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1. 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)

   Abstract 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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2. Degrees of maps and multiscale geometry

   https://doi.org/10.1017/fmp.2023.33  

   Berdnikov, Aleksandr; Guth, Larry; Manin, Fedor (January 2024, Forum of Mathematics, Pi)

   Abstract We study the degree of an L-Lipschitz map between Riemannian manifolds, proving new upper bounds and constructing new examples. For instance, if $$X_k$$ is the connected sum of k copies of $$\mathbb CP^2$$for$$k \ge 4$$, then we prove that the maximum degree of an L-Lipschitz self-map of $$X_k$$ is between $$C_1 L^4 (\log L)^{-4}$$ and $$C_2 L^4 (\log L)^{-1/2}$$. More generally, we divide simply connected manifolds into three topological types with three different behaviors. Each type is defined by purely topological criteria. For scalable simply connected n-manifolds, the maximal degree is $$\sim L^n$$. For formal but nonscalable simply connectedn-manifolds, the maximal degree grows roughly like $$L^n (\log L)^{-\theta (1)}$$. And for nonformal simply connected n-manifolds, the maximal degree is bounded by $$L^\alpha $$ for some $$\alpha < n$$. 

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3. Transversal C_k -factors in subgraphs of the balanced blow-up of C_k

   https://doi.org/10.1017/S0963548322000086  

   Ergemlidze, Beka; Molla, Theodore (November 2022, Combinatorics, Probability and Computing)

   Abstract For a subgraph$$G$$of the blow-up of a graph$$F$$, we let$$\delta ^*(G)$$be the smallest minimum degree over all of the bipartite subgraphs of$$G$$induced by pairs of parts that correspond to edges of$$F$$. Johansson proved that if$$G$$is a spanning subgraph of the blow-up of$$C_3$$with parts of size$$n$$and$$\delta ^*(G) \ge \frac{2}{3}n + \sqrt{n}$$, then$$G$$contains$$n$$vertex disjoint triangles, and presented the following conjecture of Häggkvist. If$$G$$is a spanning subgraph of the blow-up of$$C_k$$with parts of size$$n$$and$$\delta ^*(G) \ge \left(1 + \frac 1k\right)\frac n2 + 1$$, then$$G$$contains$$n$$vertex disjoint copies of$$C_k$$such that each$$C_k$$intersects each of the$$k$$parts exactly once. A similar conjecture was also made by Fischer and the case$$k=3$$was proved for large$$n$$by Magyar and Martin. In this paper, we prove the conjecture of Häggkvist asymptotically. We also pose a conjecture which generalises this result by allowing the minimum degree conditions in each bipartite subgraph induced by pairs of parts of$$G$$to vary. We support this new conjecture by proving the triangle case. This result generalises Johannson’s result asymptotically. 

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4. Bipartite-ness under smooth conditions

   https://doi.org/10.1017/S0963548323000019  

   Jiang, Tao; Longbrake, Sean; Ma, Jie (February 2023, Combinatorics, Probability and Computing)

   Abstract Given a family$$\mathcal{F}$$of bipartite graphs, theZarankiewicz number$$z(m,n,\mathcal{F})$$is the maximum number of edges in an$$m$$by$$n$$bipartite graph$$G$$that does not contain any member of$$\mathcal{F}$$as a subgraph (such$$G$$is called$$\mathcal{F}$$-free). For$$1\leq \beta \lt \alpha \lt 2$$, a family$$\mathcal{F}$$of bipartite graphs is$$(\alpha,\beta )$$-smoothif for some$$\rho \gt 0$$and every$$m\leq n$$,$$z(m,n,\mathcal{F})=\rho m n^{\alpha -1}+O(n^\beta )$$. Motivated by their work on a conjecture of Erdős and Simonovits on compactness and a classic result of Andrásfai, Erdős and Sós, Allen, Keevash, Sudakov and Verstraëte proved that for any$$(\alpha,\beta )$$-smooth family$$\mathcal{F}$$, there exists$$k_0$$such that for all odd$$k\geq k_0$$and sufficiently large$$n$$, any$$n$$-vertex$$\mathcal{F}\cup \{C_k\}$$-free graph with minimum degree at least$$\rho (\frac{2n}{5}+o(n))^{\alpha -1}$$is bipartite. In this paper, we strengthen their result by showing that for every real$$\delta \gt 0$$, there exists$$k_0$$such that for all odd$$k\geq k_0$$and sufficiently large$$n$$, any$$n$$-vertex$$\mathcal{F}\cup \{C_k\}$$-free graph with minimum degree at least$$\delta n^{\alpha -1}$$is bipartite. Furthermore, our result holds under a more relaxed notion of smoothness, which include the families$$\mathcal{F}$$consisting of the single graph$$K_{s,t}$$when$$t\gg s$$. We also prove an analogous result for$$C_{2\ell }$$-free graphs for every$$\ell \geq 2$$, which complements a result of Keevash, Sudakov and Verstraëte. 

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5. 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)

   Abstract 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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