2
votes
1answer
45 views

Decomposition of modular group elements

The modular group $PSL_2(\mathbb{Z})$ acts on the hyperbolic half-space $H$ by $$h\cdot z=\frac{az+b}{cz+d},\;z\in H,\;h=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in PSL_2(\mathbb{Z})$$ with ...
2
votes
1answer
48 views

Why a modular form is a highest weight vector of a discrete series summand of $L_2(SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R}))$?

It is said that a modular form is a highest weight vector of a discrete series summand of $L_2(SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R}))$. Why a modular form is a highest weight vector of a ...
3
votes
1answer
106 views

Real life applications of Maass wave forms

Explaining my work on Maass wave forms to friends and family (all non-mathematician) typically earns me blank faces. So I wonder whether there is some good example to explain their meaning to laymen. ...
3
votes
2answers
121 views

How important are automorphic representations among admissible ones?

I'm currently studying automorphic representations on Bump's book "Automorphic forms and representations" and on Gelbart's "Automorphic forms on Adele groups". And I have some problems in ...
0
votes
1answer
53 views

Problem about decomposition of modular space into eigenspace

It is said we can use the operator $$ \pi_\chi=\frac{1}{\phi(N)}\sum_{d\in\mathbb{Z}_N^*}\chi(d)^{-1}\langle d\rangle $$ to project function in $\mathcal{M}_k(\Gamma_1(N))$ into the $\chi-$eigenspace ...
8
votes
1answer
110 views

Flatness of residual representations associated to modular forms

Let $f\in S_k(\Gamma_1(N),\chi)$ be a Hecke eigenform of weight $k\geq 2$, $p$ an odd prime not dividing $N$, and $K_f$ the number field generated by the Hecke eigenvalues of $f$. Fixing a prime ...
7
votes
1answer
177 views

How can I determine in practice whether two elliptic curves over $\mathbb{Q}$ have isomorphic $p$-torsion?

Let $E_1$ and $E_2$ be elliptic curves over $\mathbb{Q}$ with good, ordinary reduction at an odd prime $p$. I'm wondering how to determine whether $E_1[p]$ and $E_2[p]$ are isomorphic ...
15
votes
2answers
277 views

Proving finite dimensionality of modular forms using representation theory?

It is well known how to use algebraic geometry (differentials, divisors, and Riemann-Roch) in order to prove the finite dimensionality of the vector space of modular forms of some fixed weight and ...
5
votes
1answer
266 views

Connection between Hecke operators and Hecke algebras

Hecke operators are things that act on modular forms and give rise to a lot of interesting arithmetical results: http://en.wikipedia.org/wiki/Hecke_operator On the other hand on the wikpedia page ...