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Topic: Abelian variety of CM-type

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Abstracts (Salman Abdulali) Abelian varieties of type III and the Hodge conjecture A Kuga fiber variety f : A → V is an abelian scheme parametrized by an arithmetic variety and constructed from a symplectic representation of an algebraic group. We deduce the general Hodge conjecture for certain 6-dimensional abelian varieties of type III, and the usual Hodge and Tate conjectures for certain 4-dimensional abelian varieties of type III. personal.ecu.edu /abdulalis/abstracts.html

Complex multiplication See also: abelian variety of CM-type, Lubin-Tate formal group, Drinel'd shtuka. Kronecker first postulated that the values of elliptic functions at torsion points should be enough to generate all abelian extensions for imaginary quadratic fields, an idea that went back to Eisenstein in some cases, and even to Gauss. This became known as the Kronecker Jugendtraum ; and was certainly be what would have prompted Hilbert's remark above, since it makes explicit class field theory in the way the roots of unity do for abelian extensions of the rational number field. brainyencyclopedia.com /encyclopedia/c/co/complex_multiplication.html

Lecture 1: Heights on a projective space CM points are nontrivial fixed points of Hecke operators. This includes a Manin-Mumford type conjecture and an equidistribution theorem on Galois orbits of preperiodic points with respect to the invariant measures on Julia sets. Then, the height of a point is like the degree of some curve in a polarized variety. www.math.uci.edu /~mfried/htmlfiles/zhang.html

question Polarizations of type $(1,...,1,2)$}\end{center} \vvv \noindent \hskip1cm {\bf Theorem 4.1.} {\it Let $(A,Z)$ be an $n$ dimensional abelian variety with a polarization of type $(1,...,1,2)$ (i.e. A surprising consequence is that if $(X,L)$ is a general polarized abelian variety of type $(1,...,1,2)$, and $D$ is a divisor in the linear series $mL$, then the pair $(A, (1/m)D)$ is log canonical. {\bf 98} (1973), 178-185 \bibitem{KV} {\it Y. Kawamata, E. Viehweg}, {{On a characterization of abelian varieties in the classification theory of algebraic varieties}}, Compositio Math. www.math.utah.edu /~hacon/question

Arithmetic of abelian varieties - Wikipedia, the free encyclopedia In terms of the ring End(A) there is a definition of abelian variety of CM-type that singles out the richest class. Reduction of an abelian variety A modulo a prime ideal of (the integers of)K - say, a prime number p - to get an abelian variety A In mathematics, the arithmetic of abelian varieties is the study of the number theory of an www.wikipedia.org /wiki/Arithmetic_of_elliptic_curves

zmath.html?first=1&maxdocs=3&type=tex&an=1004.14003&format=complete An Abelian variety of $S_n$-type is an Abelian variety $A_n$ with CM type $(E_n, H)$. A particular CM type $H \subset G_0\backslash G$ is defined. The author proves: \par Proposition 4.6: Suppose that a simple Abelian variety $A$ is $N$-dominated, say $N = d(A)$, and that every Hodge cycle on $A$ up to codimension $N$ is algebraic. zmath.impa.br /cgi-bin/zmen/ZMATH/en/zmath.html?first=1&maxdocs=3&type=tex&an=1004.14003&format=complete

BookWebPro —m‘ŒŸõ Preface vii (2) Preface to Complex Multiplication of Abelian ix (4) Varieties and Its Applications to Number Theory (1961) Notation and Terminology xiii CHAPTER I. Preliminaries on Abelian Varieties 3 (32) 1. Analytic theory of abelian varieties 19 (6) 4. Families of Abelian Varieties 151 (22) and Modular Functions 23. bookwebpro.kinokuniya.co.jp /booksea.cgi?ISBN=0691016569&USID=

Algebraic Number Theory Archive math.NT/0411291 : 12 Nov 2004, On the Manin-Mumford conjecture for abelian varieties with a prime of supersingular reduction, by Tetsushi Ito. math.NT/0408069 : 4 Aug 2004, The arithmetic of Prym varieties in genus 3, by Nils Bruin. ANT-0146 : 19 Oct 1998, Bounding the torsion in CM elliptic curves, by Dipendra Prasad and C.S. Yogananda. front.math.ucdavis.edu /ANT

185 By the famous theorem due to D.~T.~L\hataccent{e} and C.~P.~Ramanujam \cite{le-raman}, the topological type of an analytic germ is constant along the stratum $\mu=\const$, therefore the germs of $H_0$ and $H_1$ at the origin are topologically equivalent, in particular, the germs of analytic curves $\{H_0=0\}$ and $\{H_1=0\}$ in $(\C^2,0)$ are homeomorphic. On the other hand, the assertion of the theorem on zeros of Abelian integrals, derived from the explicit form of the system \eqref{pf}, uses pre-normalization in terms of the {\em coefficients\/} of the Hamiltonian, more precisely, the $\ell^1$-norms of its nonhomogeneity (the difference between $H$ and its principal homogeneous part). Isolated zeros of the Abelian integral of an arbitrary polynomial 1-form $\omega=\sum_i c_i\omega_i$ correspond to isolated intersections of the above curve with the hyperplane $\sum c_i I_i=0$. home.imf.au.dk /esn/preprints/185

Analytic Jacobians of Hyperelliptic Curves The dimension of the Jacobian A as a complex abelian variety. The CM type Phi is given by the second and fourth complex embeddings. The analytic Jacobian of the curve is an abelian torus and is constructed as follows. www.mpim-bonn.mpg.de /external-documentation/magma/text1223.htm

Midwest Algebraic Geometry Conference A classical theorem of Wirtinger states that the general principally polarized abelian variety of dimension 5 is a Prym variety of a double unramified covering of a curve of genus 6. We'll discuss how the vanishing theorems can be used to study the singularities of divisors on an abelian variety or a Fano manifold. In the case of a variety with isolated singularities, our metrics have the same order of growth as Saper's metrics, whose L 2-cohomology he proved is isomorphic to the intersection cohomology of the variety. www.nd.edu /~rosen/MAGC97/magc97/magc97.html

Abstracts for MWANT 2003 We also prove that the smallest dimension of a CM abelian variety over $K$ is exactly the ideal class number of $K$ and classify when a CM abelian variety over $K$ has the smallest dimension. We prove that these abelian varieties have the striking property that the vanishing order of their $L$-function at the center is dictated by the root number of the associated Hecke character. These abelian varieties are characterized as of smallest dimension and smallest conductor, and such that the abelian varieties themselves descend to $\Bbb Q$. www.math.uic.edu /~jeremy/abstracts.html

Math JS Milne Preprints For an abelian variety A and its dual B over a local field of prime characteristic, prove that A(K) is dual to the Weil-Chatelet group of B. Prove the full conjecture of Birch and Swinnerton-Dyer in the case of a constant abelian variety over a global field of prime characteristic; in particular, give the first examples of nonzero abelian varieties whose Tate-Shafarevich groups are known to be finite. -cohomologies of an algebraic variety (with Niranjan Ramachandran) www.jmilne.org /math/Preprints

connect.tex Suppose $A$ is an abelian variety defined over a field $F$ of characteristic zero, $K$ is a CM-field, $\iota : K \hookrightarrow \End_F^0(A)$ is an embedding such that $\iota(1) = 1$, and $C$ is an algebraically closed field containing $F$. \begin{cor} Suppose $X$ is an abelian variety defined over a finitely generated extension $F$ of $\Q$, $\lambda$ is a polarization on $X$, $n$ is an integer, $n \ge 5$, and ${\widetilde X}_n$ is a maximal isotropic subgroup of $X_n$ with respect to $e_{\lambda,n}$. The abelian varieties $J$ and $E$ have good reduction outside of $7$, and therefore the extension $\Q(A_p)/\Q$ is unramified away from $7$ and the prime $p$ (see Theorem 1 on p.~493 of \cite{serretate}). www.math.psu.edu /preprints/zarhin/connect.tex

pubblicazioni degli aderenti al Gruppo Severi varieties and branch curves of abelian surfaces of type (1,3. Curves of genus g on an abelian variety of dimension g. Abelian varieties over the field of the 20th roots of unity with good reduction everywhere. gruppi.altamatematica.it /gnsaga/pubblicazioni-2003.shtml

Chowla-Selberg formula The origin of such formulae is now seen to be in the theory of complex multiplication, and in particular in the theory of periods of an abelian variety of CM-type. Here χ is the quadratic residue symbol modulo D, where -D is the discriminant of an imaginary quadratic field. This has led to much research and generalisation. www.t131.greatnet.de /encyclopedia/c/ch/chowla_selberg_formula.html

Richard Pink: Recent Preprints Abstract: A theorem of Green, Lazarsfeld and Simpson (formerly a conjecture of Beauville and Catanese) states that certain naturally defined subvarieties of the Picard variety of a smooth projective complex variety are unions of translates of abelian subvarieties by torsion points. There are no restrictions on the characteristic of F or the type of G, and simultaneous approximation in finitely many algebraic groups is also studied. His main result from model theory, when applied to abelian varieties, can be rephrased in terms of algebraic geometry. www.math.ethz.ch /~pink/publications.html

CM-seminars This type of models has been successfully used to explore the underlying physical mechanism of structural formation, folding dynamics and protein-protein interaction. The reduction of the degrees of freedom of the involved coordinates in such a model, in comparison with the all-atom modelling approach, allows for accumulation of adequate statistics in computer simulations. Understanding some of these ideas should make programming for statistical applications more natural and productive. www.math.uwaterloo.ca /navigation/CompMath/Research/seminars/CM-notices.shtml

Publications of RIMS: Author Index to Volumes 31-40 Hazama, F. : On the general Hodge conjecture for abelian varieties of CM-type. Tamagawa, A. : The Eisenstein quotient of the Jacobian variety of a Drinfeld modular curve. Loi, P. : Commuting squares and the classification of finite depth inclusions of AFD type III_\lambda factors, \lambda \in (0,1). www.kurims.kyoto-u.ac.jp /~kenkyubu/publ/31-40.html

Citebase - Hodge classes on abelian varieties of low dimension We prove that the Hodge ring of any such abelian variety $X$ is generated by divisor classes together with the so-called Weil classes on (quotients of) $X$. [1] M. Borovoi, The Hodge group and the algebra of endomorphisms of an abelian variety, In: A. Onishchik (ed.), Problems in group theory and homological algebra, Yaroslav. Kuga-Satake varieties are abelian varieties associated to certain weight two Hodge structures, for example the second cohomology group of a K3 surface. citebase.eprints.org /cgi-bin/citations?id=oai%3AarXiv%2Eorg%3Amath%2F9901113

Number Theory at UBC by entire functions of finite exponential type (these are essentially functions with Fourier transforms that have bounded support) and applications of this problem to analytic number theory. Let A be an abelian variety with complex multiplication. But due to the lack of a suitable theory of "abelian units", a higher dimensional Coates-Wiles theorem remains unattainable. www.math.ubc.ca /people/faculty/gerg/NT/Fall2003semester.html

VIGRE Number Theory Working Group Homepage--Main One goal is to produce a written account of the complete proof of the Main Theorem of Complex Multiplication using the techniques of modern algebraic geometry, and to determine the L-series of a CM abelian variety in terms of Hecke L-series. CM types and reflex fields (in preparation): pdf. This year's topic is the theory of complex multiplication for abelian varieties. www.math.lsa.umich.edu /~bdconrad/vigregroup/vigre04.html

NMBRTHRY archives -- January 2002 (#20) These are CM abelian varieties, and the CM-type is primitive hence these abelian varieties are absolutely simple. If p is 1 mod 6, then we have the same splitting but exactly two of the factors are not simple but in fact are powers of a simple abelian variety. More generally, the decomposition upto isogeny of the jacobian of the Fermat curve with exponent n into (absolutely) simple abelian varieties is known. listserv.nodak.edu /cgi-bin/wa.exe?A2=ind0201&L=nmbrthry&F=&S=&P=2316

9.tex While it does not make any difference which of the two types of triples one chooses to define the curve $F_{a,b,c}$, it does make a difference which type of triple one uses to evaluate $q$ and hence the reduction type. Now suppose $J_{a,b,c}$ is isogenous to the product of two abelian varieties of smaller dimension. As far as we can tell, the problem lies in the use of the function $q(x)$ which computes the reduction type (see the introduction). www.maths.warwick.ac.uk /%7Emiles/HPFSD75/9.tex

anoort.tex \begin{thm}\label{thm4.1} Consider a Shimura variety defined by a Shimura datum $(G,X)$ where $G$ is a semi-simple algebraic group of adjoint type. \end{proof} \section{Images under Hecke correspondences.}\label{sec4} In this section we prove that the images under irreducible components of certain Hecke correspondences of an irreducible Hodge generic subvariety of a Shimura variety defined by a semi-simple algebraic group of adjoint type are irreducible. Moduli of abelian varieties, Progress in Mathematics~195 (2001), 133--155, Birkh\"auser. www.math.leidenuniv.nl /~edix/public_html_rennes/publications/anoort.tex

Search results Let A be an abelian variety with complex multiplication by a CM field K. Let $A\sb{{\frak P}}$ be the reduction of A modulo a prime ${\frak P}$. The idea is that (roughly speaking) the Dieudonne module, together with its Riemann form, classifies the abelian variety in finite characteristic. The calculation of the Dieudonne module of $A\sb{{\frak P}}$ with its Riemann form is important for the classification of points of Shimura varieties over finite fields. www.sci.hkbu.edu.hk /cgi-bin/zmen/zmath/zmath.html?first=1&maxdocs=100&type=html&an=674.14030&format=complete

tate.tex In [M1] it was proved that for such abelian varieties which do not have complex multiplication, the ring of Tate cycles is algebraic and generated by the classes of divisors. The advantage of this notation for us is that the infinity type of a Gr\"ossencharacter $\psi$ on $F$ thought of as a function on $G/H$ when thought of as a function on $G$ gives the infinity type of the Gr\"ossencharacter on $\tilde{F}$ obtained from $\psi$ by composing with the norm mapping from $\tilde{F}$ to $F$. An application of Lemma 6.1(1) implies that the infinity type of the Gr\"ossencharacter $\chi_1$ on $\tilde{M}_1$ is pullback from an infinity type on $M$. www.math.tifr.res.in /~dprasad/tate.tex

Search results The isogeny class of a CM-type abelian variety is defined over a finite extension of the prime field. www.sci.hkbu.edu.hk /cgi-bin/zmen/zmath/zmath.html?first=1&maxdocs=100&type=html&an=272.14009&format=complete

file132.html Q is not abelian because their elements produce at least three eigenlines in TQ by the directions of the three reflection discs through Q. This contradicts to the second part of condition (iv). The first one should recognize Hirzebruch's abelian covers of E ?E defined in [Hir] as Picard modular surfaces as it was done for Eisenstein numbers in [Ho86]. If G is not abelian, then it must be a binary dieder group 2D2 (quaternion group) or 2D4 because of the orders of elements of G, see e.g. www.mathematik.uni-osnabrueck.de /projects/carmen/AP11/test/file132.html

ja.tex The set of $A'/\CC$ with $\End_{O_K}(A')=R$ and with the same CM type as $A$ is a $\Pic(R)$-torsor. \end{center} \bigskip {\sc Moonen}: moduli space of abelian varieties; true for $S$ for which there exists a prime number $p$ at which all $s$ in $S$ have an ordinary reduction of which they are the canonical lift. Then $X$ is of Hodge type.} The proof is by induction on $n$ and $d$. www.math.leidenuniv.nl /~edix/public_html_rennes/talks/ja.tex

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