Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Computing images of Galois representations attached to elliptic curves   Let E be an elliptic curve without complex multiplication (CM) over a number
field K, and let G_E(ell) be the image of the Galois representation induced by
the action of the absolute Galois group of K on the ell-torsion subgroup of E.
We present two probabilistic algorithms to simultaneously determine G_E(ell) up
to local conjugacy for all primes ell by sampling images of Frobenius elements;
one is of Las Vegas type and the other is a Monte Carlo algorithm. They
determine G_E(ell) up to one of at most two isomorphic conjugacy classes of
subgroups of GL_2(Z/ell Z) that have the same semisimplification, each of which
occurs for an elliptic curve isogenous to E. Under the GRH, their running times
are polynomial in the bit-size n of an integral Weierstrass equation for E, and
for our Monte Carlo algorithm, quasi-linear in n. We have applied our
algorithms to the non-CM elliptic curves in Cremona's tables and the
Stein--Watkins database, some 140 million curves of conductor up to 10^10,
thereby obtaining a conjecturally complete list of 63 exceptional Galois images
G_E(ell) that arise for E/Q without CM. Under this conjecture we determine a
complete list of 160 exceptional Galois images G_E(ell) the arise for non-CM
elliptic curves over quadratic fields with rational j-invariants. We also give
examples of exceptional Galois images that arise for non-CM elliptic curves
over quadratic fields only when the j-invariant is irrational.

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The categories of this paper are: ao-sci
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The categories of this paper are: adap-org
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The categories of this paper are: cs
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The categories of this paper are: hep-ph
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The categories of this paper are: q-fin
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The categories of this paper are: econ
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The categories of this paper are: astro-ph
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The categories of this paper are: solv-int
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The categories of this paper are: comp-gas
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The categories of this paper are: math-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-lat
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The categories of this paper are: gr-qc
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The categories of this paper are: cmp-lg
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The categories of this paper are: plasm-ph
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The categories of this paper are: stat