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

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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.