GEOMETRIC WEIGHT-SHIFTING OPERATORS ON HILBERT MODULAR FORMS IN CHARACTERISTIC p

We carry out a thorough study of weight-shifting operators on Hilbert modular forms in characteristic p, generalising the author’s prior work with Sasaki to the case where p is ramified in the totally real field. In particular, we use the partial Hasse invariants and Kodaira–Spencer filtrations defined by Reduzzi and Xiao to improve on Andreatta and Goren’s construction of partial $\Theta $ -operators, obtaining ones whose effect on weights is optimal from the point of view of geometric Serre weight conjectures. Furthermore, we describe the kernels of partial $\Theta $ -operators in terms of images of geometrically constructed partial Frobenius operators. Finally, we apply our results to prove a partial positivity result for minimal weights of mod p Hilbert modular forms.

Emergence of a linear tracer source from air concentration measurements

The measurement of atmospheric concentrations by a monitoring network is potentially a useful tool investigated for the identification of the widespread sources of trace species. The paper addresses inversion strategies using base functions multiplicatively deduced from the concentrations adjoint to the measurements. This follows the concept of illumination: the various regions are well, poorly or not seen at all. With ordinary adjoint functions the illumination becomes excessive close to the detectors. This may be corrected by attributing each point of the space time domain a geometric weight. After stating the statistical implications, the choice of this renormalising function is shown to be constrained by an unambiguous formal criterion. The expectable values of the measurements are distinguished from the measurement errors. The entropic nature of the criterion is explained; it amounts to evenly distribute the information between the points organised with their weights as a "known domain" describing the influence of the measurements. Finally the criterion is shown to coincide with the definition of the pure states of quantum statistics, and this interpretation is investigated to describe the influence of the measurement uncertainties on the quality of a source estimate. The theory is illustrated by calculations performed with the experimental source ETEX1.