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Deformations of Galois representations and exceptional monodromy

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Abstract

For any simple algebraic group G of exceptional type, we construct geometric \(\ell \)-adic Galois representations with algebraic monodromy group equal to G, in particular producing the first such examples in types \(\mathrm {F}_4\) and \(\mathrm {E}_6\). To do this, we extend to general reductive groups Ravi Ramakrishna’s techniques for lifting odd two-dimensional Galois representations to geometric \(\ell \)-adic representations.

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Notes

  1. The same thing happens in [11], which served as inspiration for [14].

  2. So what follows will literally apply except in type \(\mathrm {E}_6\); type \(\mathrm {E}_6\) turns out to require a minor, merely technical, modification, carried out in Sects. 9 and 10.

  3. In fact, substantially more so, because we would need not only a Galois extension with group \(G(\mathbb {F}_{\ell })\), but also to know that the associated representation \(\bar{\rho }\) satisfied the various technical hypotheses of the lifting theorem, e.g. ordinarity at \(\ell \).

  4. A sequel to the present paper will remove this restriction.

  5. Note that in the case \(G= \mathrm {G}_2\), the composition of the principal \(\mathrm {SL}_2\) with the quasi-minuscule representation \(\mathrm {G}_2 \hookrightarrow \mathrm {GL}_7\) remains irreducible, so that potential automorphy techniques could be applied in this case. This approach does not work for the other exceptional groups.

  6. In particular, taking \(\ell \not \mid n+1\) in type \(\mathrm {A}_n\) and \(\ell \ge 7\) in all other cases suffices.

  7. In some of the discussion that follows, the reader could replace \(\mu :G \rightarrow S\) by some other map to an \(\mathcal {O}\)-torus, whose kernel may be bigger than the derived group.

  8. By the universal property, conjugation of the universal lift by any element of \(\widehat{G}(R^{\square }_{\bar{\rho }})\) induces a morphism \(R_{\bar{\rho }}^\square \rightarrow R_{\bar{\rho }}^\square \).

  9. The logic of the present section only depends on knowing the right-hand-side of this equation; we mention the rest only for motivation.

  10. Recall from Lemma 4.11 that \(\mathfrak {l}_{\alpha }\) is the span of the \(\alpha \)-coroot vector.

  11. \(K_{\phi }\) and \(K_{\psi }\) are Galois over F because \(\phi \) and \(\psi \) are cocycles for \(\Gamma _{F}\).

  12. We remark that this enlargement of G is frequently technically convenient: for instance, it is the Tannakian group appearing when one studies the ‘geometric’ Satake correspondence over finite fields or number fields.

  13. Namely, for the torus \(T_1= (T \times \mathbf {G}_m)/\langle 2 \rho ^\vee (-1), -1 \rangle \),

    $$\begin{aligned} \left( \rho ^\vee , \frac{\beta }{2}\right) \in X_{\bullet }(T_1) \otimes _{\mathbb {Z}} \mathbb {Q}, \end{aligned}$$

    where \(\beta \) denotes a generator of \(X_{\bullet }(\mathbf {G}_m)\), in fact defines an element of \(X_{\bullet }(T_1)\).

  14. This lift is of course not unique; for any choice, \(2 \tilde{\rho }^\vee \) differs from \(2\rho ^\vee \), the usual co-character of \(G^{\mathrm {der}}\), by an element of \(X_{\bullet }(Z_G)\).

  15. I’m grateful to Florian Herzig for pointing out an apparently well-known minor error in [8], that \(H^1(\mathrm {SL}_2(\mathbb {F}_{\ell }), {{\mathrm{Sym}}}^{\ell -3}(\mathbb {F}_{\ell }^2))\) is one-dimensional. Our running assumption \(\ell > 4h-1\) will ensure this doesn’t interfere with any of our arguments.

  16. Magma automatically does this when it produces a Chevalley basis.

  17. The c[i] can be found in the tables of [4]. Within Magma, they can be derived, for our Lie algebra \(\mathfrak {g}\), by computing \(\mathrm {rd}:= \mathbf {RootDatum}(\mathfrak {g})\), then \(A:=\mathbf {SimpleCoroots}(\mathrm {rd})\) (an \(l \times l\) matrix), then by forming a vector c whose entries are twice the sum of the rows of \(A^{-1}\); the ith entry of c is the desired c[i]. But we double-check in each case the bracket properties for X, Y, H, so how the c[i] are arrived at no longer matter.

  18. In Magma, when you call \(P:= \mathbf Centraliser (\mathfrak {g}, X)\) and then \(\mathbf ExtendBasis (P, \mathfrak {g})\), it hands you a basis of \(\mathfrak {g}\), in terms of the Chevalley basis, whose first l entries are a basis of P, and in fact already an H-eigenbasis.

  19. Intrinsically, \(\alpha _2\) and \(\alpha _4\) are those fixed under the outer automorphism of \(\mathrm {E}_6\). Our labeling convention is \(f(\alpha _i)=i\), where \(x[1], \ldots , x[6]\) is the ordered set of simple roots produced by Magma.

  20. Note that \(\Lambda \) is right-adjoint to \(\otimes _{\mathbb {F}_{\ell }} k\), and \(k \otimes _{\mathbb {F}_{\ell }} k\) is isomorphic to \(\prod _{\iota } k\) by the map \(x \otimes y \mapsto (\iota (x)y)_{\iota }\).

  21. The set \(\{r_{\alpha }\}_{\alpha \in \Delta (B, T)}\) does not depend on the choice of T.

  22. There are some elaborate Galois-theoretic means under hypotheses that are in practice too difficult to arrange; if the characteristic zero lifts are known to be automorphic, then suitable cases of the Ramanujan conjecture imply this non-splitness—this observation should suggest the difficulty of the problem. Current potential automorphy techniques would only help with the case \(G= \mathrm {G}_2\).

  23. In fact, it will even apply to appropriate unitary groups of any rank, using [2].

  24. Note that if we did not restrict to adjoint G, the above argument would show that \(\rho _{\lambda }(g)\) is a product \(z \cdot u\), where \(z \in Z_G\) and u is regular unipotent in G; but of course then by Jordan decomposition \(u \in G_{\rho }\) as well.

  25. In their notation, take \(S= \Sigma \) and \(T= \emptyset \).

  26. The groups \(I_{\widetilde{F}_{\tilde{w}}}\) and \(I_{\widetilde{F}_{\tilde{w}'}}\) are conjugate in \(\Gamma _{\Sigma \cup w}\), and the cocycle relation implies \(\phi (ghg^{-1})=0\) whenever \(\phi (h)=0\) and h acts trivially on M.

  27. Here recall that we write \(\mathfrak {t}^\vee = \mathfrak {l}_{\alpha ^\vee } \oplus \mathfrak {t}^\vee _{\alpha ^\vee }\) for the decomposition of \(\mathfrak {t}^\vee \) into the span \(\mathfrak {l}_{\alpha ^\vee }\) of the \(\alpha \)-coroot vector for \(G^\vee \) and \(\mathfrak {t}^\vee _{\alpha ^\vee }= \ker (\alpha )\).

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

    Stefan Patrikis

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This paper has a long history, and it gives me great pleasure to extend thanks both old and new. In 2006 with Richard Taylor’s guidance I proved a version of Ramakrishna’s lifting theorem for symplectic groups, as my Harvard undergraduate thesis. Without Richard’s encouragement and singular generosity, I would likely not have continued studying mathematics, much less been equipped to write the present paper; I am enormously grateful to him. At the time Frank Calegari and Brian Conrad both read and helpfully commented on the thesis, and I thank them as well. Moving toward the present day, I am grateful to Dick Gross, from whom I learned an appreciation of the principal \(\mathrm {SL}_2\), which turned out to be crucial for this paper; to Shekhar Khare, for his comments both on this paper and on other aspects of Ramakrishna’s work; to Florian Herzig for pointing out an error in [8]; and to the anonymous referee for a careful reading of an earlier version. Finally, I am greatly indebted to Rajender Adibhatla, who encouraged me to revise [24] for publication; this revision (superseded by the present paper) spurred me to revisit Ramakrishna’s methods.

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Patrikis, S. Deformations of Galois representations and exceptional monodromy. Invent. math. 205, 269–336 (2016). https://doi.org/10.1007/s00222-015-0635-3

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