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Register a new userGeneralized Weierstrass semigroups and their Poincare series
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We investigate the structure of the generalized Weierstrass semigroups at several points on a curve defined over a finite field. We present a description of these semigroups that enables us to deduce properties concerned with the arithmetical structure of divisors supported on the specified points and their corresponding Riemann-Roch spaces. This characterization allows us to show that the Poincare series associated with generalized Weierstrass semigroups carry essential information to describe entirely their respective semigroups.
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In this work we study the generalized Weierstrass semigroup $widehat{H} (mathbf{P}_m)$ at an $m$-tuple $mathbf{P}_m = (P_{1}, ldots , P_{m})$ of rational points on certain curves admitting a plane model of the form $f(y) = g(x)$ over $mathbb{F}_{q}$, where ${f(T),g(T)in mathbb{F}_q[T]}$. In particular, we compute the generating set $widehat{Gamma}(mathbf{P}_m)$ of $widehat{H} (mathbf{P}_m)$ and, as a consequence, we explicit a basis for Riemann-Roch spaces of divisors with support in ${P_{1}, ldots , P_{m}}$ on these curves, generalizing results of Maharaj, Matthews, and Pirsic.
The arithmetic motivic Poincare series of a variety $V$ defined over a field of characteristic zero, is an invariant of singularities which was introduced by Denef and Loeser by analogy with the Serre-Oesterle series in arithmetic geometry. They proved that this motivic series has a rational form which specializes to the Serre-Oesterle series when $V$ is defined over the integers. This invariant, which is known explicitly for a few classes of singularities, remains quite mysterious. In this paper we study this motivic series when $V$ is an affine toric variety. We obtain a formula for the rational form of this series in terms of the Newton polyhedra of the ideals of sums of combinations associated to the minimal system of generators of the semigroup of the toric variety. In particular, we deduce explicitly a finite set of candidate poles for this invariant.
We define and study non-abelian Poincare series for the group $G=mathrm{SU} (2,1)$, i.e. Poincare series attached to a Stone-Von Neumann representation of the unipotent subgroup $N$ of $G$. Such Poincare series have in general exponential growth. In this study we use results on abelian and non-abelian Fourier term modules obtained in arXiv:1912.01334. We compute the inner product of truncations of these series and those associated to unitary characters of $N$ with square integrable automorphic forms, in connection with their Fourier expansions. As a consequence, we obtain general completeness results that, in particular, generalize those valid for the classical holomorphic (and antiholomorphic) Poincare series for $mathrm{SL}(2,mathbb{R})$.
We define the concept of Tschirnhaus-Weierstrass curve, named after the Weierstrass form of an elliptic curve and Tschirnhaus transformations. Every pointed curve has a Tschirnhaus-Weierstrass form, and this representation is unique up to a scaling of variables. This is useful for computing isomorphisms between curves.
We consider the problem of determining Weierstrass gaps and pure Weierstrass gaps at several points. Using the notion of relative maximality in generalized Weierstrass semigroups due to Delgado cite{D}, we present a description of these elements which generalizes the approach of Homma and Kim cite{HK} given for pairs. Through this description, we study the gaps and pure gaps at several points on a certain family of curves with separated variables.
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