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1. Chern classes of automorphic vector bundles, II

   Esnault, H. ; Harris, M. ( October 2019 , Épijournal de géométrie algébrique)

   We prove that the ℓ-adic Chern classes of canonical extensions of automorphic vector bundles, over toroidal compactifications of Shimura varieties of Hodge type over ℚ¯𝑝, descend to classes in the ℓ-adic cohomology of the minimal compactifications. These are invariant under the Galois group of the 𝑝-adic field above which the variety and the bundle are defined. 

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2. A unipotent circle action on 𝑝-adic modular forms

   https://doi.org/10.1090/btran/52  

   Howe, Sean ( November 2020 , Transactions of the American Mathematical Society, Series B)

   null (Ed.)

   Following a suggestion of Peter Scholze, we construct an action of G m ^ \widehat {\mathbb {G}_m} on the Katz moduli problem, a profinite-étale cover of the ordinary locus of the p p -adic modular curve whose ring of functions is Serre’s space of p p -adic modular functions. This action is a local, p p -adic analog of a global, archimedean action of the circle group S 1 S^1 on the lattice-unstable locus of the modular curve over C \mathbb {C} . To construct the G m ^ \widehat {\mathbb {G}_m} -action, we descend a moduli-theoretic action of a larger group on the (big) ordinary Igusa variety of Caraiani-Scholze. We compute the action explicitly on local expansions and find it is given by a simple multiplication of the cuspidal and Serre-Tate coordinates q q ; along the way we also prove a natural generalization of Dwork’s equation τ = log ⁡ q \tau =\log q for extensions of Q p / Z p \mathbb {Q}_p/\mathbb {Z}_p by μ p ∞ \mu _{p^\infty } valid over a non-Artinian base. Finally, we give a direct argument (without appealing to local expansions) to show that the action of G m ^ \widehat {\mathbb {G}_m} integrates the differential operator θ \theta coming from the Gauss-Manin connection and unit root splitting, and explain an application to Eisenstein measures and p p -adic L L -functions. 

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3. Exterior Powers in Iwasawa Theory

   Frauke M. Bleher, Ted Chinburg ( April 2022 , Journal of the European Mathematical Society)

   The Iwasawa theory of CM fields has traditionally concerned Iwasawa modules that are abelian pro-p Galois groups with ramification allowed at a maximal set of primes over p such that the module is torsion. A main conjecture for such an Iwasawa module describes its codimension one support in terms of a p-adic L-function attached to the primes of ramification. In this paper, we study more general and potentially much smaller modules that are quotients of exterior powers of Iwasawa modules with ramification at a set of primes over p by sums of exterior powers of inertia subgroups. We show that the higher codimension support of such quotients can be measured by finite collections of characteristic ideals of classical Iwasawa modules, hence by p-adic L-functions under the relevant CM main conjectures. 

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4. -ADIC -FUNCTIONS FOR UNITARY GROUPS

   https://doi.org/10.1017/fmp.2020.4  

   EISCHEN, ELLEN ; HARRIS, MICHAEL ; LI, JIANSHU ; SKINNER, CHRISTOPHER ( January 2020 , Forum of Mathematics, Pi)

   This paper completes the construction of $p$ -adic $L$ -functions for unitary groups. More precisely, in Harris, Li and Skinner [‘ $p$ -adic $L$ -functions for unitary Shimura varieties. I. Construction of the Eisenstein measure’, Doc. Math. Extra Vol. (2006), 393–464 (electronic)], three of the authors proposed an approach to constructing such $p$ -adic $L$ -functions (Part I). Building on more recent results, including the first named author’s construction of Eisenstein measures and $p$ -adic differential operators [Eischen, ‘A $p$ -adic Eisenstein measure for unitary groups’, J. Reine Angew. Math. 699 (2015), 111–142; ‘ $p$ -adic differential operators on automorphic forms on unitary groups’, Ann. Inst. Fourier (Grenoble) 62 (1) (2012), 177–243], Part II of the present paper provides the calculations of local $\unicode[STIX]{x1D701}$ -integrals occurring in the Euler product (including at $p$ ). Part III of the present paper develops the formalism needed to pair Eisenstein measures with Hida families in the setting of the doubling method. 

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5. COMPATIBLE SYSTEMS OF GALOIS REPRESENTATIONS ASSOCIATED TO THE EXCEPTIONAL GROUP

   https://doi.org/10.1017/fms.2018.24  

   BOXER, GEORGE ; CALEGARI, FRANK ; EMERTON, MATTHEW ; LEVIN, BRANDON ; MADAPUSI PERA, KEERTHI ; PATRIKIS, STEFAN ( January 2019 , Forum of Mathematics, Sigma)

   We construct, over any CM field, compatible systems of $l$ -adic Galois representations that appear in the cohomology of algebraic varieties and have (for all $l$ ) algebraic monodromy groups equal to the exceptional group of type $E_{6}$ . 

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