2021-05-04

The reciprocity law for the twisted second moment of Dirichlet L-functions

Reference: The reciprocity law for the twisted second moment of Dirichlet L-functions https://arxiv.org/pdf/0708.2928.pdf

• What is a Dirichlet character?

• What is a Gauss sum?

• What is the completion of an L function? Guessing this has to do with continuation.

• What is Dirichlet’s trick?

• How can you break a sum up into arithmetic progressions?

The K-Theory of monoid sets

Reference: The $K'$-theory of monoid sets https://arxiv.org/pdf/1909.00297.pdf

K-Theory

• $K'(A)$ defined for partially cancellative $A{\hbox{-}}$sets.
  • Important example: the pointed monoid ${\mathbb{N}}\coloneqq\left\{{{\operatorname{pt}}, 1, t, t^2, \cdots, }\right\}$.
• Useful in toric geometry.

• The category ${\mathsf{FinSet}}_{{\scriptstyle { * } }}$ (see Finset ) of finite pointed sets is quasi-exact, and Barratt-Priddy-Quillen implies that $K({\mathsf{FinSet}}_{\scriptstyle { * } }) \simeq{\mathbb{S}}$.

  • If $A$ has no idempotents or units then $K(A) \simeq{\mathbb{S}}$.

• Group completion : comes from ${\Omega}^\infty {\Sigma}^\infty {\mathbf{B}}G_+$.

• Big theorem: Devissage. But I have no clue what this means. Seems to say when ${\mathsf{K}}(A) \cong {\mathsf{K}}(B)$?

• Cancel all of the things:

• Apparently easy theorem: ${\mathsf{K}}'({\mathbb{N}}) \simeq{\mathbb{S}}$.

• The Picard group of ${\mathbb{P}}^1$ shows up:

Stefan Schreieder, Refined unramified cohomology

Tags: #seminar_notes #algebraic_geometry

Reference: Stefan Schreieder, Refined unramified cohomology. Harvard/MIT AG Seminar talk.

• See the Chow ring and cycle class map. Understanding the image amounts to the Hodge conjecture and understanding torsion in the image $Z^i(X)$?

  • See algebraic equivalence in the Chow group.

• Gysin sequence yields a residue map ${\partial}_x: H^i( \kappa(X); A) \to H^{i-1}( \kappa(X); A)$.

• See Gersten conjecture

• Interesting parts of the Coniveau spectral sequence: something coming from unramified cohomology, and something coming from algebraic cycles mod algebraic equivalence.

• Failure of integral Hodge conjecture :

• Uses Bloch-Kato conjecture

• Allows detecting classes in $Z^2(X)$ using K-Theory methods.

• See Borel-Moore cohomology – for $X$ a smooth algebraic scheme, essentially singular homology with a degree shift?

• See pro-objects and ind-objects in an arbitrary category.

  • pro-scheme : an inverse limit of scheme.

• Filter by codimension, then obstructions to extending over higher codimension things is measured by cohomology of the Function field :

• Here ${{\partial}}$ is a residue map.

• See separated schemes of finite type.

Main theorem, works not just for smooth schemes, but in greater generality:

• Torsion in the Griffiths group is generally not finitely generated.

  • Use an Enriques surface to produce $({\mathbb{Z}}/2)^{\oplus \infty}$ in $\mathop{\mathrm{Griff}}^3$.

• See canonical class $K_S$ for a surface, Abel-Jacobi invariants?

• No Poincaré duality for Chow groups, at least not at the level of cycles. Need to pass to cohomology.

  • Dual $\beta$ of $[K_S] \in H^2(S; {\mathbb{Z}}/2)$ generates the Brauer group $\mathop{\mathrm{Br}}(S)$ of the surface. Note $\beta$ is not algebraic.

• Theorem: there exists a regular flat morphism proper $S\to \operatorname{Spec}{\mathbb{C}}{\left[\left[ t \right]\right] }$ such that $S_\eta$ is an Enriques surface, $S_0$ is a union of ruled surfaces, and $\mathop{\mathrm{Br}}(S) \twoheadrightarrow\mathop{\mathrm{Br}}(S_\eta)$.

  • $\mathop{\mathrm{Br}}(X_\eta) \cong {\mathbb{Z}}/2$ is generated by an unramified conic bundle.
  • Can extend conic smoothly over central fiber
  • Need that the Poincaré dual specializes to zero on the special fiber.

• See Zariski locally and étale locally.

• unramified cohomology is linked to Milnor K theory.

Clausen on rep theory

Reference: https://www.youtube.com/watch?v=XTOwj1LvntM

• Clausen: a baby topic in geometric representation theory is Springer correspondence.

  • Need the equivariant derived category, very difficult to define!