MODULAR TOWERS AND THE IGP OVER THE RATIONALS: Often researchers use algebraic relations in two variables x and y over a field to describe significant 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 the Davenport's problem of determining equivalence classes of polynomials according to their value sets over finite fields. Describing complex multiplication fields, generated by those x with (x,y) a special elliptic curve torsion point, is a variant. A relation with a data variable x produces a permutation group G: The Galois group. The investigator treats these as cases of the Inverse Galois Problem (IGP). One goal is to decide, from certain lists of relations, if some have the rational numbers as definition field. Serious group theory helped solve Davenport's 1965 problem and Schur's exceptional polynomial problem (1919; see below). Further, modular curves reveal complex multiplication to be a version of Schur's problem. Having command of all finite simple groups has many applications. One is to list those groups attached to exceptional polynomials. Knowing many characters of Chevalley groups showed Thompson and Voelklein where to find series of solutions to the IGP. Yet, it will require more than massive group theory to solve the whole IGP. The investigator uses Modular Towers, a modular curve generalization, to finesse unknowable group classifications. He and his students let a known group G, and a prime p dividing the order of G, seed the base level of a Modular Tower. Higher tower levels code relations with mysterious groups covering G related to p. Coherent maps between the tower levels replace details on these intricate covering groups. Work of Y. Ihara and J.P. Serre inspire showing that Modular Towers has many properties of their special, modular curve, case. For example, high tower levels for a G 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 cryptosystems aim to secure electronic transfers of data. Exceptional functions act as permutations on infinitely many finite fields. A special case of the Inverse Galois Problem over finite fields asks how to construct exceptional functions. These scrambling functions look simple. Users can easily 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 1896 and Carlitz 1965). They also produce unexpected new examples. Their best discovery is that exceptional polyomial Galois groups (excluding now well-understood cases) have the affine geometric property. This is useful for it gives the degrees of exceptional polynomials. Yet, it is also a difficult challenge: There can be no final description of all affine groups. The investigator with A. Mezard approaches affine groups by generalizing to all algebraic relations over finite fields Grothendieck's famous results for tame relations. Applications for exceptional functions alone include ways to manipulate data for encryption and to assure integrity. This will mean faster, more efficient and accurate file back-ups; more stable software; and more secure data transfers.
