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

模数塔和有理数上的IGP：研究人员经常在一个域上使用两个变量x和y的代数关系来描述重要数据。一些例子是具有(x，y)的值集x，其字段中的条目满足该关系。其中之一是根据有限域上多项式的值集确定多项式的等价类的Davenport问题。描述由具有(x，y)特殊椭圆曲线扭点的那些x生成的复乘法域是一种变体。与数据变量x的关系产生置换群G：伽罗华群。研究者将其视为逆Galois问题(IGP)。一个目标是从某些关系列表中确定是否有些关系具有有理数作为定义域。严肃的群论帮助解决了1965年的达文波特问题和舒尔的例外多项式问题(1919年；见下文)。此外，模数曲线显示复数乘法是舒尔问题的一种形式。掌握所有有限单群有很多应用。一种是列出那些附加到例外多项式的群。知道了切瓦利小组的许多特征，汤普森和沃克莱因知道了在哪里可以找到IGP的一系列解决方案。然而，要解决整个IGP问题，需要的不仅仅是大规模群论。研究者使用模数塔，一种模数曲线的泛化，来改进未知组的分类。他和他的学生让一个已知的群G和一个除G阶的素数p作为模塔基的种子。更高的塔层编码了与P有关的覆盖G的神秘群之间的关系。塔层之间的相干映射替换了这些复杂覆盖群的细节。Y.Ihara和J.P.Serre Inspire的工作表明，模数塔具有其特殊的模数曲线的许多性质。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，素数p=2，这扩展了Serre关于自旋结构的程序。研究人员将新的类自旋结构应用于p为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，以确保数据的电子传输。例外函数在无穷多个有限域上起无序作用。有限域上的伽罗华逆问题的一个特例是如何构造例外函数。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)。尽管如此，找到他们一直很困难。弗里德(UCI)、南加州大学古拉尼克(Guralnick)和剑桥大学(Cambridge)的萨克塞尔(Saxl)几乎将它们归类。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。他们最好的发现是例外的多项式伽罗华群(不包括现在已知的情形)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，这也是一项艰巨的挑战：对于阿拉夫群体，不可能有最终的描述。A.Mezard的研究人员通过将有限域上的所有代数关系推广到有限域上的所有代数关系来逼近仿射群，Grothendieck关于驯服关系的著名结果。仅例外函数的应用程序就包括处理数据以进行加密和确保完整性的方法。这将意味着更快、更高效、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.