scholarly journals On the order sequence of an embedding of the Ree curve

2020 ◽  
Vol 20 (2) ◽  
pp. 149-162
Author(s):  
Dane C. Skabelund
Keyword(s):  
Rational Points ◽  
Linear Series ◽  
Weierstrass Points ◽  
Order Sequence

AbstractIn this paper we compute the Weierstrass order-sequence associated with a certain linear series on the Deligne–Lusztig curve of Ree type. As a result, we show that the set of Weierstrass points of this linear series consists entirely of 𝔽q-rational points.

scholarly journals -algebras associated with curves and rational functions on . I

2014 ◽  
Vol 150 (4) ◽  
pp. 621-667 ◽  
Author(s):  
Robert Fisette ◽  
Alexander Polishchuk
Keyword(s):  
Homotopy Class ◽  
Rational Functions ◽  
Coherent Sheaf ◽  
Linear Series ◽  
Projective Curve ◽  
Canonical Embedding ◽  
Rational Map ◽  
Projective Embedding ◽  
Smooth Projective Curve ◽  
Weierstrass Points

AbstractWe consider the natural$A_{\infty }$-structure on the$\mathrm{Ext}$-algebra$\mathrm{Ext}^*(G,G)$associated with the coherent sheaf$G=\mathcal{O}_C\oplus \mathcal{O}_{p_1}\oplus \cdots \oplus \mathcal{O}_{p_n}$on a smooth projective curve$C$, where$p_1,\ldots,p_n\in C$are distinct points. We study the homotopy class of the product$m_3$. Assuming that$h^0(p_1+\cdots +p_n)=1$, we prove that$m_3$is homotopic to zero if and only if$C$is hyperelliptic and the points$p_i$are Weierstrass points. In the latter case we show that$m_4$is not homotopic to zero, provided the genus of$C$is greater than$1$. In the case$n=g$we prove that the$A_{\infty }$-structure is determined uniquely (up to homotopy) by the products$m_i$with$i\le 6$. Also, in this case we study the rational map$\mathcal{M}_{g,g}\to \mathbb{A}^{g^2-2g}$associated with the homotopy class of$m_3$. We prove that for$g\ge 6$it is birational onto its image, while for$g\le 5$it is dominant. We also give an interpretation of this map in terms of tangents to$C$in the canonical embedding and in the projective embedding given by the linear series$|2(p_1+\cdots +p_g)|$.

scholarly journals Counting and equidistribution in quaternionic Heisenberg groups

2021 ◽  
pp. 1-38
Author(s):  
JOUNI PARKKONEN ◽  
FRÉDÉRIC PAULIN
Keyword(s):  
Hyperbolic Geometry ◽  
Quaternion Algebra ◽  
Rational Points ◽  
Hyperbolic Spaces ◽  
Heisenberg Groups ◽  
Arithmetic Groups ◽  
Hermitian Forms ◽  
Dimension 2 ◽  
Counting Formula ◽  
The Relationship

Abstract We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension 2. We prove a Mertens counting formula for the rational points over a definite quaternion algebra A over ${\mathbb{Q}}$ in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over A in quaternionic Heisenberg groups.

scholarly journals Arithmetic of higher-dimensional orbifolds and a mixed Waring problem

2021 ◽  
Author(s):  
Tim Browning ◽  
Shuntaro Yamagishi
Keyword(s):  
Circle Method ◽  
Rational Points ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Waring’S Problem ◽  
Waring Problem ◽  
Waring's Problem ◽  
Main Conjecture ◽  
Higher Dimensional ◽  
Thin Set

AbstractWe study the density of rational points on a higher-dimensional orbifold $$(\mathbb {P}^{n-1},\Delta )$$ ( P n - 1 , Δ ) when $$\Delta $$ Δ is a $$\mathbb {Q}$$ Q -divisor involving hyperplanes. This allows us to address a question of Tanimoto about whether the set of rational points on such an orbifold constitutes a thin set. Our approach relies on the Hardy–Littlewood circle method to first study an asymptotic version of Waring’s problem for mixed powers. In doing so we make crucial use of the recent resolution of the main conjecture in Vinogradov’s mean value theorem, due to Bourgain–Demeter–Guth and Wooley.

Allozyme Frequencies in a Linear Series of Song Dialect Populations

Evolution ◽  
1982 ◽  
Vol 36 (5) ◽  
pp. 1020 ◽  
Author(s):  
Myron Charles Baker ◽  
Daniel B. Thompson ◽  
Gregory L. Sherman ◽  
Michael A. Cunningham ◽  
Diana F. Tomback
Keyword(s):  
Linear Series ◽  
Song Dialect

scholarly journals Le complémentaire des puissances -ièmes dans un corps de nombres est un ensemble diophantien

2015 ◽  
Vol 151 (10) ◽  
pp. 1965-1980 ◽  
Author(s):  
Jean-Louis Colliot-Thélène ◽  
Jan Van Geel
Keyword(s):  
Prime Number ◽  
Rational Points ◽  
Parameter Family ◽  
Symmetric Power ◽  
Symmetric Powers

For $n=2$ the statement in the title is a theorem of B. Poonen (2009). He uses a one-parameter family of varieties together with a theorem of Coray, Sansuc and one of the authors (1980), on the Brauer–Manin obstruction for rational points on these varieties. For $n=p$, $p$ any prime number, A. Várilly-Alvarado and B. Viray (2012) considered analogous families of varieties. Replacing this family by its $(2p+1)$th symmetric power, we prove the statement in the title using a theorem on the Brauer–Manin obstruction for rational points on such symmetric powers. The latter theorem is based on work of one of the authors with Swinnerton-Dyer (1994) and with Skorobogatov and Swinnerton-Dyer (1998), work generalising results of Salberger (1988).

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