Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On congruences mod ${\mathfrak p}^m$ between eigenforms and their
  attached Galois representations   Given a prime $p$ and cusp forms $f_1$ and $f_2$ on some $\Gamma_1(N)$ that
are eigenforms outside $Np$ and have coefficients in the ring of integers of
some number field $K$, we consider the problem of deciding whether $f_1$ and
$f_2$ have the same eigenvalues mod ${\mathfrak p}^m$ (where ${\mathfrak p}$ is
a fixed prime of $K$ over $p$) for Hecke operators $T_{\ell}$ at all primes
$\ell\nmid Np$.
  When the weights of the forms are equal the problem is easily solved via an
easy generalization of a theorem of Sturm. Thus, the main challenge in the
analysis is the case where the forms have different weights. Here, we prove a
number of necessary and sufficient conditions for the existence of congruences
mod ${\mathfrak p}^m$ in the above sense.
  The prime motivation for this study is the connection to modular mod
${\mathfrak p}^m$ Galois representations, and we also explain this connection.

13
The categories of this paper are: quant-ph
35
The categories of this paper are: mtrl-th
29
The categories of this paper are: funct-an
25
The categories of this paper are: patt-sol
19
The categories of this paper are: eess
22
The categories of this paper are: chao-dyn
36
The categories of this paper are: cmp-lg
23
The categories of this paper are: solv-int
27
The categories of this paper are: comp-gas
4
The categories of this paper are: cond-mat
28
The categories of this paper are: alg-geom
14
The categories of this paper are: math-ph
1
The categories of this paper are: cs
18
The categories of this paper are: econ
37
The categories of this paper are: supr-con
8
The categories of this paper are: hep-ex