More Like this

1. 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. 

   more » « less
2. 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. 

   more » « less
3. Hardness and approximation of submodular minimum linear ordering problems

   https://doi.org/10.1007/s10107-023-02038-z  

   Farhadi, Majid; Gupta, Swati; Sun, Shengding; Tetali, Prasad; Wigal, Michael C (December 2023, Mathematical Programming)

   Abstract The minimum linear ordering problem (MLOP) generalizes well-known combinatorial optimization problems such as minimum linear arrangement and minimum sum set cover. MLOP seeks to minimize an aggregated cost$$f(\cdot )$$ $f(·)$due to an ordering$$\sigma $$ $\sigma$of the items (say [n]), i.e.,$$\min _{\sigma } \sum _{i\in [n]} f(E_{i,\sigma })$$ ${min}_{\sigma }{\sum }_{i\in \left[n\right]}f\left(E_{i,\sigma }\right)$, where$$E_{i,\sigma }$$ $E_{i,\sigma }$is the set of items mapped by$$\sigma $$ $\sigma$to indices [i]. Despite an extensive literature on MLOP variants and approximations for these, it was unclear whether the graphic matroid MLOP was NP-hard. We settle this question through non-trivial reductions from mininimum latency vertex cover and minimum sum vertex cover problems. We further propose a new combinatorial algorithm for approximating monotone submodular MLOP, using the theory of principal partitions. This is in contrast to the rounding algorithm by Iwata et al. (in: APPROX, 2012), using Lovász extension of submodular functions. We show a$$(2-\frac{1+\ell _{f}}{1+|E|})$$ $(2-\frac{1+{\ell }_f}{1+\left|E\right|})$-approximation for monotone submodular MLOP where$$\ell _{f}=\frac{f(E)}{\max _{x\in E}f(\{x\})}$$ ${\ell }_f=\frac{f\left(E\right)}{{max}_{x\in E}f\left(\left\{x\right\}\right)}$satisfies$$1 \le \ell _f \le |E|$$ $1\le {\ell }_f\le \left|E\right|$. Our theory provides new approximation bounds for special cases of the problem, in particular a$$(2-\frac{1+r(E)}{1+|E|})$$ $(2-\frac{1+r\left(E\right)}{1+\left|E\right|})$-approximation for the matroid MLOP, where$$f = r$$ $f=r$is the rank function of a matroid. We further show that minimum latency vertex cover is$$\frac{4}{3}$$ $\frac{4}{3}$-approximable, by which we also lower bound the integrality gap of its natural LP relaxation, which might be of independent interest. 

   more » « less
4. 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. 

   more » « less
5. Long–Time and Chaotic Dynamics in Microscopic Systems with Singular Interactions

   https://doi.org/10.1007/s00205-026-02189-1  

   Béjar-López, Alexis; Blaustein, Alain; Jabin, Pierre-Emmanuel; Soler, Juan (August 2026, Archive for Rational Mechanics and Analysis)

   Abstract This paper investigates the long–time dynamics of interacting particle systems subject to singular interactions. We consider a microscopic system ofNinteracting point particles, where the time evolution of the joint distribution$$f_N(t)$$ $f_N\left(t\right)$is governed by the Liouville equation. Our primary objective is to analyze the system’s behavior over extended time intervals, focusing on the stability, the potential chaotic dynamics and the impact of singularities. In particular, we aim to derive reduced models in the regime$$N \gg 1$$ $N\gg 1$, exploring both the mean-field approximation and configurations far from chaos, where the mean-field approximation no longer holds. These reduced models do not always emerge but in these cases we prove that it is possible to derive uniform bounds in$$ L^2 $$ $L^2$, both over time and with respect to the number of particles, on the marginals$$ \left( f_{k,N}\right) _{1\leqq k \leqq N}$$ ${\left(f_{k,N}\right)}_{1\leqq k\leqq N}$, irrespective of the initial state’s chaotic nature. Furthermore, we extend previous results by considering a wide range of singular interaction kernels$$ K \in W^{\frac{-2}{d+2}, d+2} $$ $K\in W^{\frac{-2}{d+2},d+2}$in dimension$$d\geqq 2$$ $d\geqq 2$, surpassing the traditional$$ L^d $$ $L^d$regularity barriers. Finally, we address the highly singular case of$$ K \in H^{-1} $$ $K\in H^{-1}$under a threshold temperature regime, offering new insights into the behavior of such systems. 

   more » « less