How many ways can an integer n be expressed as a sum of positive integers? This question is the cornerstone of Partition Theory, and it is surprisingly difficult to answer. For example, if we let p(n) be the number of these expressions of n, even the parity of p(n) remains something of a mystery, despite the fact that it has been studied for over a century. In particular, although empirical evidence (the first several million values) seems to indicate that Po(N) = [the number of odd values of p(n) up to N] is asymptotic to N/2, no one has even been able to show that Po(N) is larger than the square root of N for N sufficiently large. Many advances in discovering the mod 2 behavior of p(n) have been made over the past several years, and most of them have required properties of l-adic Galois representations and the theory of modular forms. However, one lower bound for Po(N) (which was the state-of-the-art for a brief period) was proven using only elementary generating function techniques and results from classical analytic number theory. In this talk, we develop the history of the mystery, and we prove the latter lower bound. The talk will be aimed at the partition theoretically uninitiated, and a great deal of background will be provided.

Let Fq be the finite field and : XY an Fq cover of normal varieties. We call exceptional if it maps 1-1 on Fqt points for an infinity of t. We say over Q is exceptional if it is exceptional mod infinitely many p. When X=Y, and is over Q, we have a map: exceptional p period of mod p. RSA cryptography uses x xk (k odd) and Euler's Theorem gives us its periods.

We give a paragraph of history: Schur (1921) posed a list of all Q exceptional polynomials. This inspired Davenport and Lewis (1961) to propose that a geometric property C D-L C would imply a polynomial is exceptional. Both were right (1969). Serre's O(pen) I(mage) T(heorem) produces most remaining exceptional Q rational functions (1977).

We use the D-L generalization to show exceptional covers (of Y over Fq) form a category with fiber products: the (Y,Fq) exceptional tower. Using that we can generate subtowers that connect the tower to two famous results.

I. Denef-Loeser-Nicaise motives: They attach a "motivic Poincare series" to any problem over Q. A generalization of exceptional covers produces (we say Weil) relations among Poincare series over (Y,Fq). The easiest converse question is this: If the zeta functions of X and P1 have a special Weil relation, is X an exceptional cover?

II. Serre's O(pen) I(mage) T(heorem): Rational functions from the OIT generate two (P1,Fq) exceptional subtower. The C(omplex) M(ultiplication) part of the OIT produces exceptional covers. We see their periods from the CM analog of Euler's Theorem. Periods of the subtower from the G(eneral) L(inear) part of the OIT give our most serious challenge.

The conjectures in the title were formulated in the late 1970's as vast generalizations of the classical theorem of Stickelberger. They make a subtle connection between the Z[G(L/k)]-module structure of the Quillen K-groups K*(OL) in an abelian extension L/k of number fields and the values at negative integers of the associated G(L/k)-equivariant L-functions.

These conjectures are known to hold true if the base field k is Q, due to work of Coates-Sinnott and Kurihara. In this talk, we will provide evidence in support of these conjectures over arbitrary totally real number fields k.

Let alpha be an automorphism of a hyperelliptic curve C of genus g,
and let alpha' be the automorphism of P^1 induced by alpha.
Let n be the order of alpha and let n' be the order of alpha'.
We show that the triple (g,n,n') completely determines the
characteristic polynomial of the automorphism alpha^* of the
Jacobian of C, unless n is even, n=n', and (2g+2)/n is even,
in which case there are two possibilities. We give explicit
formulas for the characteristic polynomial in all cases.

The T-adic L-function is a unversal L-function which
interpolates classical L-functions of all p-power order
exponential sums associated to a polynomial f(x) defined
over a finite field. We study its T-adic analytic properties
(analytic continuation and its T-adic Newton polygon).
The T-adic Newton polygon provides a uniform improvement
to previous results on p-adic Newton polygon of exponential sums
in the non-ordinary case. This is joint work with Chunlei Liu.