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

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  • 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

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  • \(K'(A)\) 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:.

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  • 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 \({\mathsf{K}}(A) \cong {\mathsf{K}}(B)\)?

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  • Cancel all of the things:

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

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

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Stefan Schreieder, Refined unramified cohomology

Tags: #projects/notes/seminars #AG

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 \({\partial}_x: H^i( \kappa(X); A) \to H^{i-1}( \kappa(X); A)\).

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  • See Gersten conjecture

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  • 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 :

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  • 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 :

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  • Here \({{\partial}}\) is a residue map.

  • See separated schemes of finite type.

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

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  • 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!
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