arXiv subjects

Guillaume Tahar

Publications and source records attributed to Guillaume Tahar.

41 records · Page 3Linked to original sources

arXiv2019

Chamber structure of modular curves $X_{1}(N)$

Modular curves $X_{1}(N)$ parametrize elliptic curves with a point of order $N$. They can be identified with connected components of projectivized strata $\mathbb{P}\mathcal{H}(a,-a)$ of meromorphic differentials. As strata of meromorphic differentials, they have a canonical walls-and-chambers structure defined by the topological changes in the flat structure defined by the meromorphic differentials. We provide formulas for the number of chambers and an effective means for drawing the incidence graph of the chamber structure of any modular curve $X_{1}(N)$. This defines a family of graphs with specific combinatorial properties. This approach provides a geometrico-combinatorial computation of the genus and the number of punctures of modular curves $X_{1}(N)$. Although the dimension of a stratum of meromorphic differentials depends only on the genus and the numbers of the singularities, the topological complexity of the stratum crucially depends on the order of the singularities.

math.AG↗

arXiv2017

Veech groups of flat surfaces with poles

Flat surfaces that correspond to meromorphic $1$-forms or to meromorphic quadratic differentials containing poles of order two and higher are surfaces of infinite area. We classify groups that appear as Veech groups of translation surfaces with poles. We characterize those surfaces such that their $GL^{+}(2,\mathbb{R})$-orbit or their $SL(2,\mathbb{R})$-orbit is closed. Finally, we provide a way to determine the Veech group for a typical infinite surface in a given chamber of a stratum.

math.GT↗

arXiv2017

Counting saddle connections in flat surfaces with poles of higher order

Flat surfaces that correspond to $k$-differentials on compact Riemann surfaces are of finite area provided there is no pole of order $k$ or higher. We denote by \textit{flat surfaces with poles of higher order} those surfaces with flat structures defined by a $k$-differential with at least one pole of order at least $k$. Flat surfaces with poles of higher order have different geometrical and dynamical properties than usual flat surfaces of finite area. In particular, they can have a finite number of saddle connections. We give lower and upper bounds for the number of saddle connections and related quantities. In the case $k=1\ or\ 2$, we provide a combinatorial characterization of the strata for which there can be an infinite number of saddle connections.

math.GT↗

arXiv2017

Ordered algebraic structures and classification of semifields

Semifields are semirings in which every nonzero element has a multiplicative inverse. A rough classification uses the characteristic of the semifield, that is the isomorphism type of the semifield generated by the two neutral elements. For every characteristic, we provide a structure theorem that reduces the classification of semifields to the classification of better-known algebraic structures. Every semifield of characteristic $p$ is actually a field. There is an equivalence between semifields of characteristic one and lattice-ordered groups. Strict semifields of characteristic zero are quotients of cancellative semifields and there is an equivalence between concellative strict semifields and a particular class of partially ordered rings.

math.AG↗

arXiv2016

On the number of saddle connections in translation surfaces with poles

Translation surfaces with poles correspond to meromorphic differentials on compact Riemann surfaces. They appear in compactifications of strata of the moduli space of Abelian differentials and in the study of stability conditions. Such structures have different geometrical and dynamical properties than usual translation surfaces. In particular, they always have infinite area and can have a finite number of saddle connections. We provide a combinatorial characterization of the strata for which there can be an infinite number of saddle connections and give lower and upper bounds for the number of saddle connections and related quantities.

math.GT↗