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

模块塔和IGP在理性上：研究人员经常使用两个变量x和y在一个字段中的代数关系来描述重要数据。一些示例是具有（x，y）的值x的集合，其中字段中的条目满足关系。其中之一是达文波特问题的确定等价类的多项式根据其价值集有限域。描述复乘域是一种变体，它是由具有特殊椭圆曲线扭点的复乘域x生成的。与一个数据变量x的关系产生一个置换群G：伽罗瓦群。研究者把这些作为逆伽罗瓦问题（IGP）的情况下。一个目标是决定，从某些listsof关系，如果一些有理数作为definitionfield.Serious群论帮助解决达文波特的1965年问题和舒尔的例外多项式问题（1919年;见下文）。此外，模曲线揭示了复数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。一个是列出那些附加到exceptionalpolynomial的群。知道Chevalley群的许多性质，就能给Stephonson和Voelklein指明在哪里找到IGP的一系列解。然而，要解决整个IGP问题，需要的不仅仅是大量群论。研究者使用模块化塔，一个模块化的曲线概括，以精细化不可知的群体分类。他和他的学生们让一个已知的群G和一个素数p来划分G的阶数，作为模块塔的底层。更高的塔层代码与覆盖G相关的p的复杂群的关系。塔层之间的连贯映射取代了这些复杂覆盖群的细节。Y的工作。Ihara和J. P. 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的高塔水平应该没有有理点。对于某些群G和素数p=2，这扩展了塞尔关于自旋结构的程序。调查人员适用于新的自旋结构的所有情况下，当p是2和所有素数p.实用的密码学和IGP有限领域：现代cryptosystemsaim安全的电子数据传输。例外函数作用于无穷多个有限域上。有限域上伽罗瓦逆问题的一个特殊情况是如何构造概念函数。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，都来自19世纪世纪）。然而，找到他们仍然很困难。Fried（UCI）、Guralnick（USC）和Saxl（剑桥）几乎将它们分类。这解决了老问题（Dickson 1896和Carlitz 1965）。它们还产生了意想不到的新例子。他们最好的发现是特殊的多项式伽罗瓦群（不包括现在已经很好理解的情况）具有仿射几何性质。这是有用的，因为它给出了例外多项式的次数。然而，这也是一个困难的挑战：不可能有对allaffine组的最终描述。调查人员与A。Mezard方法仿射群一般化到所有代数关系有限fieldsGrothendieck的著名结果驯服的关系。仅例外功能的应用就包括操纵数据、加密和确保完整性的方法。这将意味着更快、更有效和更准确的文件备份;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.