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Deformations of Galois representations and exceptional monodromy, II: raising the level

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Abstract

Building on lifting results of Ramakrishna, Khare and Ramakrishna proved a purely Galois-theoretic level-raising theorem for odd representations \(\bar{\rho } :\mathrm {Gal}(\overline{\mathbb {Q}}/\mathbb {Q}) \rightarrow \mathrm {GL}_2(\overline{\mathbb {F}}_{\ell })\). In this paper, we generalize these techniques from type A1 to general (semi-)simple groups. We then strengthen our previous results on constructing geometric Galois representations with exceptional monodromy groups, achieving such constructions for almost all \(\ell \), rather than a density-one set, and achieving greater flexibility in the Hodge numbers of the lifts; the latter improvement requires the new level-raising result.

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Notes

  1. We remind the reader precisely what this means with our notational conventions: q is a place of F with a fixed extension, also denoted q, to \(\widetilde{F}\), and \(\bar{\rho }|_{\Gamma _{\widetilde{F}_q}}\) satisfies a Ramakrishna-type deformation condition with respect to some root \(\alpha \), which is now fixed. The tangent space of this deformation condition is denoted \(L_q^{\mathrm {Ram}}\), and the corresponding tangent space for unramified deformations is denoted \(L_q^{\mathrm {un}}\).

  2. For \(\ell \) sufficiently large relative to \(\Sigma \), this condition will be shown always to hold in the forthcoming thesis of Booher ([1]).

  3. See [21, Proposition 4.4, Theorem 5.5]. Note, however, that his final claim is only valid—or at least proven—for elliptic curves, not general weight two modular forms.

  4. We note there is a typo in [12, Theorem 8.4]: the integers \(r_{\alpha }\) there must satisfy \(r_{\alpha } \equiv 2 \pmod {\ell -1}\), since, in the notation of that argument, \(\alpha \circ \bar{\rho }_T= \bar{\kappa }^2\).

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  1. The University of Utah, Salt Lake City, UT, 84112, USA

    Stefan Patrikis

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  1. Stefan Patrikis
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Correspondence to Stefan Patrikis.

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I am grateful to Chandrashekhar Khare and Bjorn Poonen for helpful conversations. I also thank the anonymous referee for many helpful comments.

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Patrikis, S. Deformations of Galois representations and exceptional monodromy, II: raising the level. Math. Ann. 368, 1465–1491 (2017). https://doi.org/10.1007/s00208-016-1459-1

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