More Like this

1. Counting points on smooth plane quartics

   https://doi.org/10.1007/s40993-022-00397-8  

   Costa, Edgar; Harvey, David; Sutherland, Andrew V. (March 2023, Research in Number Theory)

   Abstract We present efficient algorithms for counting points on a smooth plane quartic curve X modulo a prime p . We address both the case where X is defined over  $${\mathbb {F}}_p$$ F p and the case where X is defined over $${\mathbb {Q}}$$ Q and p is a prime of good reduction. We consider two approaches for computing $$\#X({\mathbb {F}}_p)$$ # X ( F p ) , one which runs in $$O(p\log p\log \log p)$$ O ( p log p log log p ) time using $$O(\log p)$$ O ( log p ) space and one which runs in $$O(p^{1/2}\log ^2p)$$ O ( p 1 / 2 log 2 p ) time using $$O(p^{1/2}\log p)$$ O ( p 1 / 2 log p ) space. Both approaches yield algorithms that are faster in practice than existing methods. We also present average polynomial-time algorithms for $$X/{\mathbb {Q}}$$ X / Q that compute $$\#X({\mathbb {F}}_p)$$ # X ( F p ) for good primes $$p\leqslant N$$ p ⩽ N in $$O(N\log ^3 N)$$ O ( N log 3 N ) time using O ( N ) space. These are the first practical implementations of average polynomial-time algorithms for curves that are not cyclic covers of $${\mathbb {P}}^1$$ P 1 , which in combination with previous results addresses all curves of genus $$g\leqslant 3$$ g ⩽ 3 . Our algorithms also compute Cartier–Manin/Hasse–Witt matrices that may be of independent interest. 

   more » « less
2. Variation of canonical height for\break Fatou points on ℙ ¹

   https://doi.org/10.1515/crelle-2022-0078  

   DeMarco, Laura; Mavraki, Niki Myrto (December 2022, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let f : ℙ 1 → ℙ 1 {f:\mathbb{P}^{1}\to\mathbb{P}^{1}} be a map of degree > 1 {>1} defined over a function field k = K ⁢ ( X ) {k=K(X)} , where K is a number field and X is a projective curve over K . For each point a ∈ ℙ 1 ⁢ ( k ) {a\in\mathbb{P}^{1}(k)} satisfying a dynamical stability condition, we prove that the Call–Silverman canonical height for specialization f t {f_{t}} at point a t {a_{t}} , for t ∈ X ⁢ ( ℚ ¯ ) {t\in X(\overline{\mathbb{Q}})} outside a finite set, induces a Weil height on the curve X ; i.e., we prove the existence of a ℚ {\mathbb{Q}} -divisor D = D f , a {D=D_{f,a}} on X so that the function t ↦ h ^ f t ⁢ ( a t ) - h D ⁢ ( t ) {t\mapsto\hat{h}_{f_{t}}(a_{t})-h_{D}(t)} is bounded on X ⁢ ( ℚ ¯ ) {X(\overline{\mathbb{Q}})} for any choice of Weil height associated to D . We also prove a local version, that the local canonical heights t ↦ λ ^ f t , v ⁢ ( a t ) {t\mapsto\hat{\lambda}_{f_{t},v}(a_{t})} differ from a Weil function for D by a continuous function on X ⁢ ( ℂ v ) {X(\mathbb{C}_{v})} , at each place v of the number field K . These results were known for polynomial maps f and all points a ∈ ℙ 1 ⁢ ( k ) {a\in\mathbb{P}^{1}(k)} without the stability hypothesis,[21, 14],and for maps f that are quotients of endomorphisms of elliptic curves E over k and all points a ∈ ℙ 1 ⁢ ( k ) {a\in\mathbb{P}^{1}(k)} . [32, 29].Finally, we characterize our stability condition in terms of the geometry of the induced map f ~ : X × ℙ 1 ⇢ X × ℙ 1 {\tilde{f}:X\times\mathbb{P}^{1}\dashrightarrow X\times\mathbb{P}^{1}} over K ; and we prove the existence of relative Néron models for the pair ( f , a ) {(f,a)} , when a is a Fatou point at a place γ of k , where the local canonical height λ ^ f , γ ⁢ ( a ) {\hat{\lambda}_{f,\gamma}(a)} can be computed as an intersection number. 

   more » « less
3. 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 (March 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$$|E|\ge C_d q^{d-\frac{1}{d-1}}$$ $\left|E\right|\ge C_d{q}^{d-\frac{1}{d-1}}$. 

   more » « less
4. A New Family of Algebraically Defined Graphs with Small Automorphism Group

   https://doi.org/10.37236/10707  

   Lazebnik, Felix; Taranchuk, Vladislav (January 2022, The Electronic Journal of Combinatorics)

   Let $$p$$ be an odd  prime, $q=p^e$, $$e \geq 1$$, and $$\mathbb{F} = \mathbb{F}_q$$ denote the finite field of $$q$$ elements.  Let $$f: \mathbb{F}^2\to \mathbb{F}$$ and  $$g: \mathbb{F}^3\to \mathbb{F}$$  be functions, and  let $$P$$ and $$L$$ be two copies of the 3-dimensional vector space $$\mathbb{F}^3$$. Consider a bipartite graph $$\Gamma_\mathbb{F} (f, g)$$ with vertex partitions $$P$$ and $$L$$ and with edges defined as follows: for every $$(p)=(p_1,p_2,p_3)\in P$$ and every $$[l]= [l_1,l_2,l_3]\in L$$, $$\{(p), [l]\} = (p)[l]$$ is an edge in $$\Gamma_\mathbb{F} (f, g)$$ if $$p_2+l_2 =f(p_1,l_1) \;\;\;\text{and}\;\;\; p_3 + l_3 = g(p_1,p_2,l_1).$$The following question  appeared in Nassau: Given $$\Gamma_\mathbb{F} (f, g)$$,  is it always possible to find a function $$h:\mathbb{F}^2\to \mathbb{F}$$ such that the graph $$\Gamma_\mathbb{F} (f, h)$$  with the same vertex set as $$\Gamma_\mathbb{F} (f, g)$$ and with edges $(p)[l]$  defined in a similar way  by the system $$p_2+l_2 =f(p_1,l_1) \;\;\;\text{and}\;\;\; p_3 + l_3 = h(p_1,l_1),$$ is isomorphic to $$\Gamma_\mathbb{F} (f, g)$$ for infinitely many $$q$$?  In this paper we show that the  answer to the question is negative and the graphs $$\Gamma_{\mathbb{F}_p}(p_1\ell_1, p_1\ell_1p_2(p_1 + p_2 + p_1p_2))$$ provide such an example for $$p \equiv 1 \pmod{3}$$. Our argument is based on proving that the automorphism group of these graphs has order $$p$$, which is the smallest possible order of the automorphism group of graphs of the form $$\Gamma_{\mathbb{F}}(f, g)$$. 

   more » « less
5. A rational map with infinitely many points of distinct arithmetic degrees

   https://doi.org/10.1017/etds.2019.30  

   LESIEUTRE, JOHN; SATRIANO, MATTHEW (January 2019, Ergodic Theory and Dynamical Systems)

   Let be a dominant rational self-map of a smooth projective variety defined over $$\overline{\mathbb{Q}}$$ . For each point $$P\in X(\overline{\mathbb{Q}})$$ whose forward $$f$$ -orbit is well defined, Silverman introduced the arithmetic degree $$\unicode[STIX]{x1D6FC}_{f}(P)$$ , which measures the growth rate of the heights of the points $$f^{n}(P)$$ . Kawaguchi and Silverman conjectured that $$\unicode[STIX]{x1D6FC}_{f}(P)$$ is well defined and that, as $$P$$ varies, the set of values obtained by $$\unicode[STIX]{x1D6FC}_{f}(P)$$ is finite. Based on constructions by Bedford and Kim and by McMullen, we give a counterexample to this conjecture when $$X=\mathbb{P}^{4}$$ . 

   more » « less