How to implement a modular form

AbstractWe present a model for Fourier expansions of arbitrary modular forms. This model takes precisions and symmetries of such Fourier expansions into account. The value of this approach is illustrated by studying a series of examples. An implementation of these ideas is provided by the author. We discuss the technical background of this implementation, and we explain how to implement arbitrary Fourier expansions and modular forms. The framework allows us to focus on the considerations of a mathematical nature during this procedure. We conclude with a list of currently available implementations and a discussion of possible computational research

A generalization of manifolds with corners

In conventional Differential Geometry one studies manifolds, locally modelled on ${\mathbb R}^n$, manifolds with boundary, locally modelled on $[0,\infty)\times{\mathbb R}^{n-1}$, and manifolds with corners, locally modelled on $[0,\infty)^k\times{\mathbb R}^{n-k}$. They form categories ${\bf Man}\subset{\bf Man^b}\subset{\bf Man^c}$. Manifolds with corners $X$ have boundaries $\partial X$, also manifolds with corners, with $\mathop{\rm dim}\partial X=\mathop{\rm dim} X-1$. We introduce a new notion of 'manifolds with generalized corners', or 'manifolds with g-corners', extending manifolds with corners, which form a category $\bf Man^{gc}$ with ${\bf Man}\subset{\bf Man^b}\subset{\bf Man^c}\subset{\bf Man^{gc}}$. Manifolds with g-corners are locally modelled on $X_P=\mathop{\rm Hom}_{\bf Mon}(P,[0,\infty))$ for $P$ a weakly toric monoid, where $X_P\cong[0,\infty)^k\times{\mathbb R}^{n-k}$ for $P={\mathbb N}^k\times{\mathbb Z}^{n-k}$. Most differential geometry of manifolds with corners extends nicely to manifolds with g-corners, including well-behaved boundaries $\partial X$. In some ways manifolds with g-corners have better properties than manifolds with corners; in particular, transverse fibre products in $\bf Man^{gc}$ exist under much weaker conditions than in $\bf Man^c$. This paper was motivated by future applications in symplectic geometry, in which some moduli spaces of $J$-holomorphic curves can be manifolds or Kuranishi spaces with g-corners (see the author arXiv:1409.6908) rather than ordinary corners. Our manifolds with g-corners are related to the 'interior binomial varieties' of Kottke and Melrose in arXiv:1107.3320 (see also Kottke arXiv:1509.03874), and to the 'positive log differentiable spaces' of Gillam and Molcho in arXiv:1507.06752.Comment: 97 pages, LaTeX. (v3) final version, to appear in Advances in Mathematic

On a semitopological polycyclic monoid

We study algebraic structure of the $\lambda$-polycyclic monoid $P_{\lambda}$ and its topologizations. We show that the $\lambda$-polycyclic monoid for an infinite cardinal $\lambda\geqslant 2$ has similar algebraic properties so has the polycyclic monoid $P_n$ with finitely many $n\geqslant 2$ generators. In particular we prove that for every infinite cardinal $\lambda$ the polycyclic monoid $P_{\lambda}$ is a congruence-free combinatorial $0$-bisimple $0$-$E$-unitary inverse semigroup. Also we show that every non-zero element $x$ is an isolated point in $(P_{\lambda},\tau)$ for every Hausdorff topology $\tau$ on $P_{\lambda}$, such that $(P_{\lambda},\tau)$ is a semitopological semigroup, and every locally compact Hausdorff semigroup topology on $P_\lambda$ is discrete. The last statement extends results of the paper [33] obtaining for topological inverse graph semigroups. We describe all feebly compact topologies $\tau$ on $P_{\lambda}$ such that $\left(P_{\lambda},\tau\right)$ is a semitopological semigroup and its Bohr compactification as a topological semigroup. We prove that for every cardinal $\lambda\geqslant 2$ any continuous homomorphism from a topological semigroup $P_\lambda$ into an arbitrary countably compact topological semigroup is annihilating and there exists no a Hausdorff feebly compact topological semigroup which contains $P_{\lambda}$ as a dense subsemigroup