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1. IWASAWA THEORY FOR p -TORSION CLASS GROUP SCHEMES IN CHARACTERISTIC p

   https://doi.org/10.1017/nmj.2022.30  

   BOOHER, JEREMY ; CAIS, BRYDEN ( June 2023 , Nagoya Mathematical Journal)

   We investigate a novel geometric Iwasawa theory for${\mathbf Z}_p$-extensions of function fields over a perfect fieldkof characteristic$p>0$by replacing the usual study ofp-torsion in class groups with the study ofp-torsion class groupschemes. That is, if$\cdots \to X_2 \to X_1 \to X_0$is the tower of curves overkassociated with a${\mathbf Z}_p$-extension of function fields totally ramified over a finite nonempty set of places, we investigate the growth of thep-torsion group scheme in the Jacobian of$X_n$as$n\rightarrow \infty $. By Dieudonné theory, this amounts to studying the first de Rham cohomology groups of$X_n$equipped with natural actions of Frobenius and of the Cartier operatorV. We formulate and test a number of conjectures which predict striking regularity in the$k[V]$-module structure of the space$M_n:=H^0(X_n, \Omega ^1_{X_n/k})$of global regular differential forms as$n\rightarrow \infty .$For example, for each tower in a basic class of${\mathbf Z}_p$-towers, we conjecture that the dimension of the kernel of$V^r$on$M_n$is given by$a_r p^{2n} + \lambda _r n + c_r(n)$for allnsufficiently large, where$a_r, \lambda _r$are rational constants and$c_r : {\mathbf Z}/m_r {\mathbf Z} \to {\mathbf Q}$is a periodic function, depending onrand the tower. To provide evidence for these conjectures, we collect extensive experimental data based on new and more efficient algorithms for working with differentials on${\mathbf Z}_p$-towers of curves, and we prove our conjectures in the case$p=2$and$r=1$.

    

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2. Theta functions, fourth moments of eigenforms and the sup-norm problem II

   https://doi.org/10.1017/fmp.2024.9  

   Khayutin, Ilya ; Nelson, Paul D ; Steiner, Raphael S ( January 2024 , Forum of Mathematics, Pi)

   Letfbe an$L^2$-normalized holomorphic newform of weightkon$\Gamma _0(N) \backslash \mathbb {H}$withNsquarefree or, more generally, on any hyperbolic surface$\Gamma \backslash \mathbb {H}$attached to an Eichler order of squarefree level in an indefinite quaternion algebra over$\mathbb {Q}$. Denote byVthe hyperbolic volume of said surface. We prove the sup-norm estimate$$\begin{align*}\| \Im(\cdot)^{\frac{k}{2}} f \|_{\infty} \ll_{\varepsilon} (k V)^{\frac{1}{4}+\varepsilon} \end{align*}$$

   with absolute implied constant. For a cuspidal Maaß newform$\varphi $of eigenvalue$\lambda $on such a surface, we prove that$$\begin{align*}\|\varphi \|_{\infty} \ll_{\lambda,\varepsilon} V^{\frac{1}{4}+\varepsilon}. \end{align*}$$

   We establish analogous estimates in the setting of definite quaternion algebras.

    

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3. A TOPOLOGICAL APPROACH TO UNDEFINABILITY IN ALGEBRAIC EXTENSIONS OF Q

   https://doi.org/10.1017/bsl.2023.37  

   EISENTRÄGER, KIRSTEN ; MILLER, RUSSELL ; SPRINGER, CALEB ; WESTRICK, LINDA ( December 2023 , The Bulletin of Symbolic Logic)

   For any subset$Z \subseteq {\mathbb {Q}}$, consider the set$S_Z$of subfields$L\subseteq {\overline {\mathbb {Q}}}$which contain a co-infinite subset$C \subseteq L$that is universally definable inLsuch that$C \cap {\mathbb {Q}}=Z$. Placing a natural topology on the set${\operatorname {Sub}({\overline {\mathbb {Q}}})}$of subfields of${\overline {\mathbb {Q}}}$, we show that ifZis not thin in${\mathbb {Q}}$, then$S_Z$is meager in${\operatorname {Sub}({\overline {\mathbb {Q}}})}$. Here,thinandmeagerboth mean “small”, in terms of arithmetic geometry and topology, respectively. For example, this implies that only a meager set of fieldsLhave the property that the ring of algebraic integers$\mathcal {O}_L$is universally definable inL. The main tools are Hilbert’s Irreducibility Theorem and a new normal form theorem for existential definitions. The normal form theorem, which may be of independent interest, says roughly that every$\exists $-definable subset of an algebraic extension of${\mathbb Q}$is a finite union of single points and projections of hypersurfaces defined by absolutely irreducible polynomials.

    

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4. Fixed point sets and the fundamental group I: semi-free actions on G -CW-complexes

   https://doi.org/10.1017/prm.2023.63  

   Cappell, Sylvain ; Weinberger, Shmuel ; Yan, Min ( August 2023 , Proceedings of the Royal Society of Edinburgh: Section A Mathematics)

   Smith theory says that the fixed point set of a semi-free action of a group$G$on a contractible space is${\mathbb {Z}}_p$-acyclic for any prime factor$p$of the order of$G$. Jones proved the converse of Smith theory for the case$G$is a cyclic group acting semi-freely on contractible, finite CW-complexes. We extend the theory to semi-free group actions on finite CW-complexes of given homotopy types, in various settings. In particular, the converse of Smith theory holds if and only if a certain$K$-theoretical obstruction vanishes. We also give some examples that show the geometrical effects of different types of$K$-theoretical obstructions.

    

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5. The eleventh cohomology group of $\overline {\mathcal {M}}_{g,n}$

   https://doi.org/10.1017/fms.2023.59  

   Canning, Samir ; Larson, Hannah ; Payne, Sam ( January 2023 , Forum of Mathematics, Sigma)

   We prove that the rational cohomology group$H^{11}(\overline {\mathcal {M}}_{g,n})$vanishes unless$g = 1$and$n \geq 11$. We show furthermore that$H^k(\overline {\mathcal {M}}_{g,n})$is pure Hodge–Tate for all even$k \leq 12$and deduce that$\# \overline {\mathcal {M}}_{g,n}(\mathbb {F}_q)$is surprisingly well approximated by a polynomial inq. In addition, we use$H^{11}(\overline {\mathcal {M}}_{1,11})$and its image under Gysin push-forward for tautological maps to produce many new examples of moduli spaces of stable curves with nonvanishing odd cohomology and nontautological algebraic cycle classes in Chow cohomology.

    

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