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Constructing elliptic curves from Galois representations
- Part of
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- Journal:
- Compositio Mathematica / Volume 154 / Issue 10 / October 2018
- Published online by Cambridge University Press:
- 29 August 2018, pp. 2045-2054
- Print publication:
- October 2018
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- Article
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Given a non-isotrivial elliptic curve over an arithmetic surface, one obtains a lisse
$\ell$-adic sheaf of rank two over the surface. This lisse sheaf has a number of straightforward properties: cyclotomic determinant, finite ramification, rational traces of Frobenius elements, and somewhere not potentially good reduction. We prove that any lisse sheaf of rank two possessing these properties comes from an elliptic curve.
