Special Frobenius Traces in Galois Representations

This thesis studies Frobenius traces in Galois representations from two different directions. In the first problem we explore how often they vanish in Artin-type representations. We give an upper bound for the density of the set of vanishing Frobenius traces in terms of the multiplicities of the irreducible components of the adjoint representation. Towards that, we construct an infinite family of representations of finite groups with an irreducible adjoint action.

In the second problem we partially extend for Hilbert modular forms a result of Coleman and Edixhoven that the Hecke eigenvalues a_p of classical elliptical modular newforms f of weight 2 are never extremal, i.e., a_p is strictly less than 2[square root]p. The generalization currently applies only to prime ideals p of degree one, though we expect it to hold for p of any odd degree. However, an even degree prime can be extremal for f. We prove our result in each of the following instances: when one can move to a Shimura curve defined by a quaternion algebra, when f is a CM form, when the crystalline Frobenius is semi-simple, and when the strong Tate conjecture holds for a product of two Hilbert modular surfaces (or quaternionic Shimura surfaces) over a finite field.

Studies of temperature sensitive mutants of polyoma virus

Polyoma virus can undergo two different types of interactions with susceptible cells; one type of interaction leads to the production of new infectious virus and eventual cell death while the other leads to a neoplastically transformed cell which is able to continue to divide under conditions that inhibit the multiplication of uninfected normal cells. In order to study the viral genes involved in both of these virus-cell interactions the isolation of temperature sensitive mutants of polyoma virus was undertaken.

Two strains (TS-a, TS-b) which were temperature sensitive in their plaque forming ability at 38.5˚C, but not at 31.5˚C, were isolated from a mutagenized stock of the polyoma wild type virus (PY). TS-a was studied in further detail.

TS-a grown at 31.5˚C was found to be indistinguishable from PY in a number of physical characteristics including the heat sensitivity of the completed viral components. TS-a was inhibited in its ability to produce infectious virus in mouse cells when incubated at 38.5˚C; this inhibition could be overcome by infection with high multiplicities.

The nature of the intracellular temperature sensitive step of TS-a was analysed to some degree. It was found that this step occurs after uncoating of the infecting virus particles and about the time of new viral DNA synthesis. New infectious viral DNA does not appear to be made at the nonpermissive temperature; in contrast noninfectious capsids are made at 38.5˚C, but in amounts smaller than a full yield, such as made by TS-a at 31.5˚C or by PY at both the high and low temperature.

TS-a has also been found to be temperature sensitive in its transforming ability in vitro. Cells transformed at 31.5˚C by TS-a retain their transformed characteristics upon cultivation at 38.5˚C. Thus the temperature sensitive function seems to be important for the initiation of transformation, but not essential for the maintenance of the transformed state. TS-a also appears to be temperature sensitive in the production of tumors in newborn hamsters.