Southern California Number Theory Day at UCI, 10/14/2006

Abstracts

David Solomon

L-functions at s = 1 and reciprocity laws for Stark units

Let k be a totally real number field of degree r, S an appropriate finite set of places and L a finite abelian extension of k with group G.

If L is also totally real, then, roughly speaking, Stark's conjecture predicts that the r-th derivatives of the L-functions L_S(s,\chi) at s = 0 for all characters \chi of G are determined by an equivariant r x r regulator of certain S-units of L. These 'Stark Units' are cyclotomic units if L/Q is abelian but are known to exist in very few other nontrivial cases.

Now suppose that L is of CM type. The values of L_S(s,\chi) at s = 0 define an ideal in the 'minus part' ZG^- of ZG which generalises the Stickelberger ideal. Using instead the values at s = 1 for odd characters, I construct a p-adic map from the r-th exterior power of the p-semilocal units of L into Q_pG^- (p an odd prime). I conjecture that the image of this map lies in Z_pG. If p does not divide |G| it does so, and is closely related to the generalised Stickelberger ideal.

When L contains pⁿ-th roots of 1, a refined conjecture ties minus to plus parts by asserting that the map is congruent modulo pⁿ to one defined using the Hilbert symbols of the above mentioned Stark units attached to the maximal real subfield of L.

Kannan Soundararajan

Many solutions to the S-unit equation a+b = c

I will discuss solutions to a+b = c where a, b, c are composed entirely of primes from a given finite set S. In joint work with Konyagin, we show the existence of special sets S for which this equation has many solutions. Moreover we show that the equation a+1=c also has many solutions for some sets S. I will also discuss joint work with Lagarias where we show, assuming the Generalized Riemann Hypothesis, that there are many solutions to a+b = c when S is the set of the first s primes.