L-Functions

This chapter turns to L-functions. It first covers motivic and cohomological L-functions. There is a well-known conjectural dictionary between cohomological cuspidal automorphic representations of GLn and pure rank n motives. The chapter briefly reviews this dictionary while recasting it in the context of strongly inner Hecke summands on the one hand and pure effective motives on the other. Afterward, the critical points for L-functions and the combinatorial lemma are explored. In particular, the chapter reviews the Rankin–Selberg L-functions. A proof of combinatorial lemma is also given. The chapter then provides the main result on special values of L-functions. It concludes with some remarks.

Generalised Mycielski Graphs and the Borsuk–Ulam Theorem

Stiebitz determined the chromatic number of generalised Mycielski graphs using the topological method of Lovász, which invokes the Borsuk–Ulam theorem. Van Ngoc and Tuza used elementary combinatorial arguments to prove Stiebitz's theorem for 4-chromatic generalised Mycielski graphs, and asked if there is also an elementary combinatorial proof for higher chromatic number. We answer their question by showing that Stiebitz's theorem can be deduced from a version of Fan's combinatorial lemma. Our proof uses topological terminology, but is otherwise completely discrete and could be rewritten to avoid topology altogether. However, doing so would be somewhat artificial, because we also show that Stiebitz's theorem is equivalent to the Borsuk–Ulam theorem.

Nonemptiness of Brill–Noether loci in M(2,L)

Let [Formula: see text] be a smooth projective complex curve of genus [Formula: see text]. We investigate the Brill–Noether locus consisting of stable bundles of rank 2 and determinant [Formula: see text] of odd degree [Formula: see text] having at least [Formula: see text] independent sections. This locus possesses a virtual fundamental class. We show that in many cases this class is nonzero, which implies that the Brill–Noether locus is nonempty. For many values of [Formula: see text] and [Formula: see text] the result is best possible. We obtain more precise results for [Formula: see text]. Appendix A contains the proof of a combinatorial lemma which we need.

A Ramsey Theorem with an Application to Sequences in Banach Spaces

The notion of a maximally conditional sequence is introduced for sequences in a Banach space. It is then proved using Ramsey theory that every basic sequence in a Banach space has a subsequence which is either an unconditional basic sequence or a maximally conditional sequence. An apparently novel, purely combinatorial lemma in the spirit of Galvin's theorem is used in the proof. An alternative proof of the dichotomy result for sequences in Banach spaces is also sketched, using the Galvin–Prikry theorem.