Time-Harmonic Solutions for Maxwell’s Equations in Anisotropic Media and Bochner–Riesz Estimates with Negative Index for Non-elliptic Surfaces

We solve time-harmonic Maxwell’s equations in anisotropic, spatially homogeneous media in intersections of $$L^p$$ L p -spaces. The material laws are time-independent. The analysis requires Fourier restriction–extension estimates for perturbations of Fresnel’s wave surface. This surface can be decomposed into finitely many components of the following three types: smooth surfaces with non-vanishing Gaussian curvature, smooth surfaces with Gaussian curvature vanishing along one-dimensional submanifolds but without flat points, and surfaces with conical singularities. Our estimates are based on new Bochner–Riesz estimates with negative index for non-elliptic surfaces.

$$\pi _1$$ of Miranda moduli spaces of elliptic surfaces

We give finite presentations for the fundamental group of moduli spaces due to Miranda of smooth Weierstrass curves over $${\mathbf {P}}^1$$ P 1 which extend the classical result for elliptic curves to the relative situation over the projective line. We thus get natural generalisations of $$SL_2{{\mathbb {Z}}}$$ S L 2 Z presented in terms of $$\Bigg (\begin{array}{ll} 1&{}1\\ 0&{}1\end{array} \Bigg )$$ ( 1 1 0 1 ) , $$\Bigg (\begin{array}{ll} 1&{}0\\ {-1}&{}1\end{array} \Bigg )$$ ( 1 0 - 1 1 ) on one hand and the first examples of fundamental groups of moduli stacks of elliptic surfaces on the other.Our approach exploits the natural $${\mathbb {Z}}_2$$ Z 2 -action on Weierstrass curves and the identification of $${\mathbb {Z}}_2$$ Z 2 -fixed loci with smooth hypersurfaces in an appropriate linear system on a projective line bundle over $${{\mathbf {P}}}^1$$ P 1 . The fundamental group of the corresponding discriminant complement can be presented in terms of finitely many generators and relations using methods in the Zariski tradition.