Trace functions, a survey

At last count, my series of works with Étienne Fouvry and Philippe Michel on trace functions and their applications consists of seven research papers or preprints, amounting to a bit more than 200 pages. To these are added a number of works-in-progress or partial notes (some with results we did not need or use and so took out of earlier drafts of our papers, some with worked-out examples or remarks, etc). We have a relatively firm project of writing a monograph account of the whole theory and applications, which we view in part as a way of making accessible some of the deep consequences of the Riemann Hypothesis over finite fields of Deligne, but this is clearly a long-term project. In the meantime, we have written a first short survey, the first draft of which can be found on my web page. This is in fact the written (and slightly expanded) account of a Colloquio de Giorgi that I gave at the

Scuola Normale Superiore in Pisa earlier this year, and we have included an unusual representation of the fundamental domain of the modular group acting on the upper half-plane as an homage and acknowledgement of this occasion.