Derived Categories and the Analytic Approach to General Reciprocity Laws. Part III

Building on the scaffolding constructed in the first two articles in this series, we now proceed to the geometric phase of our sheaf (-complex) theoretic quasidualization of Kubota\u27s formalism for n-Hilbert reciprocity. Employing recent work by Bridgeland on stability conditions, we extend our yoga of t-structures situated above diagrams of specifically designed derived categories to arrangements of metric spaces or complex manifolds. This prepares the way for proving n-Hilbert reciprocity by means of singularity analysis

A Generalized Contou-Carr\`ere Symbol and its Reciprocity Laws in Higher Dimensions

We generalize the theory of Contou-Carr\`ere symbols to higher dimensions. To an $(n+1)$-tuple $f_0,\dots,f_n \in A((t_1))\cdots((t_n))^{\times}$, where $A$ denotes a commutative algebra over a field $k$, we associate an element $(f_0,\dots,f_n) \in A^{\times}$, compatible with the higher tame symbol for $k = A$, and earlier constructions for $n = 1$, by Contou-Carr\`ere, and $n = 2$ by Osipov--Zhu. Our definition is based on the notion of \emph{higher commutators} for central extensions of groups by spectra, thereby extending the approach of Arbarello--de Concini--Kac and Anderson--Pablos Romo. Following Beilinson--Bloch--Esnault for the case $n=1$, we allow $A$ to be arbitrary, and do not restrict to artinian $A$. Previous work of the authors on Tate objects in exact categories, and the index map in algebraic $K$-theory is essential in anchoring our approach to its predecessors. We also revisit categorical formal completions, in the context of stable $\infty$-categories. Using these tools, we describe the higher Contou-Carr\`ere symbol as a composition of boundary maps in algebraic $K$-theory, and conclude the article by proving a version of Parshin--Kato reciprocity for higher Contou-Carr\`ere symbols.Comment: 55 pages, introduction completely rewritte

The notion of dimension in geometry and algebra

This talk reviews some mathematical and physical ideas related to the notion of dimension. After a brief historical introduction, various modern constructions from fractal geometry, noncommutative geometry, and theoretical physics are invoked and compared.Comment: 29 pages, a revie

Explicit Reciprocity Laws in Iwasawa Theory -- A survey with some focus on the Lubin-Tate setting

Starting from Gau{\ss}' and Legendre's quadratic reciprocity law we want to sketch how it gave rise to the development of higher and generalized reciprocity laws and over all explicit reciprocity formulas in Iwasawa theory

Prismatic FF-crystals and Lubin-Tate (φq,Γ)(\varphi_q,\Gamma)-modules

Let $L/\mathbb{Q}_p$ be a finite extension. We introduce $L$-typical prisms, a mild generalization of prisms. Following ideas of Bhatt, Scholze, and Wu, we show that certain vector bundles, called Laurent $F$-crystals, on the $L$-typical prismatic site of a formal scheme $X$ over $\mathrm{Spf}\mathcal{O}_L$ are equivalent to $\mathcal{O}_L$-linear local systems on the generic fiber $X_\eta$. We also give comparison theorems for computing the \'etale cohomology of a local system in terms of the cohomology of its corresponding Laurent $F$-crystal. In the case $X = \mathrm{Spf}\mathcal{O}_K$ for $K/L$ a $p$-adic field, we show that this recovers the Kisin-Ren equivalence between Lubin-Tate $(\varphi_q,\Gamma)$-modules and $\mathcal{O}_L$-linear representations of $G_K$ and the results of Kupferer and Venjakob for computing Galois cohomology in terms of Herr complexes of $(\varphi_q,\Gamma)$-modules. We can thus regard Laurent $F$-crystals on the $L$-typical prismatic site as providing a suitable notion of relative $(\varphi_q,\Gamma)$-modules

Linked Exact Triples of Triangulated Categories and a Calculus of t-Structures

We introduce a new formalism of exact triples of triangulated categories arranged in certain types of diagrams. We prove that these arrangements are well-behaved relative to the process of gluing and ungluing t-structures defined on the indicated categories and we connect our con. structs to· a problem (from number theory) involving derived categories. We also briefly address a possible connection with a result of R. Thomason

Kant on Perception, Experience and Judgements Thereof

It is commonly thought that the distinction between subjectively valid judgements of perception and objectively valid judgements of experience in the Prolegomena is not consistent with the account of judgement Kant offers in the B Deduction, according to which a judgement is ‘nothing other than the way to bring given cognitions to the objective unity of apperception’. Contrary to this view, I argue that the Prolegomena distinction maps closely onto that drawn between the mathematical and dynamical principles in the System of Principles: Kant’s account of the Prolegomena distinction strongly suggests that it is the Analogies of Experience that make it possible for judgements of perception to give rise to judgements of experience. This means that judgements of perception are objectively valid with regard to the quantity and quality of objects, and subjectively valid with regard to the relation they posit between objects. If that is the case, then the notion of a judgement of perception is consistent with the B Deduction account of judgement

Rankin--Eisenstein classes and explicit reciprocity laws

We construct three-variable $p$-adic families of Galois cohomology classes attached to Rankin convolutions of modular forms, and prove an explicit reciprocity law relating these classes to critical values of L-functions. As a consequence, we prove finiteness results for the Selmer group of an elliptic curve twisted by a 2-dimensional odd irreducible Artin representation when the associated $L$-value does not vanish.Comment: Final version, to appear in Cambridge J Math; small correction to acknowlegement