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Pierre Deligne: A Poet of Arithmetic Geometry

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The Abel Prize 2013-2017

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Abstract

This is a report on the work of Pierre Deligne.

Dix choses soupçonnées seulement, dont aucune (la conjecture de Hodge disons) n’entraîne conviction, mais qui mutuellement s’éclairent et se complètent et semblent concourir à une même harmonie encore mystérieuse, acquièrent dans cette harmonie force de vision. Alors même que toutes les dix finiraient par se révéler fausses, le travail qui a abouti à cette vision provisoire n’a pas été fait en vain, et l’harmonie qu’il nous a fait entrevoir et qu’il nous a permis de pénétrer tant soit peu n’est pas une illusion, mais une réalité, nous appelant à la connaître. — A. Grothendieck, Récoltes et Semailles, Deuxième partie,I B 4 1.

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Notes

  1. 1.

    As Serre observed (loc. cit., 2.10), when Poincaré duality is available on the base, this degeneration can also be proved by an extension of Blanchard’s method in [33].

  2. 2.

    The published version [254] doesn’t treat derived functors either.

  3. 3.

    Picard stacks appear in deformation theory: in [61] Deligne sketched a method to use them to give an alternate proof of the theorems of ([121],VII) on deformations of torsors and group schemes—a program that has not yet been carried out. See [44, 208, 255] for recent developments.

  4. 4.

    Le déterminant d’une courbe, undated.

  5. 5.

    A scheme, for n ≥ 3, by Serre’s rigidity lemma.

  6. 6.

    Nagata’s original proof is obscure to today’s readers. A modern presentation was given by Deligne in [D112, 2010].

  7. 7.

    One can achieve this by a sequence of blow-ups ([138], 7.2).

  8. 8.

    Lattices Γ 1 and Γ 2 are called commensurable if Γ 1 ∩ Γ 2 is of finite index in Γ 1 and in Γ 2.

  9. 9.

    They come from the three descriptions of \(E_r^{pq}\): as a subobject of a quotient of K p+q, as a quotient of a subobject of K p+q, as a quotient of a subobject of \(E^{pq}_{r-1}\).

  10. 10.

    In this section, sheaves on and cohomology groups of schemes of finite type over C are taken with respect to the classical topology (on the associated complex analytic spaces), and we omit the superscript “an” for brevity.

  11. 11.

    Δ(n) is the simplicial set [p]↦Hom([p],  [n]).

  12. 12.

    The n-th component of C(f) is the disjoint union of X n, Y i for i < n, and Spec C.

  13. 13.

    In the terminology of (loc. cit., p. 260), \(\mathscr {M} \otimes \mathbf {R}\) is a formal consequence of H ∗(M, R).

  14. 14.

    I.e., an involution σ such that the real form \((G^{\mathrm {ad}}_{\mathbf {R}})^{\sigma }\) of \(G^{\mathrm {ad}}_{\mathbf {R}}\) relative to the complex conjugation \(g \mapsto \sigma (\overline {g})\) is compact, in the sense that \((G^{\mathrm {ad}}_{\mathbf {R}})^{\sigma }(\mathbf {R})\) is compact.

  15. 15.

    He calls it “hope”.

  16. 16.

    [Weil I] = [D27, 1974], [Weil II] = [D46, 1980].

  17. 17.

    For w ∈Z, a q-Weil number (resp. q-Weil integer) of weight w is an algebraic number (resp. algebraic integer) all of whose conjugates are of absolute value q w∕2.

  18. 18.

    The hypothesis p > 2 enables to apply the results of Katz in ([7], XVIII) – actually p > 2 or n odd would suffice for this reference; see also the comment after Theorem 31.

  19. 19.

    According to Katz [143], privately communicated to P. Deligne in 1971.

  20. 20.

    We will consider only constructible \(\overline {\mathbf {Q}}_{\ell }\)-sheaves, and omit “constructible” in the sequel.

  21. 21.

    See ([Weil II], 6.1.11). For schemes over Spec Z, it seems that only generic variants are available ([Weil II], 6.1.2).

  22. 22.

    I.e., a simple quotient in a Jordan–Hölder filtration by lisse subsheaves.

  23. 23.

    Grothendieck proved this theorem at a talk he gave during the SGA 7 seminar, on March 26, 1968.

  24. 24.

    Whose proof in loc. cit. was incorrect, see Sect. 10, The weight monodromy conjecture.

  25. 25.

    Suh [243] showed that (107) holds more generally assuming only X 0∕F q proper and smooth, and used it to prove the evenness of odd degree ℓ-adic Betti numbers of X. Generalizations of this last result to intersection cohomology are given by Sun-Zheng [247].

  26. 26.

    Here i ! stands for Ri !.

  27. 27.

    Drinfeld and Kedlaya [84] recently showed that, moreover, for X 0∕F q smooth, the slopes of the minimal Newton polygon have gaps ≤ 1, which result also extends to the general case ([265], 2.7).

  28. 28.

    Note that for \(\sigma \in \mathrm {Gal}(\overline {\mathbf {F}}_p/{\mathbf {F}}_p)\), the isomorphism \([\sigma ]: \mathrm {Spec} \,\overline {\mathbf {F}}_p \to \mathrm {Spec} \,\overline {\mathbf {F}}_p\) deduced by transportation of structure is given by [σ]∗ x = σ −1 x.

  29. 29.

    A variant of this construction was later considered by Langlands, with G a replaced by SL2, [164].

  30. 30.

    See Langlands’s comments on his website, on Langlands’s Notes on Artin L-functions.

  31. 31.

    This theorem says that, if (S, s, η) is a henselian trait and X∕η is separated and of finite type, \(\overline {\eta }\) a geometric point over η, and n an integer invertible on S, an open subgroup I 1 of the inertia group I acts unipotently on \(H^*(X_{\overline {\eta }},\mathbf {Z}/n\mathbf {Z})\) (resp. \(H^*_c(X_{\overline {\eta }},\mathbf {Z}/n\mathbf {Z})\)).

  32. 32.

    Grothendieck gave an unconditional proof for \(H^*_c(X_{\overline {\eta }},\mathbf {Z}/n\mathbf {Z})\) by another argument, of arithmetic nature, see (loc. cit., I).

  33. 33.

    A purely algebraic proof was found later [126].

  34. 34.

    “dimtot” stands for “dimension totale”.

  35. 35.

    The same holds for χ, as \(\chi _c(U,\mathscr {G}) = \chi (U,\mathscr {G})\) by Laumon [168].

  36. 36.

    For surfaces, this variant is re-interpreted in [141] as an equality of conductors for the restriction to every curve.

  37. 37.

    More precisely, F −1(B (q)), for F : G → G (q) the relative Frobenius.

  38. 38.

    At least, for F of characteristic zero: the case of positive characteristic was treated later by Badulescu [17].

  39. 39.

    I.e., such that Gal(E 1∕F)e = 1, where E 1 is the Galois closure of E.

  40. 40.

    I.e., a k-linear, exact functor, with an isomorphism \(\omega (X) \otimes \omega (Y) \stackrel {\sim }{\to } \omega (X \otimes Y)\) compatible with the associativity and commutativity data; such a functor necessarily has values in the category of finite dimensional K-vector spaces.

  41. 41.

    A fiber functor is then defined as an exact k-linear functor ω from \(\mathscr {A}\) to the category of quasi-coherent \(\mathscr {O}_S\)-modules, with a compatible isomorphism \(\omega (X) \otimes _{\mathscr {O}_S} \omega (Y) \stackrel {\sim }{\to } \omega (X \otimes Y)\); it is shown that it has values in the category of locally free \(\mathscr {O}_S\)-modules.

  42. 42.

    I.e., a category having the data and satisfying the axioms of a Tannakian k-linear category, with End(1) = k, but without the requirement of existence of a fiber functor.

  43. 43.

    I.e., dual objects exist and satisfy the same axioms as in a tensor category.

  44. 44.

    I.e., f : X → Y  such that Tr(fu) = 0 for all u : Y → X.

  45. 45.

    Developed in [D76, 1994], but used by him and other authors much earlier.

  46. 46.

    Assuming that the local F-semisimplified representations of the local decomposition groups are compatible, cf. (Sect. 6.2, (a)).

  47. 47.

    The crystalline data and axioms in loc. cit. are in a rudimentary form, reflecting the status of the p-adic comparison theorems at the time; Deligne made a caveat on this.

  48. 48.

    I.e., the datum, for every fiber functor ω on a scheme S, an affine group scheme G ω on S, functorial in ω, and compatible with base change S′→ S.

  49. 49.

    Except for the crystalline ones.

  50. 50.

    k is the inverse of the dual Coxeter number h ∨.

  51. 51.

    For the adjoint representation, such formulas had been found by P. Vogel; according to Deligne, that was the beginning of the story.

  52. 52.

    I.e., objects have duals, hence a dimension with value in End(1) = Q(t).

  53. 53.

    Private communication, June 2017.

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Acknowledgements

I warmly thank Alexander Beilinson, Francis Brown, Pierre Deligne, Hélène Esnault, Ragni Piene, Takeshi Saito, Jean-Pierre Serre, Claire Voisin, and Weizhe Zheng for useful comments on preliminary versions of this report, and Gérard Laumon for helpful discussions and his assistance in collecting the references.

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    Luc Illusie

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  1. Department of Mathematical Sciences, NTNU Norwegian University of Science and Technology, Trondheim, Norway

    Helge Holden

  2. Department of Mathematics, University of Oslo, Oslo, Norway

    Ragni Piene

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Illusie, L. (2019). Pierre Deligne: A Poet of Arithmetic Geometry. In: Holden, H., Piene, R. (eds) The Abel Prize 2013-2017. The Abel Prize. Springer, Cham. https://doi.org/10.1007/978-3-319-99028-6_2

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