Arithmetic on algebraic varieties

DOI10.1007/BF01084472zbMath0284.14004MaRDI QIDQ5917385

A. N. Parshin

Publication date: 1973

Published in: Journal of Soviet Mathematics (Search for Journal in Brave)

Mathematics Subject Classification ID

Arithmetic ground fields for curves (14H25) History of algebraic geometry (14-03) Arithmetic problems in algebraic geometry; Diophantine geometry (14Gxx) Formal groups, (p)-divisible groups (14L05) Abelian varieties and schemes (14Kxx) General field theory (12Exx)

Related Items (2)

Aleksei Nikolaevich Parshin (on his 70th birthday) ⋮ Geometry of algebraic varieties

Cites Work

• Completely continuous endomorphisms of \(p\)-adic Banach spaces
• Diophantine sets over polynomial rings
• On the zeta function of a hypersurface. II
• On the zeta function of a hypersurface. III
• Schemata over local rings. II
• Formal complex multiplication in local fields
• Reduction of models over a discrete valuation ring
• On Artin's L-functions of the algebraic curves uniformized by certain automorphic functions
• Algebraic values of meromorphic functions
• One-parameter formal Lie groups over p-adic integer rings
• On the relative theory of Tamagawa numbers
• On the field of moduli of an abelian variety with complex multiplication
• On the endomorphism algebra of Jacobian varieties attached to the fields of elliptic modular functions
• Moduli and fibre systems of abelian varieties
• Abelian varieties attached to automorphic forms
• On Dirichlet series and abelian varieties attached to automorphic forms
• Über die rationale Darstellbarkeit der Heckeschen Operatoren
• On Mordell's conjecture for algebraic curves over function fields
• Théoremes de finitude en cohomologie galoisienne
• On the Galois cohomology of tamely ramified fields of algebraic functions
• The Brauer group of a ringed space
• Discontinuous groups and abelian varieties
• On pencils of curves of genus two
• Correction to 'One-parameter formal Lie groups over p-adic integer rings'
• The cohomology of certain elliptic curves over local and quasi-local fields
• The parity of the rank of the Mordell-Weil group
• Punktfolgen auf Kurven und Flächen
• Un théorème sur les homomorphismes de schemas abeliens
• Endomorphisms of Abelian varieties over finite fields
• On the moduli of Abelian varieties with multiplications
• On purely transcendental fields of automorphic functions of several variables
• Division points on curves
• On heights of algebraic subspaces and diophantine approximations
• Cohomology of artinian group schemes over local fields
• Über die Fundamentalgruppe einer punktierten Riemannschen Fläche bei Charakteristik \(p>0\)
• The cohomological dimension of certain Grothendieck topologies
• Algebraic functions and analogue of the geometry of numbers: The Riemann- Roch theorem
• On rational points of homogeneous spaces over finite fields
• Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen
• On discrete subgroups of the two by two projective linear group over \(p\)-adic fields
• The Tate-*Safarevic group of a constant abelian variety
• Sur la formule de Siegel dans la théorie des groupes classiques
• The Thue-Siegel-Roth theorem in characteristic p
• On zeta functions of quadratic forms
• Quasi-functions and heights on abelian varieties
• On the algebraic theory of elliptic modular functions
• Finite subgroups and isogenies of one-parameter formal Lie groups
• On the zeta function of a fibre variety whose fibres are abelian varieties
• On the Davenport-Hasse curves
• Remarque sur une formule de Shimura-Taniyama
• Algebraic values of meromorphic functions. II
• A note of Shimura's paper: Discontinuous groups and Abelian varieties
• Formal groups and zeta-functions
• Rational surfaces over perfect fields
• Good reduction of abelian varieties
• On the solutions of analytic equations
• On the zeta function of a hypersurface. I
• Über das Verhalten des Geschlechts eines Funktionenkörpers einer Variablen bei Konstantenreduktion
• Variétés abeliennes ordinaires sur un corps fini
• Deformations and liftings of finite, commutative group schemes
• Einige Fortsetzungssätze in der \(p\)-adischen Analysis
• Hecke polynomials as congruence \(\zeta\) functions in elliptic modular case
• Algebraic approximation of structures over complete local rings
• Automorphic functions and number theory
• On the Lubin-Tate reciprocity law
• On Tate's duality theorems in Galois cohomology
• A mean value theorem in adele geometry
• Duality theorems for curves over p-adic fields
• A class number associated with the product of an elliptic curve with itself
• The congruence monodromy problems
• A \(p\)-adic proof of Weil's conjectures
• Local Euler characteristics
• Zur Reduktionstheorie algebraischer Funktionenkörper vom Transzendenzgrad 1: Existenz einer regulären Reduktion zu vorgegebenem Funktionenkörper als Restklassenkörper
• The elementary theory of finite fields
• On the field of definition for a field of automorphic functions
• Algebraic varieties without deformation and the Chow variety
• On the Shafarevich-Tate group of an elliptic curve
• The role of modular functions in a class-number problem
• Algebraic cycles on abelian varieties of complex multiplication type
• Construction of class fields and zeta functions of algebraic curves
• Isogeny classes of abelian varieties over finite fields
• Extensions of abelian varieties defines over a finite field
• On a convolution of L-series
• Galois cohomology and birational invariant of algebraic varieties
• Galois cohomology of finitely generated modules
• Extensions de corps hilberteins
• Hecke polynomials of modular groups and congruence zeta functions of fibre varieties
• On the jacobian varieties of the fields of elliptic modular functions. II
• On Mordell's conjecture for the curve over function field with arbitrary constant field
• On the algebraic curves uniformized by arithmetical automorphic functions
• On the zeta-functions of the algebraic curves uniformized by certain automorphic functions
• A remark on the Sato-Tate conjecture
• Automorphic forms on GL (2)
• Unverzweigte galoissche \(p\)-Erweiterungen algebraischer Funktionenkörper mit endlichem Konstantenkörper
• Solving diophantine problems modulo every prime
• On the Galois cohomology of the projective linear group and its applications to the construction of generic splitting fields of algebras
• Schemata over local rings
• Transcendental points on group varieties
• Cohomology of abelian varieties over function fields
• On the Tamagawa number of algebraic tori
• Arithmetic on Curves of Genus 1. I. On a conjecture of Selmer.
• Arithmetic on curves of genus 1. II. A general result.
• Arithmetic on Curves of Genus 1 : III. The Tate-Šafarevič and Selmer Groups
• Arithmetic on Curves of Genus 1. IV. Proof of the Hauptvermutung.
• Arithmetic on Curves of Genus 1 (V). Two Counter-Examples
• The Gauss-Bonnet theorem and the Tamagawa number
• Integral Points on Certain Elliptic Curves
• Irreducible Subvarieties and Rational Points
• Diophantine Problems Over Local Fields I
• Diophantine Equations with Special Reference To Elliptic Curves
• Rational points in Henselian discrete valuation rings
• Grothendieck topologies over complete local rings
• Diophantine Approximations on Toruses
• A field of cohomological dimension 1 which is not $C_1$
• Proof of Some Conjectures on Cohomological Dimension
• Shorter Notes: Abelian Formal Groups
• The Cohomology Groups of Tori in Finite Galois Extensions of Number Fields
• Arithmetic on curves of genus 1. VII. The dual exact sequence.
• Splitting of Algebras by Function Fields of One Variable
• On the Specialization of Birationally Equivalent Curves
• Elliptic Curves and Wild Ramification
• On the Hasse principle for cubic surfaces
• A Remark on Mordell's Conjecture
• Quadratic forms ‘à la’ local theory
• Formal moduli for one-parameter formal Lie groups
• A Corollary to a Conjecture of Birch and Swinnerton-Dyer
• Contributions to the theory of diophantine equations I. On the representation of integers by binary forms
• Contributions to the theory of Diophantine equations II. The Diophantine equation y ² = x ³ + k
• On the Galois Cohomology Group of the Ring of Integers in a Global Field and its Adele Ring
• Approximation in algebraic function fields of one variable
• Abelian curves of small conductor.
• On a Theorem of Dem'Janenko
• Conjectures concerning elliptic curves
• An Application of the Power Residue Theory to Some Abelian Functions
• On the Zeta Function of Biprojective Complete Intersections
• A Note on Galois Cohomology Groups of Algebraic Tori
• On the Jacobian varieties of Davenport-Hasse curves
• Rational points on algebraic varieties over large number fields
• A Classification of Almost Full Formal Groups
• Zur G<scp>ALOIS</scp>‐Kohomologie A<scp>BEL</scp>scher Mannigfaltigkeiten
• On the Group of Rational Points on Elliptic Curves Over Function Fields
• A Result on the Weil Zeta Function
• How the Number of Points of An Elliptic Curve Over a Fixed Prime Field Varies
• On Rationality of Algebraic Function Fields
• ON CERTAIN FORMAL LIE GROUPS OVER A $p$-ADIC INTEGER RINGS
• The Period-Index Problem for Elliptic Curves
• The Automorphisms of the Orthogonal Groups Ω n (V) Over Fields
• Principal Homogeneous Spaces for Finite Group Schemes
• Über die Galoissche Gruppe der algebraischen Abschließung eines Potenzreihenkörpers mit endlichem Konstantenkörper
• The Tate height of points on an abelian variety. Its variants and applications
• On subfields of hyperelliptic fields. I
• Curves Over Discrete Valuation Rings
• On the Zeta Function of an Algebraic Variety
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• On a Conjecture of Andre Weil
• NOTE ON ISOGENIES OF ONE-PARAMETER FORMAL LIE GROUPS OVER LOCAL INTEGER RINGS
• A p-adic Fixed Point Formula
• An effective p-adic analogue of a theorem of Thue
• Problems in the theory of automorphic forms to Salomon Bochner in gratitude
• Weber's class invariants
• Arithmetic on Curves of genus 1. VI. The Tate-Safarevic group can be arbitrarily large.
• Arithmetic on curves of genus 1. VIII. On conjectures of Birch and Swinnerton-Dyer.
• On the Connected Identity Component of the Adele-Class Group of an Algebraic Torus
• A CRITERION FOR THE p-ADIC SOLUBILITY OF DIOPHANTINE EQUATIONS
• A decision method for $p$-adic integral zeros of diophantine equations
• Theta Functions and Abelian Varieties over Valuation Fields of Rank one I
• On Theta Functions and Abelian Varieties over Valuation Fields of Rank One: (II) Theta Functions and Abelian Functions of Characteristicp(>0)
• On the relative theory of Tamagawa numbers
• Zeroes of Polynomials Over Finite Fields
• On heights in number fields
• ON MODULAR CORRESPONDENCES FOR SP(N, Z ) AND THEIR CONGRUENCE RELATIONS

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