Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Higher Eisenstein elements, higher Eichler formulas and rank of Hecke
  algebras   Let $N$ and $p$ be primes such that $p$ divides the numerator of
$\frac{N-1}{12}$. In this paper, we study the rank $g_p$ of the completion of
the Hecke algebra acting on cuspidal modular forms of weight $2$ and level
$\Gamma_0(N)$ at the $p$-maximal Eisenstein ideal. We give in particular an
explicit criterion to know if $g_p \geq 3$, thus answering partially a question
of Mazur.
  In order to study $g_p$, we develop the theory of \textit{higher Eisenstein
elements}, and compute the first few such elements in four different Hecke
modules. This has applications such as generalizations of the Eichler mass
formula in characteristic $p$.

37
The categories of this paper are: supr-con
20
The categories of this paper are: acc-phys
22
The categories of this paper are: chao-dyn
18
The categories of this paper are: econ
34
The categories of this paper are: chem-ph
19
The categories of this paper are: eess
7
The categories of this paper are: hep-th
36
The categories of this paper are: cmp-lg
10
The categories of this paper are: q-bio
3
The categories of this paper are: physics
5
The categories of this paper are: gr-qc
31
The categories of this paper are: ao-sci
27
The categories of this paper are: comp-gas
15
The categories of this paper are: nucl-ex
12
The categories of this paper are: nucl-th
17
The categories of this paper are: q-fin