What is God saying to you in 2021? One powerful prophetic word can change your whole life. Review: What an exact and accurate word. So encouraging. (Mary F. U.S.A. Abelian Varieties David Mumford What it is about. This is the only complete book about abelian varieties written from a modern point of view. It is the canonical reference. Electronic Version. Here is a 83 MB scan of the book in PDF format. (The book has long been out of print.) MathSciNe Abelian varieties are from a complex analytic point of view the simplest possible spaces — just tori and thus groups. But the curious thing is that tori don't fit easily into projective space. The sidebar shows the Kummer quartic with its sixteen double points: a 2-dimensional principally polarized Abelian variety mod inversion mapped to 3-space via |2Θ|. The interaction of an ample line bundle with the group structure on an Abelian variety is the subject of the first paper below as well. David Mumford was awarded the 2007 AMS Steele Prize for Mathematical Exposition. According to the citation: Abelian Varieties remains the definitive account of the subject the classical theory is beautifully intertwined with the modern theory, in a way which sharply illuminates both [It] will remain for the foreseeable future a classic to which the reader returns over and over This chapter reviews the theory of abelian varieties emphasizing those points of particular interest to arithmetic geometers. In the main it follows Mum-ford's book [16] except that most results are stated relative to an arbitrary base field, some additional results are proved, and étale cohomology is included. Many proofs have had to be omitted or only sketched. The reader is assumed to be familier with [10, Chaps. II, III] and (for a few sections that can be skipped) some étale.
ABELIAN VARIETIES BRYDEN CAIS A canonical reference for the subject is Mumford's book [6], but Mumford generally works over an algebraically closed ﬁeld (though his arguments can be modiﬁed to give results over an arbitrary base ﬁeld). Milne's article [4] is also a good source and allows a general base ﬁeld. These notes borrow heavily from van der Geer and Moonen [5] Typos in the new printing of Mumford's \Abelian Varieties Below is a list of typographical errors which I found. I include punctuation errors which were introduced, but I do not attempt to list all of the original punctuation errors (misplaced commas, etc.) The page numbering and line numbering below refers to the new version of the book, no Abelian Varieties (Tata Institute of Fundamental Research) by David Mumford (Author), C. P. Ramanujam (Contributor), Yuri Manin (Contributor) & 3.9 out of 5 stars 4 ratings. ISBN-13: 978-8185931869. ISBN-10: 8185931860. Why is ISBN important? ISBN. This bar-code number lets you verify that you're getting exactly the right version or edition of a book. The 13-digit and 10-digit formats both. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory. For example, they occur naturally when studying line bundles over an algebraic variety or the arithmetic of a number field I divided my papers up into five categories - Moduli Spaces, Abelian Varieties, Curves and Surfaces, Particular Examples and Diverse. There is some overlap, e.g. moduli of curves is not in the curves section and moduli of abelian varieties is not in the moduli section. Following a long tradition in classical geometry (seen clearly in the work of Coxeter), I have always loved finding new and sometimes exotic examples and have collected eight papers of this sort in the Examples section
Milne, Abelian Varieties (mainly Sections 1, 2, 7, 8, 9, 10, 11, 12, 16, 19) Milne, Jacobian Varieties (Sections 1-6 and 10) Mumford. Abelian Varieties; Mumford, Curves and their Jacobians; Rosen, Abelian Varieties over C; Swinnerton-Dyer, Analytic theory of abelian varieties; Modular Abelian Varieties Rational points on abelian varieties The basic result, the Mordell-Weil theorem in Diophantine geometry, says that A (K), the group of points on A over K, is a finitely-generated abelian group. A great deal of information about its possible torsion subgroups is known, at least when A is an elliptic curve
an abelian variety of dimension1 g>1. In general, it is not possible to write down explicit 1The case gD2is something of an exception to this statement. Every abelian variety of dimension 2is the Jacobian variety of a curve of genus 2, and every curve of genus 2has an equation of the form Y2Z4 Df0X6 Cf1X5ZCC f6Z6 A good reference for today is Mumford's Abelian varieties (MR282985) or Milne's notes. 1 General facts about abelian varieties Fix a ﬁeld k. Many of the results about abelian varieties over C continue to hold over k. However, the proofs are quite di↵erent and more complicated. We give some indications as to how the theory is developed, but omit most of the arguments. 1.1. MATH 731: TOPICS IN ALGEBRAIC GEOMETRY I { ABELIAN VARIETIES BHARGAV BHATT Course Description. The goal of the rst half of this class is to introduce and study the basic structure theory of abelian varieties, as covered in (say) Mumford's book. In the second half of the course, we shall discuss derived categories and the Fourier{Mukai transform, and give some geometric applications. Contents.
We will denote the Mumford{Tate group of Awith G MT(A). Intermezzo: Elliptic curves and abelian varieties over Q Let Ebe an elliptic curve, de ned by an equation Y2 = X3 + aX+ b. If the coe cients aand blie in Q, then we say that Eis de ned over Q. Similarly, if an abelian variety is the solution set of polynomials wit Algebraic Theory of Abelian Varieties via Schemes 小林真一 1 前書き この講演ではMumford のAbelian varieties [Mum] の2章Algebraic theory via varieties と 3章Algebraic theory via schemes について解説する. 内容はアーベル多様体の純代数的な取 り扱いについてである. これにより基礎体の標数が正の場合にもアーベル多様体が扱える Our main reference is Abelian Varieties, by Mumford. We will 1. classify abelian varieties over finite fieldsF and algebraic closures of finite fieldsF (Honda-Tate Theory). We will also classify -divisible groups up to isogeny (Dieudonn´e, Manin). With some more work, we can get classification up to isomorphism. Studying a variety over finite fields helps us understand abelian varieties over. In his book Abelian Varieties, David Mumford defines an abelian variety over an algebraically closed field k k to be a complete algebraic group over k k. Remarkably, any such thing is an abelian algebraic group. The assumption of connectedness is necessary for that conclusion. Automatic abeliannes
A complete connected group variety is called an abelian variety . As we shall see, they are projective and (fortunately) commutative. Their group laws will be written additively. An afﬁne group variety is called a linear algebraic group. Each such variety can be realized as a closed subgroup of GLnfor some n(Waterhouse1979, 3.4). 2Rigidit David Mumford, Abelian varieties, Oxford Univ. Press 1970 A. Polishchuk, Abelian varieties, theta functions and the Fourier transform , Cambridge Univ. Press 2003 M. Demazure , P. Gabriel , Groupes algebriques , tome 1 (later volumes never appeared), Mason and Cie, Paris 1970 - has functor of points point of view (listed also under scheme theory); for review se abelian varieties have a distinguished role to play { in some sense Hodge theory is a formal algebraic generalization of the theory of abelian varieties, and the miracle is that this \formal generalization itself carries a lot of geometric content. 2. The Mumford-Tate Group of a Polarized Q-Hodge Structure To every polarized Q-Hodge structure V we will associate a nontrivial \invariant, the.
Mumford compactiﬁcation for curves [DM69], whereas there are quite a lot of compactiﬁcations of the moduli of abelian varieties ([AMRT75], [FC90]). PLANAR CUBIC CURVES 3 Nevertheless in the present article we will construct one and only one new compactiﬁcation SQ g,K of the moduli of abelian varieties. This compact-iﬁcation is natural enough because, as we will see below, there are. Abelian varieties: Milne's notes and the book draft of van der Geer and Moonen. 1 - Elliptic Curves 2 - Smoothness 3 - ECs over C, j-invariant 4 - Modular Curve 5 - ECs are Cubics 6 - Cubics are ECs - Part 1 7 - Cohomology and Base change 8 - Cubics are ECs - Part 2 9 - Complements on Flatness, Relative Curves 10 - Torsion and Tate module 11 - Endomorphism It was constructed by Grothendieck & 1961/62, and also described by Mumford (1966) and Kleiman (2005). The Picard variety is dual to the Albanese variety of classical algebraic geometry. In the cases of most importance to classical algebraic geometry, for a non-singular complete variety V over a field of characteristic zero, the connected component of the identity in the Picard scheme is an abelian variety written Pic 0 ( V ) D. Mumford, Families of abelian varieties, In:Algebraic Groups and Discontinuous Subgroups, Proc. Symp. Pure Math., Vol. 9, Amer. Math. Soc., Providence, Rhode Island (1966), pp. 347-352. Google Scholar 11 All abelian varieties over a eld are projective (i.e. admit an ample line bundle). Proof. Mumford [5] proves this in the case where the eld is algebraically closed. A trick shows that if Xis a proper k-scheme and X k is projective, then Xis projective. 3. Theorem 3. Let Aand A0be abelian varieties over a eld k. For any prime ', the natural map t ': Z ' Z Hom k(A;A 0) !Hom k(A['1];A0.
So it follows that all Hodge cycles on an abelian variety have canonical l -adic realizations that are defined over the same finite extension of k. In particular, an open sub-group of the Galois group fixes all Hodge cycles; that is, it maps into the Mumford-Tate group of A. Share. Improve this answer. edited Dec 1 '11 at 15:23 David Mumford. Abelian varieties, pages viii+242. Tata Institute of Fundamental. Research Studies in Mathematics, No. 5. Published for the Tata. Institute of Fundamental Research, Bombay, 1970. Christina Birkenhake and Herbert Lange. Complex abelian varieties. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 302. Springer-Verlag, Berlin, 2004 ON MUMFORD'S CONSTRUCTION OF DEGENERATING ABELIAN VARIETIES VALERY ALEXEEV AND IKU NAKAMURA (Received May 8, 1998) Abstract. For a one-dimensional family of abelian varieties equipped with principal theta divisors a canonical limit is constructed as a pair consisting of a reduced projective variety and a Cartier divisor on it. Properties of such pairs are established
group and Mumford's theta group then yields a system of canonical projective coordinates for the abelian variety. It is these coordinates for the target abelian variety and the target point that one may try to compute explicitly. 1.2 Main results We ﬁrst give a criterion when the target abelian variety B= A/Gadmits a principal polarization Mumford, Algebraic Geometry I, Complex Projective Varieties Mumford, Lectures on Curves on Algebraic Surfaces Mumford, Abelian Varieties Back to Ching-Li Chai's Home Page. representations are of Mumford's type. Abelian varieties with a Galois representation of this kind are the examples of the smallest dimension where the Mumford{Tate conjecture is unsettled. Some evidence supporting the Mumford{Tate conjecture is provided by the fact that if X is an abelian variety over a number eld such that one associated '-adic representation is of Mumford's type, then. Abelian varieties Shimura varieties (A g,1) A = Cn/Γ + polarization S = Γ\X + complex structure X hermitian symmetric space X ' G(R)/Z G(R)K ∞, G Q a Q-reductive group torsion points CM points torsion subvarieties component of translate by Hecke operator of X = B +P, P ∈ A tors, B abelian subvariety a Shimura subvariety S H H, H Q = T.Hder Q [n] Hecke operator Manin-Mumford conjecture.
The main reference for this post is Mumford's Abelian varieties. 10. The Poincaré line bundle and the biduality map. The first step in understanding the biduality of abelian varieties is to understand the universal line bundle on . By definition, parametrizes line bundles on algebraically equivalent to zero, so there is a universal line bundle , called the Poincaré line bundle, on . The. Mumford conjecture for an abelian variety over a number ﬁeld, when it has supersingular reduction at a prime dividing p, by combining the methods of Bogomolov, Hrushovski, and Pink-Roessler. Our proof here is quite simple and short, and neither p-adic Hodge theory nor model theory is used. The observation is that a power of a lift of the Frobenius element at a supersingular prime acts on the. Theorem (Manin-Mumford Conjecture, proved by Raynaud in 1983) Let X/K be a curve of genus g ≥ 2. Then X(K¯)∩J(K¯) tors is ﬁnite. There are many diﬀerent proofs of this theorem in the literature. We will present an elegant short proof due to Ken Ribet based on the notion of almost rational torsion points. Torsion Points on Abelian Varieties Matthew Baker Almost rational torsion points. In: Moduli of abelian varieties (Texel Island, 1999), 255-298, Progress in Math. 195, Birkhäuser, 2001. Corrigendum. Bas Edixhoven, Ben Moonen and Frans Oort Open problems in algebraic geometry. Bull. Sci. Math. 125 (2001), 1- Ben Moonen and Yuri Zarhin Hodge classes on abelian varieties of low dimension. Math. Ann. 315 (1999), 711-733
Mumford treats abelian varieties first from a complex analytic point of view, before moving onto an old-style variety-theoretic manner, before finally dealing with the modern scheme-theoretic language. Christina Birkenhake and Herbert Lange, Complex abelian varieties. If there is something about complex analytic abelian varieties you would like to know, this book probably contains it. In. On Mumford's construction of degenerating abelian varieties Valery Alexeev, Iku Nakamura 1999 Tohoku mathematical journal On Mumford's construction of degenerating abelian varieties. Tohoku mathematical journal (1999) 399-420 MLA; Harvard; CSL-JSON; BibTeX; Internet Archive. We are a US 501(c)(3) non-profit library, building a global archive of Internet sites and other cultural artifacts. This is a reprinting of the revised second edition (1974) of David Mumford's classic 1970 book. It gives a systematic account of the basic results about abelian varieties. It includes expositions of analytic methods applicable over the ground field of complex numbers, as well as of scheme-theoretic methods used to deal with inseparable isogenies when the ground field has positive characteristic Abelian varieties of the same kind, along with a choice of basis for H1(A;Z), are parameterized by the period domain D= G(R)=K. 1. Note: points of Dare classes of gHin terms of ˚. ˚ gH= g˚g 1. Main idea: family comes from the Mumford-Tate domain: D ˚= M(R)=M(R)\ KˆD. This should have the properties that for all Hodge structures H02D ˚, 1. MT(H0) ˆM 2.any Hodge tensor for Ais a Hodge.
Abelian varieties are special examples of projective varieties. As such they can be described by a set of homogeneous polynomial equations. The theory of abelian varieties originated in the beginning of the ninetheenth centrury with the work of Abel and Jacobi. The subject of this book is the theory of abelian varieties over the field of complex numbers, and it covers the main results of the. ABELIAN VARIETIES AND AX-LINDEMANN-WEIERSTRASS 3 2. Abelianvarieties In this section we will deﬁne abelian varieties and their morphisms and state their basic properties, and those of their torsion points. We work over an arbi-trary base ﬁeld, although some of the theorems will include a condition on th The Mumford--Tate conjecture is a precise way of saying that certain extra structure on the -adic étale cohomology groups of~ (namely, a Galois representation) and certain extra structure on the singular cohomology groups of~ (namely, a Hodge structure) convey the same information. The main result of this paper says that if and~ are abelian. Abelian Varieties. Now back in print, the revised edition of this popular study gives a systematic account of the basic results about abelian varieties. Mumford describes the analytic methods and results applicable when the ground field k is the complex field C and discusses the scheme-theoretic methods and results used to deal with inseparable. Seminar on abelian varieties Prof. Dr. Uwe Jannsen, Dr. Yigeng Zhao Wednesday, 10-12 h, M 006 Introduction The aim of this seminar is to study some basic theories of abelian varieties over an algebraic closed eld k, which are one of the most important and most studied objects in arithmetic geometry. Abelian varieties are proper algebraic varieties that carry a group structure, and are a higher.
SYZYGIES OF ABELIAN VARIETIES GIUSEPPE PARESCHI Let Abe an ample line bundle on an abelian variety X(over an algebraically closed eld). A theorem of Koizumi ([Ko], [S]), developing Mumford's ideas and results ([M1]), states that ifm 3 the line bundle L= A membeds Xin projective space as a projectively normal variety. Moreover, a celebrated theorem of Mumford ([M2]), slightly re ned by Kempf. Around Hodge, Tate and Mumford-Tate conjectures on abelian varieties Victoria Cantoral-Farf an Advisor : Prof. Marc Hindry Preliminaries De nition. An abelian variety is a projective algebraic variety that is also an algebraic group with a group law which is commutative. Example. An elliptic curve is an abelian variety of dimension 1 with a group law which is explained here: Let de ne V = H 1.
Abelian Varieties by David Mumford, 9788185931869, available at Book Depository with free delivery worldwide ABELIAN VARIETIES (MATH 731) BHARGAV BHATT Goal. The goal of the ﬁrst half of this class is to introduce and study the basic structure theory of abelian varieties, as covered in (say) Mumford's book. In the second half of the course, we shall discuss derived categories and the Fourier-Mukai transform, and give some geometric applications. Prerequisites. I will use the language of schemes. relative Manin-Mumford conjecture for one dimensional families of semi-abelian surfaces. Applications include special cases of the Zilber-Pink conjecture for curves in a mixed Shimura variety of dimension four, as well as the study of polynomial Pell equations with non-separable discriminants. Contents 1 Introduction Abelian varieties over local and global fields. TCC course, Spring 2016. Lecture notes (If you have any comments, please email me.). Course description: This course is a selection of topics on abelian varieties and rational points on them.. Assessment: By essay.You need to submit your essay to me by email by 1 May 2016.You are asked to write a text of about15 pages on a subject related to.
For certain abelian varieties A, we show that the usual Hodge conjecture for all powers of A implies the general Hodge conjecture for . Mathematics Subject Classiﬁcation (1991): 14C30. Key words: Hodge conjecture, algebraic cycle, abelian variety, Kuga ﬁber variety 1. Introduction The arithmetic ﬁltration on the cohomology of a smooth complex projective variety X is deﬁned by letting F. Abelian varieties, l-adic representations, and l-independence M. Larsen* and R. Pink Let A be an abelian variety of dimension g over a global ﬁeld K. Let K¯ denote a separable closure of K. If ' is a rational prime distinct from the characteristic of K, the Galois group Gal(K/K¯ ) acts on the group A[' n] ∼=(Z/' Z)2g of 'n-torsion points of A(K¯). Therefore, it acts continuously. Browse Our Great Selection of Books & Get Free UK Delivery on Eligible Orders
D. Mumford: The Red book of varieties and schemes. Springer LN 1358. U. Goertz, T. Wedhorn: Algebraic Geometry I. Vieweg. R. Hartshorne: Algebraic Geometry GTM 52. Springer. D. Eisenbud and J. Harris, The Geometry of Schemes, GTM 197, Springer. Vakil: Foundations of algebraic geometry. Online lectures. Shafarevich, Basic Algebraic Geometry I + II. Schedule1 Oct 11: De nition a ne algebraic set. David Mumford: Abelian Varieties over a Perfect Field and Dieudonne` Modules: 1967: James Milne: The Conjecture of Birch and Swinnerton-Dyer for Constant Abelian Varieties over Function Fields: 1967: Nelson Max: Homemorphisms of Sn x Sl: 1967: Sandy Grabiner: Andrew Gleason: Radical Banach Algebras and Formal Power Series: 1967 : Hubert Goldschmidt: Shlomo Sternberg: Overdetermined Systems of. I also think Mumford also wants to avoid Jacobians since many times Jacobians are used to motivate the study of abelian varieties. Also, since the geometry of Jacobians is so related to the geometry of curves, one can also use the geometry of curves to study these varieties. It seems Mumford wanted a purely algebraic theory that developed the abstract general theory of abelian varieties, and. Notation and conventions. (0.1) In general, k denotes an arbitrary ﬁeld, k¯ denotes an algebraic closure of k, and k s a separable closure. (0.2) If A is a commutative ring, we sometimes simply write A for Spec(A) D. Mumford, Abelian Varieties, Oxford University Press. Plan of the course: Introduction Elliptic curves; Overview of the course; Definitions and basic properties Definition and examples; Rigidity; Rational maps into abelian varieties; Abelian varieties over the complex numbers Complex tori ; Line bundles on a complex torus; Algebraizability of tori; Group schemes Definitions; Elementary.
Welters, Gerald E. Polarized abelian varieties and the heat equations. Compositio Mathematica, Tome 49 (1983) no. 2, pp. 173-194. http://www.numdam.org/item/CM_1983. Abelian varieties | Mumford. | download | Z-Library. Download books for free. Find book abelian varieties over Z ζ well known, we have the Deligne-Mumford compactiﬁcation for curves [DM69], whereas there are quite a lot of compactiﬁcations of the moduli of abelian varieties ([AMRT75], [FC90]). Nevertheless we will construct one and only one new compactiﬁcation SQ g,K of the moduli of abelian varieties. This compactiﬁcation is natural enough because, as we will see. Then discuss the Mumford{Tate groups, abelian varieties of CM-type and [Del82, Example 3.7]. The goal of this talk is to cover [Del82, pp.39-47], but the speaker may need to rearrange the order of materials so that the audience can follow the talk easily. See also [Mil05, pp.281-284, pp.319-320, pp.335-336]. Lecture 7. Absolute Hodge cycles on abelian varieties of CM-type I. Following [Del82.
of abelian varieties By PETER NORMAN Over an algebraically closed field of characteristic p our knowledge of explicit local moduli of a polarized abelian variety was tied to separable phenomena: either the polarization had to be separable ([6]) or the abelian variety had to be ordinary ([2]). We present here a method for finding explicitly the local moduli in what was formerly the least. Relative Manin-Mumford for abelian varieties D. Masser 2010 MSC codes. 11G10, 14K15, 14K20, 11G50, 34M99. Abstract: With an eye or two towards applications to Pell's equation and to Davenport's work on integration of algebraic functions, Umberto Zannier and I have recently charac-terised torsion points on a xed algebraic curve in a xed abelian scheme of dimension bigger than one (when all. Hirzebruch-Mumford proportionality 1 Modular varieties of orthogonal type Let L be an integral indeﬁnite lattice of signature (2,n) and ( , ) the associated bilinear form. By D L we denote a connected component of the homogeneous type IV complex domain of dimension n DL = {[w] ∈ P(L⊗C) | (w,w) = 0, (w,w) > 0}+. O+(L) is the index 2 subgroup of the integral orthogonal group O(L) that. abelian varieties is not necessary: one only needs the ideas of §1, §2, §6, and §7 of the present paper. However, duality theory (at least for elliptic curves) plays a role in the construction of the string orientation of the theory of topological modular forms. Remark 0.0.4. In many parts of this paper, we work in the setting of (spectral) algebraic geometry over an E 8-ring Awhich might.
Mumford is a well-known mathematician and winner of the Fields Medal, the highest honor available in mathematics. Many of these papers are currently unavailable, and the commentaries by Gieseker, Lange, Viehweg and Kempf are being published here for the first time abelian variety of dimension g and λ : A →At is an isomorphism between A and its dual (a principal polarization), such that λ is equal to the map deﬁned by an ample line bundle, but one does not ﬁx such a line bundle. This point of view is the algebraic approach most closely tied to Hodge theory. (Moduli of pairs approach). This is the point of view taken in Alexeev's work [3]. Here. Faltings' theorem. Meets: W 13.15-15.00 in von Neumann 1.023. Starts: 15.4.2014. Description (pdf version) The main goal of the semester is to understand some aspects of Faltings' proofs of some far--reaching finiteness theorems about abelian varieties over number fields, the highlight being the Tate conjecture, the Shafarevich conjecture, and the Mordell conjecture Known cases of the Mumford-Tate conjecture. Abelian varieties of dimension \(\le 3\) K3 surfaces. Some other surfaces with \(p_{g} = 1\) A few other special cases. The conjecture is not additive. Main theorem \(A\): abelian surface \(X\): K3 surface. The Mumford-Tate conjecture is true for \(\mathrm{H}^{2}(A \times X)\) Remarks about. Keywords: torsion points, abelian varieties, Manin-Mumford conjecture 1 Introduction The Manin-Mumford conjecture is the following statement. Theorem 1.1 Let Abe an abelian variety deﬁned over Q and Xa closed subvariety of A. Denote by Tor(A) the set of torsion points of A. Then X∩Tor(A) = [i∈I X i ∩Tor(A), where Iis a ﬁnite set and each X i is the translate by an element of Aof an.
First lecture: Tuesday, February 23, 2016 First exercise class: Thursday, February 25, 2016 Content. Introduction to the theory of complex abelian varieties. Complex tori and Polarisations, Vector bundles on complex tori, cohomology of line bundles, Theta functions and Riemann's Theta relations, Hodge structures, the Hodge/Mumford-Tate group and the Hodge conjecture Variétés Abéliennes Complexes Minicours dans le Séminaire de Mathématiques des thésards Centre de Mathématiques Laurent Schwartz, École Polytechnique Première séance: Mercredi 14/10/2015, 14:30h, Salle de Conférences (CMLS Algebraic geometry Complex projective varieties 1 von Mumford, David Veröffentlicht: Berlin [u.a.], Springer, 197
Abelian scheme. A smooth group scheme over a base scheme S, the fibres of which are Abelian varieties (cf. Abelian variety ). The following is an equivalent definition: An Abelian scheme over S, or an Abelian S -scheme, is a proper smooth group S -scheme all fibres of which are geometrically connected. Intuitively, an Abelian S -scheme may be. -Mumford, David: Algebraic geometry ; 1 . Complex projective varieties Mumford, David mat 9:mu53/a53-1 2 of Mumford-Tate, Hodge, Lang and Tate for a large family of abelian va-rieties of type I and II. In addition, for this family, we prove an analogue of the open image theorem of Serre. 2000 Mathematics Subject Classiﬁcation: 11F80, 11G10 Keywords and Phrases: abelian varieties, l-adic representations 1. Introduction. Let Abe an abelian variety deﬁned over a number ﬁeld F.Let lbe an odd. Moret-Bailly, Laurent. Pinceaux de variétés Abéliennes. Astérisque, no. 129 (1985), 274 p. http://numdam.org/item/AST_1985__129__1_0
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