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

• 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 $${\mathbf{P}}^1$$ shows up:

## Stefan Schreieder, Refined unramified cohomology

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 $$Z^i(X)$$?

• 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-theoretic 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 Unsorted/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 $$({\mathbf{Z}}/2)^{\oplus \infty}$$ 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 $$[K_S] \in H^2(S; {\mathbf{Z}}/2)$$ generates the Brauer group $$\mathop{\mathrm{Br}}(S)$$ of the surface. Note $$\beta$$ is not algebraic.
• Theorem: there exists a regular flat morphism proper $$S\to \operatorname{Spec}{\mathbf{C}}{\left[\left[ t \right]\right] }$$ such that $$S_\eta$$ is an Enriques surface, $$S_0$$ is a union of ruled surfaces, and $$\mathop{\mathrm{Br}}(S) \twoheadrightarrow\mathop{\mathrm{Br}}(S_\eta)$$.

• $$\mathop{\mathrm{Br}}(X_\eta) \cong {\mathbf{Z}}/2$$ 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

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

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