{"title":"Just-likely intersections on Hilbert modular surfaces","authors":"","doi":"10.1007/s00208-023-02793-6","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>In this paper, we prove an intersection-theoretic result pertaining to curves in certain Hilbert modular surfaces in positive characteristic <em>p</em>. Specifically, let <em>C</em>, <em>D</em> be two proper curves inside a mod <em>p</em> Hilbert modular surface associated to a real quadratic field split at <em>p</em>. Suppose that the curves are generically ordinary, and that at least one of them is ample. Then, the set of points in <span> <span>\\((x,y) \\in C\\times D\\)</span> </span> with abelian surfaces parameterized by <em>x</em> and <em>y</em> isogenous to each other is Zariski dense in <span> <span>\\(C\\times D\\)</span> </span>, thereby proving a case of a just-likely intersection conjecture. We also compute the change in Faltings height under appropriate <em>p</em>-power isogenies of abelian surfaces with real multiplication over characteristic <em>p</em> global fields.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"12 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2024-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematische Annalen","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00208-023-02793-6","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we prove an intersection-theoretic result pertaining to curves in certain Hilbert modular surfaces in positive characteristic p. Specifically, let C, D be two proper curves inside a mod p Hilbert modular surface associated to a real quadratic field split at p. Suppose that the curves are generically ordinary, and that at least one of them is ample. Then, the set of points in \((x,y) \in C\times D\) with abelian surfaces parameterized by x and y isogenous to each other is Zariski dense in \(C\times D\), thereby proving a case of a just-likely intersection conjecture. We also compute the change in Faltings height under appropriate p-power isogenies of abelian surfaces with real multiplication over characteristic p global fields.
摘要 本文证明了与正特征 p 的某些希尔伯特模面中的曲线有关的交点理论结果。具体地说，设 C，D 是与实二次型场 p 分割相关的模 p 希尔伯特模面中的两条适当曲线。那么，在 \((x,y) \in C\times D\) 中以 x 和 y 为参数的无常曲面彼此同源的点集就是 \(C\times D\) 中的扎里斯基密集点，从而证明了一个 "可能交点猜想"。我们还计算了在特征 p 全局域上具有实乘法的无常曲面的适当 p-power 同源下的 Faltings 高度变化。
期刊介绍：
Begründet 1868 durch Alfred Clebsch und Carl Neumann. Fortgeführt durch Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück und Nigel Hitchin.
The journal Mathematische Annalen was founded in 1868 by Alfred Clebsch and Carl Neumann. It was continued by Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguigon, Wolfgang Lück and Nigel Hitchin.
Since 1868 the name Mathematische Annalen stands for a long tradition and high quality in the publication of mathematical research articles. Mathematische Annalen is designed not as a specialized journal but covers a wide spectrum of modern mathematics.
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