Month: June 2020

Two recent research theses from my department

~ David Roberts ~ Leave a comment

One is in complex analytic geometry, and the other is in number theory, with a dash of differential geometry. I had the pleasure of seeing both Haripriya and Ben develop from their first forays into research to producing two excellent theses, both of which won a commendation.

Abstract: Let M be an open Riemann surface. A recent result due to Forstnerič and Lárusson [8] says that, for a closed conical subvariety A \subset \mathbb{C}^n such that A \setminus \{0\} is an Oka manifold, the weak homotopy type of the space of non-degenerate holomorphic A-immersions of M into \mathbb{C}^n is the same as that of the space of holomorphic (or equivalently, continuous) maps from M into A\setminus \{0\}. In their paper, the authors sketch the proof of this theorem through claiming analogy with a related result, and invoking advanced results from complex and differential geometry, including seminal theorems from Oka theory. The work contained in this thesis was motivated by the absence of a self-contained proof for the special case where A = \mathbb{C} – as, perhaps, the first geometrically interesting case that one would consider. We remedy the absence by providing a fully detailed, self-contained proof of this case; namely, the parametric h-principle for holomorphic immersions of open Riemann surfaces into \mathbb{C}. We outline this more precisely as follows. Take a holomorphic 1-form \theta on M which vanishes nowhere. We denote by \mathcal{I}(M, \mathbb{C}) the space of holomorphic immersions of M into \mathbb{C}, and denote by \mathcal{O}(M, \mathbb{C}^*) the space of nonvanishing holomorphic functions on M. We prove, in all detail, that the continuous map

\mathcal{I}(M, \mathbb{C}) \to \mathcal{O}(M, \mathbb{C}^*), f \mapsto df/\theta,

is a weak homotopy equivalence. This gives a full description of the weak homotopy type of \mathcal{I}(M, \mathbb{C}), as the target space \mathcal{O}(M, \mathbb{C}^*) is known by algebraic topology (Remark 5.2.3).

Abstract: We present some results related to the areas of theta functions, modular forms, Gauss sums and reciprocity. After a review of background material, we recount the elementary theory of modular forms on congruence subgroups and provide a proof of the transformation law for Jacobi’s theta function using special values of zeta functions. We present a new proof, obtained during work with Michael Eastwood, of Jacobi’s theorem that every integer is a sum of four squares. Our proof is based on theta functions but emphasises the geometry of the thrice-punctured sphere.

Next, we detail some investigations into quadratic Gauss sums. We include a new proof of the Landsberg–Schaar relation by elementary methods, together with a second based on evaluations of Gauss sums. We give elementary proofs of generalised and twisted Landsberg–Schaar relations, and use these results to answer a research problem posed by Berndt, Evans and Williams. We conclude by proving some sextic and octic local analogues of the Landsberg–Schaar relation.

Finally, we give yet another proof of the Landsberg–Schaar relation based on the relationship between Mellin transforms and asymptotic expansions. This proof makes clear the relationship between the Landsberg–Schaar relation and the existence of a metaplectic Eisenstein series with certain properties. We note that one may promote this correspondence to the setting of number fields, and furthermore, that the higher theta functions constructed by Banks, Bump and Lieman are ideal candidates for future investigations of such correspondences.