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We sketch the construction of a derived enhancement of the reciprocity isomorphism of class field theory. Details will appear in a forthcoming joint paper of the authors with A. Raksit., Comment: In memory of John Coates

Fix a positive integer n and a finite field $${\mathbb {F}}_q$$ F q . We study the joint distribution of the rank $${{\,\mathrm{rk}\,}}(E)$$ rk ( E ) , the n-Selmer group $$\text {Sel}_n(E)$$ Sel n ( E ) , and the n-torsion in the Tate–Shafarevich group "Equation missing" as E varies over elliptic curves of fixed height $$d \ge 2$$ d ≥ 2 over $${\mathbb {F}}_q(t)$$ F q ( t ) . We compute this joint distribution in the large q limit. We also show that the “large q, then large height” limit of this distribution agrees with the one predicted by Bhargava–Kane–Lenstra–Poonen–Rains.

We study a symplectic variant of algebraic $K$-theory of the integers, which comes equipped with a canonical action of the absolute Galois group of $\mathbf{Q}$. We compute this action explicitly. The representations we see are extensions of Tate twists $\mathbf{Z}_p(2k-1)$ by a trivial representation, and we characterize them by a universal property among such extensions. The key tool in the proof is the theory of complex multiplication for abelian varieties., Comment: 64 pages, final accepted version

Let $G$ be a split semi-simple group over a global function field $K$. Given a cuspidal automorphic representation $\Pi$ of $G$ satisfying a technical hypothesis, we prove that for almost all primes $\ell$, there is a cyclic base change lifting of $\Pi$ along any $\mathbb{Z}/\ell\mathbb{Z}$-extension of $K$. Our proof does not rely on any trace formulas; instead it is based on modularity lifting theorems, together with a Smith theory argument to obtain base change for residual representations. As an application, we also prove that for any split semisimple group $G$ over a local function field $F$, and almost all primes $\ell$, any irreducible admissible representation of $G(F)$ admits a base change along any $\mathbb{Z}/\ell\mathbb{Z}$-extension of $F$. Finally, we characterize local base change more explicitly for a class of representations called toral supercuspidal representations., Comment: Minor revisions

In previous work, we defined certain virtual fundamental classes for special cycles on the moduli stack of Hermitian shtukas, and related them to the higher derivatives of non-singular Fourier coefficients of Siegel-Eisenstein series. In the present article, we construct virtual fundamental classes in greater generality, including those expected to relate to the higher derivatives of singular Fourier coefficients. We assemble these classes into "higher" theta series, which we conjecture to be modular. Two types of evidence are presented: structural properties affirming that the cycle classes behave as conjectured under certain natural operations such as intersection products, and verification of modularity in several special situations. One innovation underlying these results is a new approach to special cycles in terms of derived algebraic geometry., Comment: Substantial revisions

Lafforgue and Genestier-Lafforgue have constructed the global and (semisimplified) local Langlands correspondences for arbitrary reductive groups over function fields. We establish various properties of these correspondences regarding functoriality for cyclic base change: For $\mathbb{Z}/p\mathbb{Z}$-extensions of global function fields, we prove the existence of base change for mod $p$ automorphic forms on arbitrary reductive groups. For $\mathbb{Z}/p\mathbb{Z}$-extensions of local function fields, we construct a base change homomorphism for the mod $p$ Bernstein center of any reductive group. We then use this to prove existence of local base change for mod $p$ irreducible representation along $\mathbb{Z}/p\mathbb{Z}$-extensions for all large enough $p$, and that Tate cohomology realizes descent along base change, verifying a function field version of a conjecture of Treumann-Venkatesh. The proofs are based on equivariant localization arguments for the moduli spaces of shtukas. They also draw upon new tools from representation theory, including parity sheaves and Smith-Treumann theory. In particular, we use these to establish a categorification of the base change homomorphism for mod $p$ spherical Hecke algebras, in a joint appendix with Gus Lonergan., Comment: Substantial revisions

We introduce a derived enhancement of local Galois deformation rings that we call the "spectral Hecke algebra", in analogy to a construction in the Geometric Langlands program. This is a Hecke algebra that acts on the spectral side of the Langlands correspondence, i.e. on moduli spaces of Galois representations. We verify the simplest form of derived local-global compatibility between the action of the spectral Hecke algebra on the derived Galois deformation ring of Galatius-Venkatesh, and the action of Venkatesh's derived Hecke algebra on the cohomology of arithmetic groups., Comment: Results reformulated in terms of the derived Hecke algebra

Using the calculus of Steenrod operations in \'etale cohomology developed in [Feng17], we prove that the analogue of Tate's pairing on higher Brauer groups is alternating on 2-torsion. This improves upon a result of Jahn [Jahn15, Math. Annalen]., Comment: This paper is being withdrawn because it has been completely subsumed by (version 2 of) arXiv:1706.00151

In 1972, Serre showed that the adelic Galois representation associated to a non-CM elliptic curve over a number field has open image in GL_2(\hat{Z}). In Greicius' thesis, he develops necessary and sufficient criteria for determining when this representation is actually surjective and exhibits such an example. However, verifying these criteria turns out to be difficult in practice; Greicius describes tests for them that apply only to semistable elliptic curves over a specific class of cubic number fields. In this paper, we extend Greicius' methods in several directions. First, we consider the analogous problem for elliptic curves with full 2-torsion. Following Greicius, we obtain necessary and sufficient conditions for the associated adelic representation to be maximal and also develop a battery of computationally effective tests that can be used to verify these conditions. We are able to use our tests to construct an infinite family of curves over Q(alpha) with maximal image, where alpha is the real root of x^3 + x + 1. Next, we extend Greicius' tests to more general settings, such as non-semistable elliptic curves over arbitrary cubic number fields. Finally, we give a general discussion concerning such problems for arbitrary torsion subgroups., 23 pages, this version incorporates the suggestions of the referee