Moduli Spaces that are Upper Half Plane Quotients and the Inverse Galois Problem

模块化塔和有理上的IGP：研究人员经常使用两个变量x和y的代数关系来描述一个字段上的重要数据。一些例子是值x的集合具有（x,y），字段中的条目满足关系。其中之一是根据有限域上多项式的值集来确定多项式的等价类的达文波特问题。描述由具有（x,y）特殊椭圆曲线扭转点的x生成的复乘域是一个变体。与数据变量x的关系产生一个置换群G：伽罗瓦群。研究者把这些当作逆伽罗维问题（IGP）的例子。一个目标是从某些关系列表中确定是否有有理数作为定义字段。严肃的群论帮助解决了达文波特1965年的问题和舒尔的例外多项式问题（1919；见下文）。此外，模曲线揭示了复乘法是舒尔问题的一个版本。掌握所有有限单群有许多应用。一种是列出那些附加到特殊多项式上的群。了解了Chevalley群的许多特征，thompson和Voelklein找到了一系列解决IGP的方法。然而，要解决整个IGP问题，需要的不仅仅是大规模群论。研究者使用模块化塔，一种模块化曲线泛化，对不可知的群体分类进行精细处理。他和他的学生让已知群G和一个素数p除以G的阶，作为模塔的基层。更高的塔层用覆盖G与p相关的神秘群来编码关系。塔层之间的连贯映射取代了这些复杂覆盖群的细节。伊哈拉和j.p的作品。Serre启发表明，模块化塔具有其特殊的模块化曲线的许多特性。例如，没有Z/p商的aG的高塔关卡应该没有有理点。对于某些群G和素数p=2，这扩展了Serre关于自旋结构的规划。研究者将新的类自旋结构应用于p = 2时的所有情况和所有素数p。有限域上的实用密码学和IGP：现代密码系统旨在保证数据的电子传输安全。异常函数在无限多个有限域上起突变作用。有限域上逆伽罗瓦问题的一个特例是如何构造异常函数。这些加扰函数看起来很简单。用户可以很容易地应用它们（直到1993年，它们都来自19世纪）。然而，找到他们一直很困难。弗里德（UCI）、古拉尔尼克（USC）和萨克斯（剑桥）几乎把它们归为一类。这解决了老问题（Dickson 1896和Carlitz 1965）。它们还会产生意想不到的新例子。他们最好的发现是特殊的多项式伽罗瓦群（不包括现在已经被充分理解的情况）具有仿射几何性质。这很有用，因为它给出了异常多项式的次。然而，这也是一个困难的挑战：对allaffine群没有最终的描述。研究者A. Mezard通过将grothendieck关于驯服关系的著名结果推广到有限域上的所有代数关系来研究仿射群。异常功能的应用程序包括操作数据进行加密和确保完整性的方法。这将意味着更快，更有效和准确的文件备份；更稳定的软件；以及更安全的数据传输。

MODULAR TOWERS AND THE IGP OVER THE RATIONALS: Often researchers usealgebraic relations in two variables x and y over a field to describesignificant data. Some examples are sets of values x having an (x,y)with entries in the field that satisfy the relation. One of these is theDavenport's problem of determining equivalence classes of polynomialsaccording to their value sets over finite fields. Describing complexmultiplication fields, generated by those x with (x,y) a specialelliptic curve torsion point, is a variant. A relation with a datavariable x produces a permutation group G: The Galois group. Theinvestigator treats these as cases of the Inverse GaloisProblem (IGP). One goal is to decide, from certain listsof relations, if some have the rational numbers as definition field.Serious group theory helped solve Davenport's 1965 problem and Schur'sexceptional polynomial problem (1919; see below). Further, modularcurves reveal complex multiplication to be a version of Schur'sproblem. Having command of all finite simple groups has manyapplications. One is to list those groups attached to exceptionalpolynomials. Knowing many characters of Chevalley groups showed Thompsonand Voelklein where to find series of solutions to the IGP. Yet, it willrequire more than massive group theory to solve the whole IGP. Theinvestigator uses Modular Towers, a modular curve generalization, tofinesse unknowable group classifications. He and his students let aknown group G, and a prime p dividing the order of G, seed the base levelof a Modular Tower. Higher tower levels code relations with mysteriousgroups covering G related to p. Coherent maps between the tower levelsreplace details on these intricate covering groups. Work of Y. Ihara andJ.P. Serre inspire showing that Modular Towers has many properties oftheir special, modular curve, case. For example, high tower levels for aG with no Z/p quotient should have no rational points. For certain groups G, and the prime p=2, this extends Serre's program on spin structures. The investigator applies new spin-like structures to all cases when p is 2 and to all primes p. PRACTICAL CRYPTOGRAPHY AND IGP OVER FINITE FIELDS: Modern cryptosystemsaim to secure electronic transfers of data. Exceptional functions act aspermutations on infinitely many finite fields. A special case of theInverse Galois Problem over finite fields asks how to constructexceptional functions. These scrambling functions look simple. Users caneasily apply them (up to 1993, all came from the 19th century). Still,finding them has been difficult. Fried (UCI), Guralnick (USC) and Saxl(Cambridge) nearly classify them. This solves old problems (Dickson 1896and Carlitz 1965). They also produce unexpected new examples. Their bestdiscovery is that exceptional polyomial Galois groups (excluding nowwell-understood cases) have the affine geometric property. This isuseful for it gives the degrees of exceptional polynomials. Yet, it isalso a difficult challenge: There can be no final description of allaffine groups. The investigator with A. Mezard approaches affine groupsby generalizing to all algebraic relations over finite fieldsGrothendieck's famous results for tame relations. Applications forexceptional functions alone include ways to manipulate data forencryption and to assure integrity. This will mean faster, moreefficient and accurate file back-ups; more stable software; and moresecure data transfers.