DOCSLIB.ORG

Generic Vanishing, Tannakian Categories, and Equidistribution Preliminary Version – 20.3.2021

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Generic Vanishing, Tannakian Categories, and Equidistribution Preliminary Version – 20.3.2021 GENERIC VANISHING, TANNAKIAN CATEGORIES, AND EQUIDISTRIBUTION PRELIMINARY VERSION { 20.3.2021 ARTHUR FOREY, JAVIER FRESAN,´ AND EMMANUEL KOWALSKI Abstract. We prove a generic vanishing theorem for twists of perverse sheaves on a commutative algebraic group G over a finite field. Using this tool, we construct a tannakian category with convolution on G as tensor operation. Using Deligne's Riemann Hypothesis, we show how this leads to equidistribution theorems for discrete Fourier transforms of trace functions of perverse sheaves on G, generalizing the work of Katz in the case of the multiplicative group. We give some concrete examples of applications of these results. Contents 1. Introduction 1 2. Preliminaries 9 3. Generic vanishing theorems 19 4. Tannakian categories of perverse sheaves 36 5. Equidistribution theorems 47 6. Applications 55 7. Open problems 81 References 82 1. Introduction 1.1. Statement of results. Since Deligne's proof of his equidistribution theorem for traces of Frobenius of `-adic local systems on varieties over finite fields [10], it has been known that any family of exponential sums parameterized by an algebraic variety satisfies some form of equidistribution, and that the concrete expression of this equidistribution statement depends on the determination of the geometric monodromy group of the `-adic sheaf that underlies the family of exponential sums. The best known result of this kind is probably the computation by Katz [25] of these monodromy groups in the case of Kloosterman sums in many variables over finite fields, which are defined for × some fixed non-trivial additive character of Fq and a 2 Fq as (−1)n−1 X Kl (a; q) = (x + ··· + x ): n q(n−1)=2 1 n × n (x1;:::;xn)2(Fq ) x1···xn=a A. F. and E. K. are partially supported by the DFG-SNF lead agency program grant 200020L 175755. J. F. is partially supported by the grant ANR-18-CE40-0017 of the Agence Nationale de la Recherche. 1 This computation led him in particular to the proof of the average version of the Sato-Tate law for classical Kloosterman sums, namely the equidistribution of the set (Kl (a; q)) × as q ! +1 2 a2Fq with respect to the Sato-Tate measure on [−2; 2]. Further deep investigations by Katz, especially in his monograph [26], provide a cornucopia of examples of equidistribution statements. Among other things, this framework allows for the study of exponential sums of the form X S(M; ) = tM(x; Fqn ) (x); x2Fqn where tM is the trace function of a perverse sheaf M on the additive group Ga over Fq and ranges over characters of Fqn . These sums are the discrete Fourier transform 7! S(M; ) of the function x 7! tM(x; Fqn ) on the finite group Ga(Fqn ), and the key point is that they are themselves the trace functions of another perverse sheaf on the dual group parameterizing additive characters. In a more recent conceptual breakthrough, Katz [31] succeeded in proving equidistribution results for families of exponential sums parameterized by multiplicative characters, despite the fact that the set of multiplicative characters of a finite field Fq does not naturally arise as the set of Fq-points of an algebraic variety. In analogy with the above, such sums are of the form X S(M; χ) = tM(x; Fqn )χ(x); × x2Fqn except that M is now a perverse sheaf on the multiplicative group Gm over Fq and χ ranges × over characters of Fqn . Katz's beautiful insight was to replace points of algebraic varieties by fiber functors of tannakian categories as parameter spaces, and produce the groups governing equidistribution by means of the tannakian formalism (see [15] for an accessible survey). Further work of Katz generalized this to elliptic curves [33] and certain abelian varieties (unpublished). The goal of this paper is to extend these ideas to exponential sums parameterized by the characters of the points of any connected commutative algebraic group over a finite field. ¯ Let k be a finite field and k an algebraic closure of k. For each n > 1, we denote by kn the extension of k of degree n inside k¯. Let ` be a prime number different from the characteristic of k and Q` an algebraic closure of the field of `-adic numbers. Let G be a connected commutative algebraic group over k. We denote by G(b kn) the group of Q`-valued characters of G(kn) and, for each χ 2 G(b kn), by Lχ the `-adic lisse character sheaf of rank one associated to χ by means of the Lang torsor construction. By perverse sheaves, we always understand Q`-perverse sheaves. In rough outline, we establish the following results: { We prove generic vanishing theorems for the cohomology of twists of perverse sheaves on G by the sheaves Lχ associated to characters χ 2 G(kn). { Using these vanishing theorems, we construct a tannakian category of perverse sheaves on G in which the tensor product is given by the convolution coming from the group law. { We prove that the tannakian group of a semisimple object M of this category that is pure of weight zero controls the distribution properties of the analogue of the sums above, namely X S(M; χ) = tM(x; kn)χ(x); x2G(kn) where χ ranges over the set G(b kn). Under some assumptions on G, we prove the stronger result that the unitary conjugacy classes of which these sums are traces become equidis- tributed in a maximal compact subgroup of the tannakian group as n ! +1, as is custom- ary since Deligne's work. 2 The following theorems are special cases of our main results, which we formulate in simplified form in order to make it possible to present a self-contained statement at this stage. Theorem 1.1.1. Let M be a perverse sheaf on a connected commutative algebraic group G of dimension d over a finite field k. (1) (Generic vanishing) The sets i i X (kn) = χ 2 G(b kn) j Hc(Gk¯; M ⊗ Lχ) = H (Gk¯; M ⊗ Lχ) = 0 for all i 6= 0 0 0 and Hc (Gk¯; M ⊗ Lχ) is isomorphic to H (Gk¯; M ⊗ Lχ) are generic, in the sense that the estimate d−1 G(b kn) X (kn) jknj holds for n > 1, with an implied constant that only depends on M. (2) (Stratified vanishing) For −d 6 i 6 d and n > 1, the estimate i i d−i χ 2 G(b kn) j Hc(Gk¯; M ⊗ Lχ) 6= 0 or H (Gk¯; M ⊗ Lχ) 6= 0 jknj holds, with an implied constant that only depends on M. The most general vanishing statements that we prove appear as Theorems 3.1.1 and 3.1.3. Ap- plications to “stratification" estimates for exponential sums are then given in Section 6.1. Remark 1.1.2. With variations in the definition of generic set of characters, such statements were proved by Katz{Laumon [34] for powers of the additive group, Saibi [46] for unipotent groups, Gabber{Loeser [18] for tori, Weissauer [53] for abelian varieties and Kr¨amer[37] for semi-abelian varieties (see Remark 3.1.2). The very rough idea to obtain Theorem 1.1.1 is to prove for the various types of commutative groups a relative version of the vanishing theorem with a good control of the implicit constant. Hence, we also significantly improve the aforementioned results. For example, in the case of tori, Gabber{Loeser [18] prove the vanishing as stated above only under the assumption that resolution of singularities holds up to the dimension of the torus. We remove this assumption using alterations instead. For abelian varieties, we extend Weissauer's work [53] by proving a relative version of the theorem, which involves the use of Orgogozo's work [44] on constructibility and moderation. Using the vanishing theorems, and ideas going back to Gabber and Loeser, we can construct tannakian categories with the convolution on G as tensor operation. Using these, and Deligne's Riemann Hypothesis over finite fields, we obtain the following equidistribution theorem for the Fourier transforms of exponential sums on G: Theorem 1.1.3 (Equidistribution on average). Let G be a connected commutative algebraic group over k. Let ι: Q` ! C be an isomorphism. Let M be a geometrically irreducible `-adic perverse sheaf on G that is ι-pure of weight zero, with trace function tM(·; kn): G(kn) ! C. There exists an integer r > 1 and a compact subgroup K ⊂ Ur(C) of the unitary group such that the sums X S(M; χ) = tM(x; kn)χ(x) x2G(kn) for complex-valued characters χ of G(kn) become equidistributed on average in C with respect to the image by the trace of the probability Haar measure µ on K. That is, for any bounded continuous function f : C ! C, the following equality holds: 1 X 1 X Z (1.1) lim f(S(M; χ)) = f(Tr(x))dµ(x); N!+1 N jG(kn)j K 16n6N χ 3 where χ runs over all characters of G(kn). The general version of this theorem appears as Theorem 5.3.3. Under an additional assump- tion (which holds for all tori and all abelian varieties, at least), we could also deduce it from Theorem 5.4.1, which is a more precise equidistribution result for conjugacy classes of Frobenius in K. Remark 1.1.4. (1) In the classical setting of Ga and the Fourier transform, the group K is a maximal compact subgroup of the arithmetic monodromy group of the `-adic Fourier transform of M.
Load more

Recommended publications
  • The Work of Pierre Deligne
    THE WORK OF PIERRE DELIGNE W.T. GOWERS 1. Introduction Pierre Deligne is indisputably one of the world's greatest mathematicians. He has re- ceived many major awards, including the Fields Medal in 1978, the Crafoord Prize in 1988, the Balzan Prize in 2004, and the Wolf Prize in 2008. While one never knows who will win the Abel Prize in any given year, it was virtually inevitable that Deligne would win it in due course, so today's announcement is about as small a surprise as such announcements can be. This is the third time I have been asked to present to a general audience the work of the winner of the Abel Prize, and my assignment this year is by some way the most difficult of the three. Two years ago, I talked about John Milnor, whose work in geometry could be illustrated by several pictures. Last year, the winner was Endre Szemer´edi,who has solved several problems with statements that are relatively simple to explain (even if the proofs were very hard). But Deligne's work, though it certainly has geometrical aspects, is not geometrical in a sense that lends itself to pictorial explanations, and the statements of his results are far from elementary. So I am forced to be somewhat impressionistic in my description of his work and to spend most of my time discussing the background to it rather than the work itself. 2. The Ramanujan conjecture One of the results that Deligne is famous for is solving a conjecture of Ramanujan. This conjecture is not the obvious logical starting point for an account of Deligne's work, but its solution is the most concrete of his major results and therefore the easiest to explain.
    [Show full text]
  • MAPPING STACKS and CATEGORICAL NOTIONS of PROPERNESS Contents 1. Introduction 2 1.1. Introduction to the Introduction 2 1.2
    MAPPING STACKS AND CATEGORICAL NOTIONS OF PROPERNESS DANIEL HALPERN-LEISTNER AND ANATOLY PREYGEL Abstract. One fundamental consequence of a scheme being proper is that there is an algebraic space classifying maps from it to any other finite type scheme, and this result has been extended to proper stacks. We observe, however, that it also holds for many examples where the source is a geometric stack, such as a global quotient. In our investigation, we are lead naturally to certain properties of the derived category of a stack which guarantee that the mapping stack from it to any geometric finite type stack is algebraic. We develop methods for establishing these properties in a large class of examples. Along the way, we introduce a notion of projective morphism of algebraic stacks, and prove strong h-descent results which hold in the setting of derived algebraic geometry but not in classical algebraic geometry. Contents 1. Introduction 2 1.1. Introduction to the introduction2 1.2. Mapping out of stacks which are \proper enough"3 1.3. Techniques for establishing (GE) and (L)5 1.4. A long list of examples6 1.5. Comparison with previous results7 1.6. Notation and conventions8 1.7. Author's note 9 2. Artin's criteria for mapping stacks 10 2.1. Weil restriction of affine stacks 11 2.2. Deformation theory of the mapping stack 12 2.3. Integrability via the Tannakian formalism 16 2.4. Derived representability from classical representability 19 2.5. Application: (pGE) and the moduli of perfect complexes 21 3. Perfect Grothendieck existence 23 3.1.
    [Show full text]
  • Notes on Tannakian Categories
    NOTES ON TANNAKIAN CATEGORIES JOSE´ SIMENTAL Abstract. These are notes for an expository talk at the course “Differential Equations and Quantum Groups" given by Prof. Valerio Toledano Laredo at Northeastern University, Spring 2016. We give a quick introduction to the Tannakian formalism for algebraic groups. The purpose of these notes if to give a quick introduction to the Tannakian formalism for algebraic groups, as developed by Deligne in [D]. This is a very powerful technique. For example, it is a cornerstone in the proof of geometric Satake by Mirkovi´c-Vilonenthat gives an equivalence of tensor categories between the category of perverse sheaves on the affine Grassmannian of a reductive algebraic group G and the category of representations of the Langlands dual Gb, see [MV]. Our plan is as follows. First, in Section 1 we give a rather quick introduction to representations of al- gebraic groups, following [J]. The first main result of the theory says that we can completely recover an algebraic group by its category of representations, see Theorem 1.12. After that, in Section 2 we formalize some properties of the category of representations of an algebraic group G into the notion of a Tannakian category. The second main result says that every Tannakian category is equivalent to the category of repre- sentations of an appropriate algebraic group. Our first definition of a Tannakian category, however, involves the existence of a functor that may not be easy to construct. In Section 3 we give a result, due to Deligne, that gives a more intrinsic characterization of Tannakian categories in linear-algebraic terms.
    [Show full text]
  • Annual Report 2017-2018
    Annual Review 2017 | 2018 ONTENTS C 1 Overview 1 2 Profile 4 3 Research 6 4 Events 9 5 Personnel 13 6 Mentoring 17 7 Structures 18 APPENDICES R1 Highlighted Papers 20 R2 Complete List of Papers 23 E1 HIMR-run Events 29 E2 HIMR-sponsored Events 31 E3 Focused Research Events 39 E4 Future Events 54 P1 Fellows Joining in 2017|2018 59 P2 Fellows Leaving since September 2017 60 P3 Fellows Moving with 3-year Extensions 62 P4 Future Fellows 63 M1 Mentoring Programme 64 1. Overview This has been another excellent year for the Heilbronn Institute, which is now firmly established as a major national mathematical research centre. HIMR has developed a strong brand and is increasingly influential in the UK mathematics community. There is currently an outstanding cohort of Heilbronn Research Fellows doing first-rate research. Recruitment of new Fellows has been most encouraging, as is the fact that many distinguished academic mathematicians continue to work with the Institute. The research culture at HIMR is excellent. Members have expressed a high level of satisfaction. This is especially the case with the Fellows, many of whom have chosen to continue their relationships with the Institute. Our new Fellows come from leading mathematics departments and have excellent academic credentials. Those who left have moved to high-profile groups, including several to permanent academic positions. We currently have 29 Fellows, hosted by 6 universities. We are encouraged by the fact that of the 9 Fellows joining us this year, 5 are women. The achievements of our Fellows this year again range from winning prestigious prizes to publishing in the elite mathematical journals and organising major mathematical meetings.
    [Show full text]
  • Homotopical and Higher Categorical Structures in Algebraic Geometry
    Homotopical and Higher Categorical Structures in Algebraic Geometry Bertrand To¨en CNRS UMR 6621 Universit´ede Nice, Math´ematiques Parc Valrose 06108 Nice Cedex 2 France arXiv:math/0312262v2 [math.AG] 31 Dec 2003 M´emoire de Synth`ese en vue de l’obtention de l’ Habilitation a Diriger les Recherches 1 Je d´edie ce m´emoire `aBeussa la mignonne. 2 Pour ce qui est des choses humaines, ne pas rire, ne pas pleurer, ne pas s’indigner, mais comprendre. Spinoza Ce dont on ne peut parler, il faut le taire. Wittgenstein Ce que ces gens l`asont cryptiques ! Simpson 3 Avertissements La pr´esent m´emoire est une version ´etendue du m´emoire de synth`ese d’habilitation originellement r´edig´epour la soutenance du 16 Mai 2003. Le chapitre §5 n’apparaissait pas dans la version originale, bien que son contenu ´etait pr´esent´e sous forme forte- ment r´esum´e. 4 Contents 0 Introduction 7 1 Homotopy theories 11 2 Segal categories 16 2.1 Segalcategoriesandmodelcategories . 16 2.2 K-Theory ................................. 21 2.3 StableSegalcategories . .. .. 23 2.4 Hochschild cohomology of Segal categories . 24 2.5 Segalcategoriesandd´erivateurs . 27 3 Segal categories, stacks and homotopy theory 29 3.1 Segaltopoi................................. 29 3.2 HomotopytypeofSegaltopoi . 32 4 Homotopical algebraic geometry 35 4.1 HAG:Thegeneraltheory. 38 4.2 DAG:Derivedalgebraicgeometry . 40 4.3 UDAG: Unbounded derived algebraic geometry . 43 4.4 BNAG:Bravenewalgebraicgeometry . 46 5 Tannakian duality for Segal categories 47 5.1 TensorSegalcategories . 49 5.2 Stacksoffiberfunctors . .. .. 55 5.3 TheTannakianduality .......................... 58 5.4 Comparison with usual Tannakian duality .
    [Show full text]
  • Henri Darmon
    Henri Darmon Address: Dept of Math, McGill University, Burnside Hall, Montreal, PQ. E-mail: [email protected] Web Page: http://www.math.mcgill.ca/darmon Telephone: Work (514) 398-2263 Home: (514) 481-0174 Born: Oct. 22, 1965, in Paris, France. Citizenship: Canadian, French, and Swiss. Education: 1987. B.Sc. Mathematics and Computer Science, McGill University. 1991. Ph.D. Mathematics, Harvard University. Thesis: Refined class number formulas for derivatives of L-series. University Positions: 1991-1994. Princeton University, Instructor. 1994-1996. Princeton University, Assistant Professor. 1994-1997. McGill University, Assistant Professor. 1997-2000. McGill University, Associate Professor. 2000- . McGill University, Professor. 2005-2019. James McGill Professor, McGill University. Other positions: 1991-1994. Cercheur hors Qu´ebec, CICMA. 1994- . Chercheur Universitaire, CICMA. 1998- . Director, CICMA (Centre Interuniversitaire en Calcul Math´ematique Alg´ebrique). 1999- . Member, CRM (Centre de Recherches Math´ematiques). 2005-2014. External member, European network in Arithmetic Geometry. Visiting Positions: 1991. IHES, Paris. 1995. Universit´a di Pavia. 1996. Visiting member, MSRI, Berkeley. 1996. Visiting professor and guest lecturer, University of Barcelona. 1997. Visiting Professor, Universit´e Paris VI (Jussieu). 1997. Visitor, Institut Henri Poincar´e. 1998. Visiting Professor and NachDiplom lecturer, ETH, Zuric¨ h. 1999. Visiting professor, Universit`a di Pavia. 2001. Visiting professor, Universit`a di Padova. 2001. Korea Institute for Advanced Study. 2002. Visiting professor, RIMS and Saga University (Japan). 1 2003. Visiting Professor, Universit´e Paris VI, Paris. 2003. Visiting professor, Princeton University. 2004. Visiting Professor, Universit´e Paris VI, Paris. 2006. Visiting Professor, CRM, Barcelona, Spain. 2008. Visiting Professor, Universit´e Paris-Sud (Orsay).
    [Show full text]
  • Modified Mixed Realizations, New Additive Invariants, and Periods Of
    MODIFIED MIXED REALIZATIONS, NEW ADDITIVE INVARIANTS, AND PERIODS OF DG CATEGORIES GONC¸ALO TABUADA Abstract. To every scheme, not necessarily smooth neither proper, we can associate its different mixed realizations (de Rham, Betti, ´etale, Hodge, etc) as well as its ring of periods. In this note, following an insight of Kontsevich, we prove that, after suitable modifications, these classical constructions can be extended from schemes to the broad setting of dg categories. This leads to new additive invariants of dg categories, which we compute in the case of differential operators, as well as to a theory of periods of dg categories. Among other applications, we prove that the ring of periods of a scheme is invariant under projective homological duality. Along the way, we explicitly describe the modified mixed realizations using the Tannakian formalism. 1. Modified mixed realizations Given a perfect field k and a commutative Q-algebra R, Voevodsky introduced in [49, §2] the category of geometric mixed motives DMgm(k; R). By construction, this R-linear rigid symmetric monoidal triangulated category comes equipped with a ⊗-functor M(−)R : Sm(k) → DMgm(k; R), defined on smooth k-schemes of finite type, and with a ⊗-invertible object T := R(1)[2] called the Tate motive. Moreover, when k is of characteristic zero, the preceding functor can be extended from Sm(k) to the category Sch(k) of all k-schemes of finite type. Recall also the construction of Voevodsky’s big category of mixed motives DM(k; R). This R-linear symmet- ric monoidal triangulated category admits arbitrary direct sums and DMgm(k; R) identifies with its full triangulated subcategory of compact objects.
    [Show full text]
  • The Fermat Conjecture
    THE FERMAT CONJECTURE by C.J. Mozzochi, Ph.D. Box 1424 Princeton, NJ 08542 860/652-9234 [email protected] Copyright C.J. Mozzochi, Ph.D. - 2011 THE FERMAT CONJECTURE FADE IN: EXT. OXFORD GRAMMAR SCHOOL -- DAY (1963) A ten-year-old ANDREW WILES is running toward the school to talk with his friend, Peter. INT. HALL OF SCHOOL Andrew Wiles excitedly approaches PETER, who is standing in the hall. ANDREW WILES Peter, I want to show you something very exciting. Let’s go into this classroom. INT. CLASSROOM Andrew Wiles writes on the blackboard: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11...positive integers. 3, 5, 7, 11, 13...prime numbers. 23 = 2∙ 2∙ 2 35 = 3∙ 3∙ 3∙ 3∙ 3 xn = x∙∙∙x n-terms and explains to Peter the Fermat Conjecture; as he writes on the blackboard. ANDREW WILES The Fermat Conjecture states that the equation xn + yn = zn has no positive integer solutions, if n is an integer greater than 2. Note that 32+42=52 so the conjecture is not true when n=2. The famous French mathematician Pierre Fermat made this conjecture in 1637. See how beautiful and symmetric the equation is! Peter, in a mocking voice, goes to the blackboard and writes on the blackboard: 1963-1637=326. PETER Well, one thing is clear...the conjecture has been unproved for 326 years...you will never prove it! 2. ANDREW WILES Don’t be so sure. I have already shown that it is only necessary to prove it when n is a prime number and, besides, Fermat could not have known too much more about it than I do.
    [Show full text]
  • Tannakian Formalism Applied to the Category of Mixed Hodge Structures
    Tannakian Formalism Applied to the Category of Mixed Hodge Structures Licentiate Thesis Submitted to the Department of Mathematics and Statistics in Partial Fulfillment of the Requirements for the degree Licentiate of Philosophy Field of Mathematics By Antti Veilahti University of Helsinki Faculty of Science June 2008 c Copyright by Antti Veilahti 2008 All Rights Reserved ii Abstract Tannakian Formalism Applied to the Category of Mixed Hodge Structures Antti Veilahti Many problems in analysis have been solved using the theory of Hodge struc- tures. P. Deligne started to treat these structures in a categorical way. Following him, we introduce the categories of mixed real and complex Hodge structures. We describe and apply the Tannakian formalism to treat the category of mixed Hodge structures as a category of representations of a certain affine group scheme. Using this approach we give an explicit formula and method for calculating the Ext1-groups in the category and show that the higher Ext-groups vanish. Also, we consider some examples illustrating the usefulness of these calculations in analysis and geometry. iii Acknowledgements During my graduate studies I have received a lot of support from Professor Kari Vilonen, who made it possible for me to study in Northwestern University with all the wonderful mathematicians. With his excellent knowledge in many different areas he kept guiding me and helped me to find paths to the very core of modern mathematics. Especially, I would like to thank him for the discussions on Hodge theory that ultimately led me to this thesis. I would also like to thank Professor Kalevi Suominen, for reading the manu- script and his comments.
    [Show full text]
  • Arxiv:2005.14365V1 [Math.NT] 29 May 2020
    VARIATIONS IN THE DISTRIBUTION OF PRINCIPALLY POLARIZED ABELIAN VARIETIES AMONG ISOGENY CLASSES EVERETT W. HOWE Abstract. We show that for a large class of rings R, the number of principally polarized abelian varieties over a finite field in a given simple ordinary isogeny class and with endomorphism ring R is equal either to 0, or to a ratio of class numbers associated to R, up to some small computable factors. This class of rings includes the maximal order of the CM field K associated to the isogeny class (for which the result was already known), as well as the order R generated over Z by Frobenius and Verschiebung. For this latter order, we can use results of Louboutin to estimate the appropriate ratio of class numbers in terms of the size of the base field and the Frobenius angles of the isogeny class. The error terms in our estimates are quite large, but the trigonometric terms in the estimate are suggestive: Combined with a result of Vl˘adut¸on the distribution of Frobenius angles of isogeny classes, they give a heuristic argument in support of the theorem of Katz and Sarnak on the limiting distribution of the multiset of Frobenius angles for principally polarized abelian varieties of a fixed dimension over finite fields. 1. Introduction In this paper we consider the problem of estimating the number of isomorphism classes of principally polarized abelian varieties (A; λ) such that A lies in a given isogeny class of simple ordinary abelian varieties over a finite field. We approach this problem by subdividing isogeny classes into their strata, which are the subsets of an isogeny class consisting of abelian varieties sharing the same endomorphism ring.
    [Show full text]
  • Grothendieck Toposes As Unifying ‘Bridges’ in Mathematics
    Grothendieck toposes as unifying ‘bridges’ in Mathematics Mémoire pour l’obtention de l’habilitation à diriger des recherches* Olivia Caramello *Successfully defended on 14 December 2016 in front of a Jury composed by Alain Connes (Collège de France) - President, Thierry Coquand (University of Gothenburg), Jamshid Derakhshan (University of Oxford), Ivan Fesenko (University of Nottingham), Anatole Khelif (Universitè de Paris 7), Frédéric Patras (Universitè de Nice - Sophia Antipolis), Enrico Vitale (Universitè Catholique de Louvain) and Boban Velickovic (Universitè de Paris 7). 1 Contents 1 Introduction 2 2 General theory 5 2.1 The ‘bridge-building’ technique . .6 2.1.1 Decks of ‘bridges’: Morita-equivalences . .7 2.1.2 Arches of ‘bridges’: Site characterizations . 11 2.1.3 The flexibility of Logic: a two-level view . 13 2.1.4 The duality between real and imaginary . 18 2.1.5 A theory of ‘structural translations’ . 20 2.2 Theories, Sites, Toposes ....................... 21 2.3 Universal models and classifying toposes . 30 3 Theory and applications of toposes as ‘bridges’ 35 3.1 Characterization of topos-theoretic invariants . 35 3.2 De Morgan and Boolean toposes . 39 3.3 Atomic two-valued toposes . 40 3.3.1 Fraïssé’s construction from a topos-theoretic persective . 40 3.3.2 Topological Galois Theory . 45 3.3.3 Motivic toposes . 50 4 Dualities, bi-interpretations and Morita-equivalences 53 4.1 Dualities, equivalences and adjunctions for preordered structures and topological spaces . 54 4.1.1 The topos-theoretic construction of Stone-type dualities and adjunctions . 54 4.1.2 A general method for building reflections .
    [Show full text]
  • January 2019 Volume 66 · Issue 01
    ISSN 0002-9920 (print) ISSN 1088-9477 (online) Notices ofof the American MathematicalMathematical Society January 2019 Volume 66, Number 1 The cover art is from the JMM Sampler, page 84. AT THE AMS BOOTH, JMM 2019 ISSN 0002-9920 (print) ISSN 1088-9477 (online) Notices of the American Mathematical Society January 2019 Volume 66, Number 1 © Pomona College © Pomona Talk to Erica about the AMS membership magazine, pick up a free Notices travel mug*, and enjoy a piece of cake. facebook.com/amermathsoc @amermathsoc A WORD FROM... Erica Flapan, Notices Editor in Chief I would like to introduce myself as the new Editor in Chief of the Notices and share my plans with readers. The Notices is an interesting and engaging magazine that is read by mathematicians all over the world. As members of the AMS, we should all be proud to have the Notices as our magazine of record. Personally, I have enjoyed reading the Notices for over 30 years, and I appreciate the opportunity that the AMS has given me to shape the magazine for the next three years. I hope that under my leadership even more people will look forward to reading it each month as much as I do. Above all, I would like the focus of the Notices to be on expository articles about pure and applied mathematics broadly defined. I would like the authors, topics, and writing styles of these articles to be diverse in every sense except for their desire to explain the mathematics that they love in a clear and engaging way.
    [Show full text]