in arithmetic geometry connects algebraic structures to number-theoretic properties. It extends classical theorems like Riemann-Roch to higher dimensions, enabling the study of rational points on varieties and arithmetic invariants.
K-Theory's applications in arithmetic geometry include relating special values of L-functions to regulators on K-groups. This connection is crucial for understanding deep conjectures like Birch and Swinnerton-Dyer, which link elliptic curves to their L-functions.
K-Theory for Arithmetic Varieties
Grothendieck-Riemann-Roch Theorem
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Relates the Chow groups of a smooth projective variety X to the K-groups of X
Provides a powerful tool for studying arithmetic properties of varieties
Extends the classical Riemann-Roch theorem to higher dimensions
Classical Riemann-Roch theorem relates the dimension of the space of sections of a line bundle to its degree and the genus of the curve (elliptic curves, algebraic curves)
Hirzebruch-Riemann-Roch and Arithmetic Riemann-Roch Theorems
expresses the Euler characteristic of a coherent sheaf on a variety in terms of characteristic classes
Allows for the computation of arithmetic invariants (, Todd classes)
relates the height of a rational point on a variety to the degree of a certain line bundle
Enables the study of rational points on varieties (elliptic curves, algebraic surfaces)
Riemann-Roch theorem for surfaces relates the dimension of the space of sections of a line bundle to its self-intersection number and the canonical class
Crucial for understanding the arithmetic of surfaces (, )
Grothendieck-Riemann-Roch theorem for singular varieties extends the classical theorem to singular schemes
Allows for the study of arithmetic properties in more general settings (singular curves, singular surfaces)
K-Theory and L-functions
Birch and Swinnerton-Dyer Conjecture
Relates the rank of the of an elliptic curve to the order of vanishing of its L-function at s=1
Connects K-Theory and L-functions in the context of elliptic curves
Predicts that the , which measures the failure of the Hasse principle for an elliptic curve, is finite
Tate-Shafarevich group is a key object in the study of rational points on elliptic curves
Beilinson and Bloch-Kato Conjectures
relate special values of L-functions to regulators on K-groups
Provides a deep connection between K-Theory and L-functions in arithmetic geometry
describes the relationship between the Tamagawa number of a motive and its L-function
Links K-Theory and L-functions via the theory of motives (Artin motives, Grothendieck motives)
generalizes the Bloch-Kato conjecture to the equivariant setting
Strengthens the connection between K-Theory and L-functions (Dedekind zeta functions, Hecke L-functions)
K-Theory in Conjectures
Birch and Swinnerton-Dyer Conjecture and its Generalizations
predicts the rank of the Mordell-Weil group of an elliptic curve over a number field
Equal to the order of vanishing of its L-function at s=1
Tate-Shafarevich group is conjectured to be finite by the Birch and Swinnerton-Dyer conjecture
Measures the failure of the Hasse principle for an elliptic curve
Bloch-Kato conjecture relates the special values of L-functions to the orders of certain Selmer groups
Selmer groups can be studied using K-Theory (, )
Equivariant Tamagawa Number Conjecture (ETNC)
Vast generalization of the Birch and Swinnerton-Dyer conjecture
Encompasses a wide range of arithmetic conjectures (Beilinson conjectures, Bloch-Kato conjecture)
Can be formulated using K-Theory and motivic cohomology
K-Theory can be used to define and study the Euler characteristics of coherent sheaves
Appear in the formulation of various conjectures in arithmetic geometry (Birch and Swinnerton-Dyer conjecture, Beilinson conjectures)
K-Theory vs Motivic Cohomology
Relationship between K-Theory and Motivic Cohomology
Motivic cohomology is an algebro-geometric analog of singular cohomology
Closely related to K-Theory via the motivic spectral sequence (, )
Beilinson conjectures relate special values of L-functions to regulators on motivic cohomology groups
Establishes a deep connection between motivic cohomology and L-functions
Bloch-Kato conjecture can be formulated in terms of motivic cohomology
Relates the special values of L-functions to the orders of certain motivic cohomology groups
Conjectures Involving Motivic Cohomology and L-functions
relates the special values of Dedekind zeta functions of number fields to the orders of certain motivic cohomology groups
Provides another link between motivic cohomology and L-functions
Study of polylogarithms and their generalizations involves the use of motivic cohomology
Appear in the formulation of the Beilinson conjectures and other conjectures relating K-Theory and L-functions (, )
Bloch-Kato exponential map is a key tool in the study of polylogarithms
Can be interpreted as a map between certain motivic cohomology groups and Galois cohomology groups (étale cohomology, p-adic Hodge theory)