14.4 Applications to arithmetic geometry

K-Theory 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

• Hirzebruch-Riemann-Roch theorem expresses the Euler characteristic of a coherent sheaf on a variety in terms of characteristic classes
  • Allows for the computation of arithmetic invariants (Chern classes, Todd classes)
• Arithmetic Riemann-Roch theorem 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 (K3 surfaces, Enriques 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 Mordell-Weil group 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 Tate-Shafarevich group, 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

• Beilinson 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
• Bloch-Kato conjecture 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)
• Equivariant Tamagawa Number Conjecture (ETNC) 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

• Birch and Swinnerton-Dyer conjecture 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 (étale K-Theory, motivic cohomology)

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 (Atiyah-Hirzebruch spectral sequence, Bloch-Lichtenbaum 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

• Lichtenbaum conjecture 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 (Zagier conjecture, Goncharov conjecture)
• 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)