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Results tagged with modular-forms
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user 1384
Questions about modular forms and related areas
18
votes
Why are modular forms (usually) defined only for congruence subgroups?
The main point is that the basic definitions work fine but the link with arithmetic is much more "vague". Look at early papers of Tony Scholl. There are Galois representations attached to certain non- …
- 40.3k
9
votes
Accepted
If $f$ is an $p$-nonordinary eigenform of weight $k\leqslant p+1$ are there always two eigen...
It's deeper than theta cycles, I think.
I am going to assume that $N$ is prime to $p$ -- you don't say this in your question but most of my answer assumes this in a very serious way.
If $f$ is ordin …
- 40.3k
13
votes
Accepted
Can a the q-expansion of a p-adic modular form be a non-constant polynomial?
It is. I want to argue the following way: if the polynomial is non-constant then after scaling it has integral coefficients and so the reduction of the p-adic form mod p^n will be a classical form who …
- 40.3k
4
votes
SL(2,Z/N)-decomposition of space of cusp forms for Gamma(N)
If you think about this question in terms of automorphic representations then it sort of becomes trivial. The space $Sk(\Gamma(N))$ can be re-interpreted as the direct sum of $\pi^{U(N)}$, where $\pi$ …
- 40.3k
3
votes
How many L-values determine a modular form?
I think the answer to your second question is "no". For example if $k=2$ and $f$ and $g$ correspond to elliptic curves over $Q$ with positive rank, then the only critical point is $s=1$ and (at least …
- 40.3k
19
votes
2
answers
576
views
Can something finite over $\mathbb{C}(q)$ be a modular form?
If $f\in\mathbf{C}[[q]]$ is non-constant, and algebraic over $\mathbf{C}[q]$ (in the sense that it is a root of a polynomial with coefficients in in $\mathbf{C}[q]$) then can $f$ be the $q$-expansion …
- 40.3k
17
votes
Accepted
Why is there a weight 2 modular form congruent to any modular form
By "level $\ell$" I assume you mean "level $\Gamma_1(\ell)$".
Here's a proof. By the Eichler-Shimura theorem, the system of eigenvalues associated to the modular form shows up in $H^1(SL(2,\mathbf{Z} …
- 40.3k
11
votes
Accepted
Hilbert Modular Newforms
If I've understood your question correctly, you're right that $C(q,f)\not=0$ always and there is a natural representation-theoretic proof of this result (before I start let me say that I don't know ho …
- 40.3k
15
votes
Hecke algebra generated by a single element
[I took the time to chase this up so may as well post it as an answer.]
There is a (cuspidal) modular (eigen)form of level $\Gamma_0(512)$ and weight 2, which if I remember correctly was shown to me …
- 40.3k
8
votes
Number of modular lifts with prescribed parameters
I can give you a "formula" in the sense that I can give you an algorithm to compute the number in any given case. If $\ell\not=p$ is prime then an old result of Carayol and Livn\'e says that the condu …
- 40.3k
10
votes
Accepted
Level raising by prime powers
Presumably you want the form (let me call it g) of level Np^3 to be new at p, otherwise it's trivial.
Let me also assume ell isn't p.
If the form g is new at p, and has level Gamma0(p^3) at p, then …
- 40.3k
15
votes
Accepted
Galois representations attached to newforms
The right way to do this sort of question is to apply Saito's local-global theorem, which says that the (semisimplification of the) Weil-Deligne representation built from $D_{pst}(\rho_{f,p})$ by forg …
- 40.3k
4
votes
Accepted
Are Fredholm hypersurfaces affinoid?
No they're not in general affinoid. The problem is that the zero locus of the power series is computed within a space which is almost never affinoid -- for example in the modular curve case the ambien …
- 40.3k
5
votes
Image of complex conjugation by modular representations in characteristic 2
Joel -- it's difficult to work out what you're asking. Of course both possibilities can occur, as Wanax said. Furthermore both possibilities can occur even for the same modular form. For example, if y …
- 40.3k
10
votes
1
answer
515
views
Example of a non-smooth irreducible component of the generic fibre of a Hida family?
Is there a known example of a non-smooth irreducible component of the rigid generic fibre of a Hida family?
Let me explain some of the context around this question (but I'm not going to explain Hida …
- 40.3k
18
votes
Why does the definition of modularity demand weight 2?
There has been a lot written already about this question. but here is a simple answer. The Hodge--Tate weights of the Tate module of an elliptic curve are 0 and 1. The Hodge--Tate weights of the Galoi …
- 40.3k
25
votes
Accepted
Relation between Hecke Operator and Hecke Algebra
The fact that Hecke operators (double coset stuff coming from $SL_2(\mathbf{Z})$ acting on modular forms) and Hecke algebras (locally constant functions on $GL_2(\mathbf{Q}_p)$) are related has nothin …
- 40.3k
8
votes
Why are functional equations important?
I am surprised no-one seems to have mentioned one key use for functional equations: they are a key input in converse theorems. If you have a power series in q that you're trying to show is a modular f …
- 40.3k
4
votes
Modular forms reference
Just to add one more thing to what Pete said: the variety A_f that one normally attaches to f might have endomorphism ring bigger than an order in the coefficient field of f: for example if E is an el …
- 40.3k
2
votes
Companion forms
It might all depend on precisely what you mean by Serre's conjecture. Various versions are in print. Serre's original conjecture stayed away from $k=1$ and K-W resolved this version of the conjecture …
- 40.3k
7
votes
Accepted
Effective detection of CM modular forms
If the form is CM then it will be isomorphic to a quadratic twist of itself. So I think what I'd do with a form which I suspect is or is not CM is to just twist by all the (finitely many ) possible qu …
- 40.3k
11
votes
Accepted
Consequences of the geometric properties of the eigencurve
The eigencurve is an honest moduli space---it parametrises families of finite slope overconvergent modular eigenforms (or more precisely, of systems of overconvergent finite slope Hecke eigenvalues)-- …
- 40.3k
13
votes
Accepted
Does anyone want a pretty Maass form?
[these are comments, not an answer, but there were too many for the comments box]
Hey---I wrote that code too! I did it to teach myself "practical Maass forms". I wrote in pari, not sage. I didn't do …
19
votes
6
answers
2k
views
weight 4 eigenforms with rational coefficients---is it reasonable to expect they all come fr...
A weight 2 modular form which happens to be a normalised cuspidal eigenform with rational coefficients has a natural geometric avatar---namely an elliptic curve over the rationals. It seems to be a su …
- 40.3k
27
votes
2
answers
2k
views
How to explicitly compute lifting of points from an elliptic curve to a modular curve?
Say $E$ is an elliptic curve over the rationals, of conductor $N$. There's a covering of $E$ by the modular curve $X_0(N)$, and if you rig it right then you can define this map over $\mathbf{Q}$: ther …
- 40.3k
26
votes
2
answers
2k
views
Are there any Hecke operators acting on an elliptic curve with additive reduction that I don...
I could have made this question very brief but instead I've maximally gone the other way and explained a huge amount of background. I don't know whether I put off readers or attract them this way. The …
- 40.3k
46
votes
Are there mistakes in the proof of FLT?
No there are not any mistakes in these papers of any interest. In the 1990s there were a bazillion study groups and seminars across the world devoted to these papers; I personally read all three of th …
- 40.3k
13
votes
$A_5$-extension of number fields unramified everywhere
Oh, I know how I would try and build examples. First I would write down a random $A_5$ extension $K$ of $\mathbf{Q}$, ramified at some primes (in fact I would look in a table, e.g. in Buhler's thesis …
- 40.3k
46
votes
Accepted
Are there Maass forms where the expected Galois representation is $\ell$-adic?
Here's some piece of the bigger picture. Maass forms and holomorphic modular forms are both automorphic representations for $GL(2)$ over the rationals. An automorphic representation is a typically hug …
- 40.3k
29
votes
Intuition behind the Eichler-Shimura relation?
Let me highlight some issues that Emerton doesn't:
1) you seem to hint that you don't know that modular forms can be viewed as a product of a bunch of local terms. So there is an adelic story, where …
- 40.3k