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?
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What is a Gauss sum?
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What is the completion of an L function? Guessing this has to do with continuation.
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What is Dirichlet’s trick?
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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 \({\mathsf{FinSet}}_{{\scriptstyle { \ast } }}\) (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 \(K(A) \simeq{\mathbb{S}}\).
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Group completion : comes from \({\Omega}^\infty {\Sigma}^\infty {\mathbf{B}}G_+\).
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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:
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Apparently easy theorem: \({\mathsf{K}}'({\mathbb{N}}) \simeq{\mathbb{S}}\).
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The Picard group of \({\mathbf{P}}^1\) shows up:
Stefan Schreieder, Refined unramified cohomology
Tags: #projects/notes/seminars #AG
Reference: Stefan Schreieder, Refined unramified cohomology. Harvard/MIT AG Seminar talk.
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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.
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Gysin sequence yields a residue map \({\partial}_x: H^i( \kappa(X); A) \to H^{i-1}( \kappa(X); A)\).
- 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.
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Failure of integral Hodge conjecture :
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Uses Bloch-Kato conjecture
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Allows detecting classes in \(Z^2(X)\) using K-theoretic methods.
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See Borel-Moore cohomology – for \(X\) a smooth algebraic scheme, essentially singular homology with a degree shift?
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See Pro objects and Ind objects in an arbitrary category.
- pro scheme : an inverse limit of scheme.
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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.
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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 \(({\mathbf{Z}}/2)^{\oplus \infty}\) in \(\mathop{\mathrm{Griff}}^3\).
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See canonical class \(K_S\) for a surface, Abel-Jacobi invariants?
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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.
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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.
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See Zariski locally and étale locally.
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unramified cohomology is linked to Milnor K theory.
Clausen on rep theory
Reference: https://www.youtube.com/watch?v=XTOwj1LvntM
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Clausen: a baby topic in geometric representation theory is Springer correspondence.
- Need the equivariant derived category, very difficult to define!