Probability, Geometry and Integrable Systems MSRI Publications

ABSTRACT. We study the function field of a principally polarized abelian va-riety from the point of view of differential algebra. We implement in a concrete case the following result of I. Barsotti, which he derived from what he called the prostapheresis formula and showed to characterize theta functions: the log-arithmic derivatives of the theta function along one line generate the function field. We outline three interpretations of the differential algebra of theta func-tions in the study of commutative rings of partial differential operators. Henry McKean was one of the earliest contributors to the field of “integrable PDEs”, whose origin for simplicity we shall place in the late 1960s. One way in which Henry conveyed the stunning and powerful discovery of a linearizing change of variables was by choosing Isaiah 40:3-4 as an epigram for [McKean 1979]: The voice of him that crieth in the wilderness, Prepare ye the way of the Lord, make straight in the desert a highway for our God. Every valley shall be exalted and every mountain and hill shall be made low: and the crooked shall be made straight and the rough places plain. Thus, on this contribution to a volume intended to celebrate Henry’s many fundamental achievements on the occasion of his birthday, my title. I use the word line in the extended sense of “linear flow”, of course, since no projective line can be contained in an abelian variety — the actual line resides in the universal cover. This article is concerned primarily with classical theta functions, with an ap-pendix to report on a daring extension of the concept to infinite-dimensional tori, also initiated by Henry. Thirty years (or forty, if you regard the earliest experi-ments by E. Fermi, J. Pasta and S. Ulam, then M. D. Kruskal and N. J. Zabusky, as more than an inspiration in the discovery of solitons; see [Previato 2008] for references) after the ground was broken in this new field, in my view one of the main remaining questions in the area of theta functions as related to PDEs, is still that of straight lines, both on abelian varieties and on Grassmann manifolds (the two objects of greatest interest to geometers in the nineteenth century!). O