Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Ramification in Division Fields and Sporadic Points on Modular Curves   Consider an elliptic curve $E$ over a number field $K$. Suppose that $E$ has
supersingular reduction at some prime $\mathfrak{p}$ of $K$ lying above the
rational prime $p$. We completely classify the valuations of the $p^n$-torsion
points of $E$ by the valuation of a coefficient of the $p^{\text{th}}$ division
polynomial. We apply this description to find the minimum necessary
ramification at $\mathfrak{p}$ in order for $E$ to have a point of exact order
$p^n$.
  Using this bound we show that sporadic points on the modular curve $X_1(p^n)$
cannot correspond to supersingular elliptic curves without a canonical
subgroup. We generalize our methods to $X_1(N)$ with $N$ composite.

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The categories of this paper are: eess
20
The categories of this paper are: acc-phys
31
The categories of this paper are: ao-sci
9
The categories of this paper are: nlin
26
The categories of this paper are: bayes-an
8
The categories of this paper are: hep-ex
0
The categories of this paper are: hep-ph
15
The categories of this paper are: nucl-ex
33
The categories of this paper are: plasm-ph
24
The categories of this paper are: dg-ga
32
The categories of this paper are: atom-ph
3
The categories of this paper are: physics
22
The categories of this paper are: chao-dyn
28
The categories of this paper are: alg-geom
14
The categories of this paper are: math-ph
21
The categories of this paper are: adap-org