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1. Relating elliptic curve point-counting and solutions of quadratic forms with congruence conditions

   https://doi.org/10.1007/s11139-026-01341-5  

   Mondal, Koustav (March 2026, The Ramanujan Journal)

   Abstract In this paper, we analyze the theta series associated to the quadratic form$$Q(\vec {x}) :=x_1^2+x_2^2+x_3^2+x_4^2$$ $Q\left(\overrightarrow{x}\right):=x_1^2+x_2^2+x_3^2+x_4^2$ with congruence conditions on$$x_i$$ $x_i$ modulo 2, 3, 4 and 6. By employing special operators on modular, non-holomorphic Eisenstein series of weight 2, we construct a basis for Eisenstein space for levels$$2^k, k\le 7$$ $2^k,k\le 7$ ,$$3^{\ell }, \ell \le 3$$ $3^{\ell },\ell \le 3$ andp, for odd primep. Using the relation between the trace of Frobenius on an elliptic curve and the Fourier coefficients of the cusp form part of theta series corresponding toQ, we establish relation between the number of integer solutions to the equation$$Q(\vec {x}) = p$$ $Q\left(\overrightarrow{x}\right)=p$ and the number of$$\mathbb {F}_p$$ $F_p$ -rational points on the associated elliptic curve under certain congruence conditions onp. 

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2. Abelian varieties of prescribed order over finite fields

   https://doi.org/10.1007/s00208-024-03084-4  

   van_Bommel, Raymond; Costa, Edgar; Li, Wanlin; Poonen, Bjorn; Smith, Alexander (May 2025, Mathematische Annalen)

   Abstract Given a prime powerqand$$n \gg 1$$ $n\gg 1$ , we prove that every integer in a large subinterval of the Hasse–Weil interval$$[(\sqrt{q}-1)^{2n},(\sqrt{q}+1)^{2n}]$$ $[{(\sqrt{q}-1)}^{2n},{(\sqrt{q}+1)}^{2n}]$ is$$\#A({\mathbb {F}}_q)$$ $\#A\left(F_q\right)$ for some ordinary geometrically simple principally polarized abelian varietyAof dimensionnover$${\mathbb {F}}_q$$ $F_q$ . As a consequence, we generalize a result of Howe and Kedlaya for$${\mathbb {F}}_2$$ $F_2$ to show that for each prime powerq, every sufficiently large positive integer is realizable, i.e.,$$\#A({\mathbb {F}}_q)$$ $\#A\left(F_q\right)$ for some abelian varietyAover$${\mathbb {F}}_q$$ $F_q$ . Our result also improves upon the best known constructions of sequences of simple abelian varieties with point counts towards the extremes of the Hasse–Weil interval. A separate argument determines, for fixedn, the largest subinterval of the Hasse–Weil interval consisting of realizable integers, asymptotically as$$q \rightarrow \infty $$ $q\to \infty$ ; this gives an asymptotically optimal improvement of a 1998 theorem of DiPippo and Howe. Our methods are effective: We prove that if$$q \le 5$$ $q\le 5$ , then every positive integer is realizable, and for arbitraryq, every positive integer$$\ge q^{3 \sqrt{q} \log q}$$ $\ge q^{3\sqrt{q}logq}$ is realizable. 

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3. Refined Chabauty–Kim computations for the thrice-punctured line over $$\mathbb {Z}[1/6]$$

   https://doi.org/10.1007/s40993-024-00597-4  

   Lüdtke, Martin (March 2025, Research in Number Theory)

   Abstract The Chabauty–Kim method and its refined variant by Betts and Dogra aim to cut out theS-integral points$$X(\mathbb {Z}_S)$$ $X\left(Z_S\right)$ on a curve inside thep-adic points$$X(\mathbb {Z}_p)$$ $X\left(Z_p\right)$ by producing enough Coleman functions vanishing on them. We derive new functions in the case of the thrice-punctured line whenScontains two primes. We describe an algorithm for computing refined Chabauty–Kim loci and verify Kim’s Conjecture over$$\mathbb {Z}[1/6]$$ $Z[1/6]$ for all choices of auxiliary prime $$p < 10{,}000$$ $p<10,000$ . 

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4. VC-Dimension of Hyperplanes Over Finite Fields

   https://doi.org/10.1007/s00373-025-02909-6  

   Ascoli, Ruben; Betti, Livia; Cheigh, Justin; Iosevich, Alex; Jeong, Ryan; Liu, Xuyan; McDonald, Brian; Milgrim, Wyatt; Miller, Steven J; Romero_Acosta, Francisco; et al (April 2025, Graphs and Combinatorics)

   Abstract Let$$\mathbb {F}_q^d$$ $F_q^d$ be thed-dimensional vector space over the finite field withqelements. For a subset$$E\subseteq \mathbb {F}_q^d$$ $E\subseteq F_q^d$ and a fixed nonzero$$t\in \mathbb {F}_q$$ $t\in F_q$ , let$$\mathcal {H}_t(E)=\{h_y: y\in E\}$$ $H_t\left(E\right)=\{h_y:y\in E\}$ , where$$h_y:E\rightarrow \{0,1\}$$ $h_y:E\to \{0,1\}$ is the indicator function of the set$$\{x\in E: x\cdot y=t\}$$ $\{x\in E:x·y=t\}$ . Two of the authors, with Maxwell Sun, showed in the case$$d=3$$ $d=3$ that if$$|E|\ge Cq^{\frac{11}{4}}$$ $\left|E\right|\ge C{q}^{\frac{11}{4}}$ andqis sufficiently large, then the VC-dimension of$$\mathcal {H}_t(E)$$ $H_t\left(E\right)$ is 3. In this paper, we generalize the result to arbitrary dimension by showing that the VC-dimension of$$\mathcal {H}_t(E)$$ $H_t\left(E\right)$ isdwhenever$$E\subseteq \mathbb {F}_q^d$$ $E\subseteq F_q^d$ with$$|E|\ge C_d q^{d-\frac{1}{d-1}}$$ $\left|E\right|\ge C_d{q}^{d-\frac{1}{d-1}}$ . 

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5. Undecidable Translational Tilings with Only Two Tiles, or One Nonabelian Tile

   https://doi.org/10.1007/s00454-022-00426-4  

   Greenfeld, Rachel; Tao, Terence (December 2023, Discrete & Computational Geometry)

   Abstract We construct an example of a group$$G = \mathbb {Z}^2 \times G_0$$ $G=Z^2\times G_0$ for a finite abelian group $$G_0$$ $G_0$ , a subsetEof $$G_0$$ $G_0$ , and two finite subsets$$F_1,F_2$$ $F_1,F_2$ of G, such that it is undecidable in ZFC whether$$\mathbb {Z}^2\times E$$ $Z^2\times E$ can be tiled by translations of$$F_1,F_2$$ $F_1,F_2$ . In particular, this implies that this tiling problem isaperiodic, in the sense that (in the standard universe of ZFC) there exist translational tilings ofEby the tiles$$F_1,F_2$$ $F_1,F_2$ , but no periodic tilings. Previously, such aperiodic or undecidable translational tilings were only constructed for sets of eleven or more tiles (mostly in $$\mathbb {Z}^2$$ $Z^2$ ). A similar construction also applies for$$G=\mathbb {Z}^d$$ $G=Z^d$ for sufficiently large d. If one allows the group$$G_0$$ $G_0$ to be non-abelian, a variant of the construction produces an undecidable translational tiling with only one tile F. The argument proceeds by first observing that a single tiling equation is able to encode an arbitrary system of tiling equations, which in turn can encode an arbitrary system of certain functional equations once one has two or more tiles. In particular, one can use two tiles to encode tiling problems for an arbitrary number of tiles. 

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