L-values and Extensions

Description

This lecture covers de Rham theory, L-values, and non-split extensions. It delves into Riemann's C-function, Euler product, analytic continuation, functional equation, and shared features of L-functions. The significance of certain L-values and extensions is explored, shedding light on mixed Hodge structures, p-adic Galois representations, and mixed motives. Special value formulas for quadratic characters and an extension to singular cohomology are discussed, along with examples related to Hecke characters, modular forms, and symmetric squares of modular forms. The lecture concludes with a discussion on Euler systems, Galois representations, and future research directions.

Official source

https://mediaspace.epfl.ch/media/0_ydehjkuf

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Related concepts (42)

Cyclotomic character

In number theory, a cyclotomic character is a character of a Galois group giving the Galois action on a group of roots of unity. As a one-dimensional representation over a ring R, its representation space is generally denoted by R(1) (that is, it is a representation χ : G → AutR(R(1)) ≈ GL(1, R)). Fix p a prime, and let GQ denote the absolute Galois group of the rational numbers. The roots of unity form a cyclic group of order , generated by any choice of a primitive pnth root of unity ζn.

Hecke character

In number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class of L-functions larger than Dirichlet L-functions, and a natural setting for the Dedekind zeta-functions and certain others which have functional equations analogous to that of the Riemann zeta-function. A name sometimes used for Hecke character is the German term Größencharakter (often written Grössencharakter, Grossencharacter, etc.).

Hilbert modular form

In mathematics, a Hilbert modular form is a generalization of modular forms to functions of two or more variables. It is a (complex) analytic function on the m-fold product of upper half-planes satisfying a certain kind of functional equation. Let F be a totally real number field of degree m over the rational field. Let be the real embeddings of F. Through them we have a map Let be the ring of integers of F. The group is called the full Hilbert modular group.

Siegel modular form

In mathematics, Siegel modular forms are a major type of automorphic form. These generalize conventional elliptic modular forms which are closely related to elliptic curves. The complex manifolds constructed in the theory of Siegel modular forms are Siegel modular varieties, which are basic models for what a moduli space for abelian varieties (with some extra level structure) should be and are constructed as quotients of the Siegel upper half-space rather than the upper half-plane by discrete groups.

Modular curve

In number theory and algebraic geometry, a modular curve Y(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL(2, Z). The term modular curve can also be used to refer to the compactified modular curves X(Γ) which are compactifications obtained by adding finitely many points (called the cusps of Γ) to this quotient (via an action on the extended complex upper-half plane).