Related papers: Explicit reciprocity laws for diagonal classes: hi…

We describe all inequalities among generalized diagonals in positive semi-definite matrices. These turn out to be governed by a simple partial order on the symmetric group. This provides an analogue of results of Drake, Gerrish, and…

We construct a central extension of the smooth Deligne cohomology group of a compact oriented odd dimensional smooth manifold, generalizing that of the loop group of the circle. While the central extension turns out to be trivial for a…

In this note we introduce higher order polar loci as natural generalizations of the classical polar loci, replacing the role of tangent spaces by that of higher order osculating spaces. The close connection between polar loci and dual…

An element $g$ in a group $G$ is called \emph{reciprocal} if there exists $h \in G$ such that $g^{-1}=hgh^{-1}$. The reciprocal elements are also known as `real elements' or `reversible elements' in the literature. We classify the…

We prove a variant of the reciprocity laws for CM abelian varieties, CM K3 surfaces, and CM points on Shimura varieties. Given a CM object over the complex numbers, our variation describes the set of all models over a given number field $F$…

Given a complex simply connected simple algebraic group $G$ of exceptional type and a maximal parabolic subgroup $P \subset G$, we classify all triples $(G,P,H)$ such that $H \subset G$ is a maximal reductive subgroup acting spherically on…

For each subset of primes in a totally real field above a rational prime $p$, there is the notion of partially classical Hilbert modular forms, where the empty set recovers the overconvergent forms and the full set of primes above $p$…

Let $(\mathfrak{g},[p])$ be a finite dimensional restricted Lie algebra over a perfect field $\mathbbm{k}$ of characteristic $p\!\ge \!3$. By combining methods from recent work of Benson-Carlson \cite{BC20} with those of \cite{CF21,Fa17} we…

We prove a commutative algebra result which has consequences for congruences between automorphic forms modulo prime powers. If C denotes the congruence module for a fixed automorphic Hecke eigenform \pi_0 we prove an exact relation between…

Let $E/\mathbb{Q}$ be an elliptic curve having multiplicative reduction at a prime $p$. Let $(g,h)$ be a pair of eigenforms of weight $1$ arising as the theta series of an imaginary quadratic field $K$, and assume that the triple-product…

Let $p$ be a prime and let $K$ be a finite extension of the field ${\bf Q}_p$ of $p$-adic numbers such that the group ${}_pK^\times$ has order $p$. The ${\bf F}_p$-space $K^\times\!/K^{\times p}$ carries a natural filtration coming from the…

Given a smooth proper dg-algebra $A$, a perfect dg $A$-module $M$, and an endomorphism $f$ of $M$, we define the Hochschild class of the pair $(M,f)$ with values in the Hochschild homology of $A$. Our main result is a Riemann-Roch type…

We revisit several known versions of the Dehn--Sommerville relations in the context of: homology manifolds, semi-Eulerian complexes, general simplicial complexes, balanced semi-Eulerian complexes and general completely balanced complexes.…

We classify 1-tilting classes over an arbitrary commutative ring. As a consequence, we classify all resolving subcategories of finitely presented modules of projective dimension at most 1. Both these collections are in 1-1 correspondence…

We exhibit a higher-level analogue of the Bernstein-Gelfand-Gelfand (BGG) reciprocity for twisted current algebras for each positive integer, which recovers the original one (established by Bennett, Berenstein, Chari, Ion, Khoroshkin,…

We use non-abelian fundamental groups to define a sequence of higher reciprocity maps on the adelic points of a variety over a number field satisfying certain conditions in Galois cohomology. The non-abelian reciprocity law then states that…

We give an explicit version of Shimura's reciprocity law for singular values of Siegel modular functions. We use this to construct the first examples of class invariants of quartic CM fields that are smaller than Igusa invariants. Our…

Following up a previous article of the authors which studies the interpolation of certain anticyclotomic $p$-adic $L$-functions associated to quaternionic modular forms in a Hida family, we extend the work of F. Castella on the…

We reformulate the problem of bounding the total rank of the homology of perfect chain complexes over the group ring $\mathbb{F}_p[G]$ of an elementary abelian $p$-group $G$ in terms of commutative algebra. This extends results of Carlsson…

We prove that every degree-g polynomial in the $\psi$-classes on $\overline{\mathcal M}_{g, n}$ can be expressed as a sum of tautological classes supported on the boundary with no $\kappa$-classes. Such equations, which we refer to as…