Abstract
Let k be a finitely generated field of characteristic 0 embedded into \({\mathbb{C},\,X}\) a smooth, separated and geometrically connected scheme over k with generic point \({\eta}\) and \({f : Y \rightarrow X}\) a smooth proper morphism. Let \({f^{an}_{\mathbb{C}} : Y^{an}_{\mathbb{C}} \rightarrow X^{an}_{\mathbb{C}}}\) denote the associated morphism of complex analytic spaces. For \({x \in X(\mathbb{C})}\), write H for the Betti cohomology of \({Y^{an}_{\mathbb{C}, x}}\) with coefficients in \({\mathbb{Q}}\) and for \({x \in X}\), write \({{\rm H}_{\ell}}\) for the ℓ-adic cohomology of \({Y_{\overline{x}}}\) with coefficients in \({\mathbb{Q}_{\ell}}\) (Under our assumptions on \({f : Y \rightarrow X,\,{\rm H}}\) and \({{\rm H}_{\ell}}\) are independent of x). For every prime ℓ, let \({X^{ex}_{\ell}}\) be the set of all \({x \in X}\) where the Zariski closure \({G_{\ell, x}}\) of the image of the Galois representation \({\Gamma_{k(x)} \rightarrow {\rm GL}({\rm H}_{\ell})}\) has dimension strictly smaller than the dimension of \({G_{\ell,\eta}}\). By previous works of A. Tamagawa and the author, \({X^{ex}_{\ell}}\) is ‘small’ in the sense that if X is a curve then for every integer \({\delta \geq 1}\) the set of all \({x \in X^{ex}_{\ell}}\) with \({[k(x) : k] \leq \delta}\) is finite. Set \({X^{ex} := {\bigcap}_{\ell} X^{ex}_{\ell}}\). The Tate conjectures predict that for every \({x \in X}\) the \({G_{\ell, x}}\) are defined over \({\mathbb{Q}}\), reductive and independent of ℓ hence, in particular, that the sets \({X^{ex}_{\ell}}\) are independent of ℓ. Let \({\overline{G}}\) denote the Zariski closure of the image of the monodromy representation \({\pi_{1} (X_{\mathbb{C}}^{an}; \, x) \rightarrow {\rm GL}({\rm H})}\). Then \({\overline{G}}\) is a semi-simple algebraic group of rank—say—r. The main result of this note is that for \({x \notin X^{ex},\,G_{\ell, x} \cap \overline{G}_{\mathbb{Q}_{\ell}}}\) is a semi-simple algebraic group of rank r. This implies in particular that: (1) If \({\overline{G}_{\overline{\mathbb{Q}}}}\) has only simple factors of type \({A_{n}}\) then \({X^{ex}_{\ell}}\) is independent of ℓ; (2) For every prime ℓ and \({x \notin X^{ex}}\) the unipotent radical of \({G_{\ell, x}}\) coincides with the unipotent radical of \({G_{\ell, \eta}}\) and, in particular, is independent of \({x \notin X^{ex}}\); (3) For every prime ℓ, if there exists \({x_{\ell} \in X}\) such that \({G_{\ell, x_{\ell}}}\) is reductive then for every \({x \notin X^{ex},\,G_{\ell, x}}\) is reductive. (3) applies in particular when \({{\rm H}}\) is a geometrically irreducible \({\overline{G}}\)-module. This implies, for instance, that apart from a few exceptional cases, for every r-tuple \({\underline{d}=(d_{1}, \dots, d_{r})}\) of integers \({\geq 2}\) there exists a non-singular complete intersection in \({\mathbb{P}^{n+r}_{\mathbb{Q}}}\) with multi-degree \({\underline{d}}\) for which the Tate semi-simplicity conjecture holds (for every prime ℓ).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
André, Y.: Pour une théorie inconditionnelle des motifs. Publ. Math. I.H.E.S. 83, 5–49 (1996)
André, Y.: Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panorama et synthèse, vol. 17. S.M.F. (2004)
Bogomolov F.: Sur l’algébricité des représentations \({\ell}\)-adiques. C.R. Acad. Sci. Paris 290, 103–701 (1980)
Borel, A.: Linear Algebraic Groups—2nd Enlarged ed., G.T.M., vol. 126, Springer, Berlin (1991)
Borel A., Tits J.: Eléments unipotents et sous-groupes paraboliques de groupes réductifs I. Invent. Math. 12, 95–104 (1971)
Breuillard E., Green B., Guralnick R., Tao T.: Strongly dense free subgroups of semi-simple algebraic groups. Isr. J. Math. 192, 347–379 (2012)
Cadoret A., Tamagawa A.: A uniform open image theorem for \({\ell}\)-adic representations I. Duke Math. J. 161, 2605–2634 (2012)
Cadoret A., Tamagawa A.: A uniform open image theorem for \({\ell}\)-adic representations II. Duke Math. J. 162, 2301–2344 (2013)
Deligne P.: Théorie de Hodge, II. Inst. Hautes Etudes Sci. Publ. Math. 40, 5–57 (1971)
Deligne P.: La conjecture de Weil, II. Inst. Hautes Études Sci. Publ. Math. 52, 137–252 (1980)
Faltings, G.: Crystalline cohomology and p-adic Galois representations. In: Algebraic Analysis, Geometry, and Number Theory (Baltimore, 1988), pp. 191–224. John Hopkins University Press (1989)
Fontaine J.-M., Messing W.: p-Adic periods and p-adic étale cohomology. Contemp. Math. 67, 179–207 (1987)
Gille, P.: The Borel–de Siebenthal Theorem, Preprint Available on www.math.ens.fr/~gille/prenotes/bds (2010)
Hui C.-Y.: Specialization of monodromy and \({\ell}\)-independence. C.R. Acad Sci. Paris 350, 5–7 (2012)
Hui C.-Y.: Monodromy of Galois representations and equal rank subalgebra equivalence. Math. Res. Lett. 20, 1–24 (2013)
Humphreys, J.E.: Linear Algebraic Groups. G.T.M., vol. 21. Springer (1975)
Illusie L.: Constructibilité générique et uniformité en \({\ell}\), Preprint. Available on http://www.math.upsud.fr/~illusie/constructible3.pdf (2010)
Katz N., Laumon G.: Transformation de Fourier et majoration de sommes exponentielles. Publ. Math. I.H.E.S. 62, 361–418 (1985)
Moonen, B.: An introduction to Mumford–Tate groups, Preprint, 2004 available on http://staff.science.uva.nl/~bmoonen/MTGps
Olsson, M.: On Faltings’ method of almost étale extensions. In: Algebraic Geometry—Seattle 2005, Proceedings of the Symposium in Pure Mathematics, vol. 80 Part 2, pp. 811–936, A.M.S. (2009)
Serre, J.-P.: Letter to Ken Ribet (1 Jan 1981). Collected works IV (1985–1988). Springer (2000)
Serre, J.-P.: Letter to Ken Ribet (29 Jan 1981). Collected works IV (1985–1988). Springer (2000)
Tsuji T.: p-Adic étale cohomology and crystalline cohomology in the semistable reduction case. Invent. Math. 137, 233–411 (1999)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cadoret, A. On ℓ-independency in families of motivic ℓ-adic representations. manuscripta math. 147, 381–398 (2015). https://doi.org/10.1007/s00229-015-0737-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-015-0737-7