Locally potentially equivalent two dimensional Galois representations and Frobenius fields of elliptic curves

Manisha Kulkarni; Vijay M. Patankar; C. S. Rajan

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arXiv · 1403.5635

Locally potentially equivalent two dimensional Galois representations and Frobenius fields of elliptic curves

Manisha Kulkarni·Vijay M. Patankar·C. S. Rajan

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Abstract

We show that a two dimensional $\ell $-adic representation of the absolute Galois group of a number field which is locally potentially equivalent to a $GL(2)$-$\ell$-adic representation $\rho$ at a set of places of $K$ of positive upper density is potentially equivalent to $\rho$. For an elliptic curver $ E $ defined over a number field $ K $ and a finite place $ v $ of $ K $ of good reduction for $ E $, let $ F(E,v) $ denote the Frobenius field of $ E $ at $ v $, given by the splitting field of the characteristic polynomial of the Frobenius automorphism at $ v $ acting on the Tate module of $ E $. As an application, suppose $ E_1 $ and $ E_2 $ defined over a number field $ K $, with at least one of them without complex multiplication. We prove that the set of places $ v $ of $ K $ of good reduction such that the corresponding Frobenius fields are equal has positive upper density if and only if $ E_1 $ and $ E_2 $ are isogenous over some extension of $ K $. We show that for an elliptic curve $ E $ defined over a number field $ K $, the set of finite places of $ K $ such that the Frobenius field $ F(E, v) $ at $v$ equals a fixed imaginary quadratic field $ F $ has positive upper density if and only if $ E $ has complex multiplication by $ F $.

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Manisha Kulkarni, Vijay M. Patankar, C. S. Rajan. 2014-03-22. Locally potentially equivalent two dimensional Galois representations and Frobenius fields of elliptic curves. https://arxiv.org/abs/1403.5635

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