $a_p$'s that I mentioned above. Furthermore, see this answer here mathoverflow.net/questions/35870/… (about the two ways an elliptic curve appears)... | |
| Nov 13, 2015 at 18:43 | comment | added | user40276 | I don't know if I'm understanding you. The modular curve is not an elliptic curve in general (you can more or less easily compute its genus by computing its index of the corresponding group as a subgroup of $SL_2 (\mathbb{Z})$). Maybe you mean why a moduli have a geometrical structure (and this have nothing to do with modularity). In this case, there's the general idea of an internal object inside a category. It's common to regard a bunch of information packed together as a space (for instance, the fundamental groupoid of a manifold can be given the structure of a manifold). ... | |
| Nov 13, 2015 at 8:30 | comment | added | მამუკა ჯიბლაძე | Or say why should the form of a flock of birds resemble form of a bird? | |
| Nov 13, 2015 at 7:57 | comment | added | მამუკა ჯიბლაძე | I mean, there is no reason whatsoever that shape of a thing should resemble shape of one of its elements. For example, take the Hopf fibration. From its point of view $S^2$ is a kind of moduli space of circles - to each point of $S^2$ corresponds the fibre over this point which is a circle. But there is no natural way for $S^2$ to parametrize a circle - there are no nice maps from $S^2$ to a circle at all! | |
| Nov 13, 2015 at 7:53 | comment | added | მამუკა ჯიბლაძე | @user40276 Sorry for repeatedly bothering you, I just have a feeling we keep misunderstanding each other. It would be great if you could turn your comments into an answer. What worries me now is the last phrase in the penultimate comment of yours: neither me nor Will Sawin said that the relationship between modular curves and moduli is a coincidence! The relationship I find obscure is between the shape of the modular curve and the shape of an elliptic curve: a priori they might have entirely incomparable shapes, since the latter is (in a sense) an element of the first. | |
| Nov 12, 2015 at 10:31 | comment | added | user40276 | … know that the upper half plane quotiented by the modular group parametrizes elliptic curves, it's natural to consider bigger quotients and cocycle relations about these subgroups. | |
| Nov 12, 2015 at 10:31 | comment | added | user40276 | The Hecke operators (that classically acts on the coefficients of a modular form) can be seen as acting on these functions. Lattices codifies the complex points of an elliptic curve (and hence are points of the moduli). The whole point is that a modular form is a function defined on the upper half plane satisfying some cocycle conditions. These conditions means that they're sections a line bundle on a quotient of the upper half plane. So it's natural to consider these quotients (the modular curves). I disagree that modular curves and moduli are related by just a coincidence. Once you ... | |
| Nov 12, 2015 at 10:31 | comment | added | user40276 | The motivation for starting a formalization of this relation is coincidental as I said in my first comment. People were interested in understanding the Ramanujan work about the tau function, that is the coefficients of power series expansion of the discriminant (which coincides, by the way, with the discriminant of an elliptic curve after the q-expansion) and suddenly discovered that these coefficients codified the solutions mop $p$ of certain equations. Maybe these remarks will help. Modular forms can be seen as functions on lattices. … | |
| Nov 12, 2015 at 9:46 | comment | added | მამუკა ჯიბლაძე | As if one would seek a relationship between, say, the structure of a book and the structure of a letter from a book | |
| Nov 12, 2015 at 9:44 | comment | added | მამუკა ჯიბლაძე | Yes I understand. But as the answer of Will Sawin shows this relationship is considered coincidental. And the objects it relates (the first one being moduli space of the second) makes (for me) this relationship look sort of bizarre... | |
| Nov 12, 2015 at 8:39 | comment | added | user40276 | Because one wants to relate modular forms and elliptic curves. And, as I said before, modular forms are just the global sections of a line bundle on the modular curve. Any other curve whose jacobian splits into abelian varieties have a priori no relation to modular forms. | |
| Nov 12, 2015 at 7:37 | comment | added | მამუკა ჯიბლაძე | But the question was precisely this - why should one want to choose not any variety with the property that an action of some commutative group splits out of its Jacobian an elliptic curve but specifically the variety whose points may be viewed as some structured elliptic curves? I understand that among many such varietes one has to choose "better" ones (with most economical parametrizations) but why should such better one happen to be a moduli space of things one of which is the thing we want to parametrize? | |
| Nov 12, 2015 at 4:49 | answer | added | Will Sawin | timeline score: 4 | |
| Nov 12, 2015 at 3:24 | history | edited | Myshkin |
+top level tag (nt & ag)
|
|
| Nov 11, 2015 at 21:29 | comment | added | user40276 | ... the expected value of solutions defined on $\mathbb{F}_p$, i.e, $a_p = 1+ p - |E (\mathbb{F}_p)|$) | |
| Nov 11, 2015 at 21:27 | comment | added | user40276 | Maybe I'm not understanding your question. The point is that a modular form of weight $k$ is a global section of the $k$-th tensor power of the canonical bundle on $X_0 (N)$. This is exactly what I said in my previous comment. This is important because we want to link an analytical object (modular forms) with a geometrical one (elliptic curves). The beauty of this correspondence is that the coefficients $a_p$ of the Fourier series expansion of the modular form is an invariant of the elliptic curve (the deviation from ... | |
| Nov 11, 2015 at 20:52 | comment | added | მამუკა ჯიბლაძე | Sorry maybe I did not formulate my question clearly. I meant this: whenever an action of any commutative group (not just Hecke operators) on any variety $X$ (not just a modular curve) splits out a one-dimensional abelian subvariety from the Jacobian of $X$, you get a map to an elliptic curve. Yet the case when this $X$ is a modular curve seems to be an important one; why? | |
| Nov 11, 2015 at 20:31 | comment | added | user40276 | Well, modular curve is roughly (regarding the complex points) a quotient of the upper half plane and a congruence subgroup of $SL_2 (\mathbb{Z})$. However this subgroup is exactly the subgroup that acts on modular forms. Of course, this subgroup depends on the kind of modular forms you're considering. | |
| Nov 11, 2015 at 20:21 | comment | added | მამუკა ჯიბლაძე | @user40276 This is warmer :D Jacobian of about anything is an abelian variety more or less by definition, and if a commutative group happens to act on it there is a chance that this action splits a one-dimensional abelian variety out of it. But in this picture the modular curve seems to be there only because it furnishes this commutative group action. Are there no other reasons? | |
| Nov 11, 2015 at 19:57 | comment | added | user40276 | Now, modularity asks for the opposite direction. | |
| Nov 11, 2015 at 19:57 | comment | added | user40276 | The link between power series expansion of modular forms and moduli is given by Hecke operators. They can be seen as acting on a moduli space (of elliptic curves with additional structure on torsion points of a given order) or as acting on modular forms (because modular forms are just sections of a line bundle). Shimura constructed long time ago an abelian variety $A_f$ from a modular eigenfunction $f$ (an eingenfunction of the Hecke operator) of weight $k =2$ by decomposing the Jacobian of the moduli space $J(X_0 (N)) \cong \bigoplus_f A_f$ with the action of the Hecke algebra. | |
| Nov 11, 2015 at 19:33 | comment | added | მამუკა ჯიბლაძე | @user40276 Sorry even this (and also another question linked from there) seems to be too technical for me. Besides, I would prefer staying on the geometric side of the intuition if possible, without diving into analytic subtleties like $L$-functions, etc. I don't even understand what the modular parametrization has to do with power series expansions of Fourier series. | |
| Nov 11, 2015 at 19:24 | comment | added | user40276 | I think this all started with Ramanujan (accidentally) and, then, coincidentally people (Mordell, Hecke and others) discovered that these power series expansion on Fourier series had a geometrical meaning. See, for instance, mathoverflow.net/questions/75709/… | |
| Nov 11, 2015 at 18:53 | history | edited | მამუკა ჯიბლაძე | CC BY-SA 3.0 |
deleted 10 characters in body
|
| Nov 11, 2015 at 18:47 | history | edited | მამუკა ჯიბლაძე | CC BY-SA 3.0 |
added 52 characters in body
|
| Nov 11, 2015 at 18:42 | history | asked | მამუკა ჯიბლაძე | CC BY-SA 3.0 |