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Potential automorphy of odd-dimensional symmetric powers of elliptic curves and applications. (English) Zbl 1234.11068
Tschinkel, Yuri (ed.) et al., Algebra, arithmetic, and geometry. In honor of Y. I. Manin on the occasion of his 70th birthday. Vol. II. Boston, MA: Birkhäuser (ISBN 978-0-8176-4746-9/hbk; 978-0-8176-4747-6/ebook). Progress in Mathematics 270, 1-21 (2009).
The paper under review extends the modularity lifting theorem of R. Taylor [Publ. Math., Inst. Hautes Étud. Sci. 108, 183–239 (2008; Zbl 1169.11021)], using the combined works of Chenevier, Clozel, Guerberoff, Labese, Shin and the author. This simplifies the potential modularity result [Theorem 1 of Ann. Math. (2) 171, No. 2, 779–813 (2010; Zbl 1263.11061)] and eliminates one of the unwanted hypothesis therein.
Let \(F\) be a CM field, \(F^+\) be its maximal real subfield and \(\Gamma_{F^+}=\text{Gal}(\bar{F^+}/F^+)\) its absolute Galois group. Let \(E/F^+\) be an elliptic curve without CM. The author uses his potential modularity result for the Galois representation \(\rho_{E,\ell}^n=\text{Sym}^{n-1}\rho_{E,\ell}: \Gamma_{F^+} \rightarrow \text{GL}(n,\mathbb Q_\ell)\). Then for even \(n\), the potential modularity of \(\rho_{E,\ell}^n\) in turn is used to prove that \(L(s,\rho_{E,\ell}^n)\) is potentially automorphic for any even \(n\), provided that \(j(E)\) is non-integral.
Using a tensor product trick developed in the current article, the author converts an odd dimensional representation to an even dimensional representation, so that he can apply the results alluded to above for even \(n\). The author explains how to choose the auxiliary representation to be tensored with correctly, so as to recover the relevant automorphy result that reads as follows:
Theorem. Suppose \(E\) and \(E^\prime\) are elliptic curves over \(F^+\), and assume \(E\) and \(E^\prime\) do not become isogenous over an abelian extension of \(F^+\). Let \(m\) and \(m^\prime\) be positive integers. Then the \(L\)-function \(L(s, \rho_{E,\ell}^m\otimes\rho_{E^\prime,\ell}^{m^\prime})\) is invertible and satisfies the expected functional equation.
As a corollary to this result, the author obtains an affirmative answer to a question of Mazur and Katz:
Theorem: Suppose \(E\) and \(E^\prime\) are elliptic curves over \(F^+\), and assume \(E\) and \(E^\prime\) do not become isogenous over an abelian extension of \(F^+\). For any prime \(v\) of \(F^+\) where \(E\) and \(E^\prime\) both have good reduction, set \[ |E(k_v)|=\left(1-q_v^{\frac{1}{2}}e^{i\phi_v}\right)\left( (1-q_v^{\frac{1}{2}}e^{-i\phi_v}\right) \] \[ |E^{\prime}(k_v)|=\left(1-q_v^{\frac{1}{2}}e^{i\psi_v}\right)\left( (1-q_v^{\frac{1}{2}}e^{-i\psi_v}\right) \] where \(k_v\) is the residue field at \(v\), \(q_v=|k_v|\) and \(\phi_v,\psi_v\in [0,\pi]\).
Then the pairs \((\phi_v,\psi_v) \in [0,\pi]\times [0,\pi]\) are uniformly distributed with respect to the measure \[ \frac{4}{\pi^2}\sin^2\phi\sin^2\psi\,d\phi d\psi. \]
For the entire collection see [Zbl 1185.00042].

11F80 Galois representations
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11G05 Elliptic curves over global fields
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
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