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1. Higher uniformity of arithmetic functions in short intervals I. All intervals

   https://doi.org/10.1017/fmp.2023.28  

   Matomäki, Kaisa ; Shao, Xuancheng ; Tao, Terence ; Teräväinen, Joni ( January 2023 , Forum of Mathematics, Pi)

   We study higher uniformity properties of the Möbius function$\mu $, the von Mangoldt function$\Lambda $, and the divisor functions$d_k$on short intervals$(X,X+H]$with$X^{\theta +\varepsilon } \leq H \leq X^{1-\varepsilon }$for a fixed constant$0 \leq \theta < 1$and any$\varepsilon>0$.

   More precisely, letting$\Lambda ^\sharp $and$d_k^\sharp $be suitable approximants of$\Lambda $and$d_k$and$\mu ^\sharp = 0$, we show for instance that, for any nilsequence$F(g(n)\Gamma )$, we have$$\begin{align*}\sum_{X < n \leq X+H} (f(n)-f^\sharp(n)) F(g(n) \Gamma) \ll H \log^{-A} X \end{align*}$$

   when$\theta = 5/8$and$f \in \{\Lambda , \mu , d_k\}$or$\theta = 1/3$and$f = d_2$.

   As a consequence, we show that the short interval Gowers norms$\|f-f^\sharp \|_{U^s(X,X+H]}$are also asymptotically small for any fixedsfor these choices of$f,\theta $. As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals and show that multiple ergodic averages along primes in short intervals converge in$L^2$.

   Our innovations include the use of multiparameter nilsequence equidistribution theorems to control type$II$sums and an elementary decomposition of the neighborhood of a hyperbola into arithmetic progressions to control type$I_2$sums.

    

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2. THE PAUCITY PROBLEM FOR CERTAIN SYMMETRIC DIOPHANTINE EQUATIONS

   https://doi.org/10.1017/S000497272200096X  

   WOOLEY, TREVOR D. ( August 2023 , Bulletin of the Australian Mathematical Society)

   Let$\varphi _1,\ldots ,\varphi _r\in {\mathbb Z}[z_1,\ldots z_k]$be integral linear combinations of elementary symmetric polynomials with$\text {deg}(\varphi _j)=k_j\ (1\le j\le r)$, where$1\le k_1. Subject to the condition$k_1+\cdots +k_r\ge \tfrac {1}{2}k(k-~1)+2$, we show that there is a paucity of nondiagonal solutions to the Diophantine system$\varphi _j({\mathbf x})=\varphi _j({\mathbf y})\ (1\le j\le r)$.

    

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3. 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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4. p -ADIC L -FUNCTIONS AND RATIONAL POINTS ON CM ELLIPTIC CURVES AT INERT PRIMES

   https://doi.org/10.1017/S147474802300021X  

   Burungale, Ashay A ; Kobayashi, Shinichi ; Ota, Kazuto ( May 2024 , Journal of the Institute of Mathematics of Jussieu)

   LetKbe an imaginary quadratic field and$p\geq 5$a rational prime inert inK. For a$\mathbb {Q}$-curveEwith complex multiplication by$\mathcal {O}_K$and good reduction atp, K. Rubin introduced ap-adicL-function$\mathscr {L}_{E}$which interpolates special values ofL-functions ofEtwisted by anticyclotomic characters ofK. In this paper, we prove a formula which links certain values of$\mathscr {L}_{E}$outside its defining range of interpolation with rational points onE. Arithmetic consequences includep-converse to the Gross–Zagier and Kolyvagin theorem forE.

   A key tool of the proof is the recent resolution of Rubin’s conjecture on the structure of local units in the anticyclotomic${\mathbb {Z}}_p$-extension$\Psi _\infty $of the unramified quadratic extension of${\mathbb {Q}}_p$. Along the way, we present a theory of local points over$\Psi _\infty $of the Lubin–Tate formal group of height$2$for the uniformizing parameter$-p$.

    

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5. Almost all orbits of the Collatz map attain almost bounded values

   https://doi.org/10.1017/fmp.2022.8  

   Tao, Terence ( January 2022 , Forum of Mathematics, Pi)

   Define theCollatz map${\operatorname {Col}} \colon \mathbb {N}+1 \to \mathbb {N}+1$on the positive integers$\mathbb {N}+1 = \{1,2,3,\dots \}$by setting${\operatorname {Col}}(N)$equal to$3N+1$whenNis odd and$N/2$whenNis even, and let${\operatorname {Col}}_{\min }(N) := \inf _{n \in \mathbb {N}} {\operatorname {Col}}^n(N)$denote the minimal element of the Collatz orbit$N, {\operatorname {Col}}(N), {\operatorname {Col}}^2(N), \dots $. The infamousCollatz conjectureasserts that${\operatorname {Col}}_{\min }(N)=1$for all$N \in \mathbb {N}+1$. Previously, it was shown by Korec that for any$\theta> \frac {\log 3}{\log 4} \approx 0.7924$, one has${\operatorname {Col}}_{\min }(N) \leq N^\theta $for almost all$N \in \mathbb {N}+1$(in the sense of natural density). In this paper, we show that foranyfunction$f \colon \mathbb {N}+1 \to \mathbb {R}$with$\lim _{N \to \infty } f(N)=+\infty $, one has${\operatorname {Col}}_{\min }(N) \leq f(N)$for almost all$N \in \mathbb {N}+1$(in the sense of logarithmic density). Our proof proceeds by establishing a stabilisation property for a certain first passage random variable associated with the Collatz iteration (or more precisely, the closely related Syracuse iteration), which in turn follows from estimation of the characteristic function of a certain skew random walk on a$3$-adic cyclic group$\mathbb {Z}/3^n\mathbb {Z}$at high frequencies. This estimation is achieved by studying how a certain two-dimensional renewal process interacts with a union of triangles associated to a given frequency.

    

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