scholarly journals On the symmetric determinantal representations of the Fermat curves of prime degree

2016 ◽  
Vol 12 (04) ◽  
pp. 955-967 ◽  
Author(s):  
Yasuhiro Ishitsuka ◽  
Tetsushi Ito
Keyword(s):  
Plane Curves ◽  
Symmetric Matrices ◽  
Torsion Points ◽  
Prime Degree ◽  
Linear Forms ◽  
Jacobian Varieties ◽  
Smooth Plane ◽  
Fermat Curves ◽  
Theta Characteristics

We prove that the defining equations of the Fermat curves of prime degree cannot be written as the determinant of symmetric matrices with entries in linear forms in three variables with rational coefficients. In the proof, we use a relation between symmetric matrices with entries in linear forms and non-effective theta characteristics on smooth plane curves. We also use some results of Gross–Rohrlich on the rational torsion points on the Jacobian varieties of the Fermat curves of prime degree.

scholarly journals Nontrivial algebraic cycles in the Jacobian varieties of some quotients of Fermat curves

2016 ◽  
Vol 27 (03) ◽  
pp. 1650027 ◽  
Author(s):  
Yuuki Tadokoro
Keyword(s):  
Algebraic Cycles ◽  
Trace Map ◽  
Jacobian Varieties ◽  
Fermat Curves

We obtain the trace map images of the values of certain harmonic volumes for some cyclic quotients of [Formula: see text]th Fermat curves. These provide the algorithm showing that the algebraic cycles, called [Formula: see text]th Ceresa cycles, are not algebraically equivalent to zero in their Jacobian varieties. We apply the method to curves for prime [Formula: see text] with [Formula: see text] modulo [Formula: see text], [Formula: see text] and [Formula: see text] [Formula: see text].

The group algebra decomposition of Fermat curves of prime degree

2015 ◽  
Vol 104 (2) ◽  
pp. 145-155 ◽  
Author(s):  
Patricio Barraza ◽  
Anita M. Rojas
Keyword(s):  
Group Algebra ◽  
Prime Degree ◽  
Fermat Curves ◽  
Group Algebra Decomposition

scholarly journals Lower bounds for Clifford indices in rank three

2010 ◽  
Vol 150 (1) ◽  
pp. 23-33 ◽  
Author(s):  
H. LANGE ◽  
P. E. NEWSTEAD
Keyword(s):  
Lower Bounds ◽  
Vector Bundles ◽  
Plane Curves ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Smooth Plane ◽  
Semistable Vector Bundles

AbstractClifford indices for semistable vector bundles on a smooth projective curve of genus at least 4 were defined in previous papers by the authors. In this paper, we establish lower bounds for the Clifford indices for rank 3 bundles. As a consequence we show that, on smooth plane curves of degree at least 10, there exist non-generated bundles of rank 3 computing one of the Clifford indices.

scholarly journals The kernel of the monodromy of the universal family of degree d smooth plane curves

2020 ◽  
pp. 1-15
Author(s):  
Reid Monroe Harris
Keyword(s):  
Plane Curve ◽  
Teichmüller Space ◽  
Mapping Class ◽  
Plane Curves ◽  
Teichmuller Space ◽  
Infinite Rank ◽  
Universal Family ◽  
Smooth Plane ◽  
Quartic Curves ◽  
Countably Infinite

We consider the parameter space [Formula: see text] of smooth plane curves of degree [Formula: see text]. The universal smooth plane curve of degree [Formula: see text] is a fiber bundle [Formula: see text] with fiber diffeomorphic to a surface [Formula: see text]. This bundle gives rise to a monodromy homomorphism [Formula: see text], where [Formula: see text] is the mapping class group of [Formula: see text]. The main result of this paper is that the kernel of [Formula: see text] is isomorphic to [Formula: see text], where [Formula: see text] is a free group of countably infinite rank. In the process of proving this theorem, we show that the complement [Formula: see text] of the hyperelliptic locus [Formula: see text] in Teichmüller space [Formula: see text] has the homotopy type of an infinite wedge of spheres. As a corollary, we obtain that the moduli space of plane quartic curves is aspherical. The proofs use results from the Weil–Petersson geometry of Teichmüller space together with results from algebraic geometry.

scholarly journals Determinants of subquotients of Galois representations associated with abelian varieties

2013 ◽  
Vol 13 (3) ◽  
pp. 517-559 ◽  
Author(s):  
Eric Larson ◽  
Dmitry Vaintrob
Keyword(s):  
Elliptic Curves ◽  
Abelian Variety ◽  
Number Field ◽  
Riemann Hypothesis ◽  
Galois Representations ◽  
Abelian Varieties ◽  
Number Fields ◽  
Torsion Points ◽  
Prime Degree ◽  
Rho A

AbstractGiven an abelian variety $A$ of dimension $g$ over a number field $K$, and a prime $\ell $, the ${\ell }^{n} $-torsion points of $A$ give rise to a representation ${\rho }_{A, {\ell }^{n} } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ( \mathbb{Z} / {\ell }^{n} \mathbb{Z} )$. In particular, we get a mod-$\ell $representation ${\rho }_{A, \ell } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ({ \mathbb{F} }_{\ell } )$ and an $\ell $-adic representation ${\rho }_{A, {\ell }^{\infty } } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ({ \mathbb{Z} }_{\ell } )$. In this paper, we describe the possible determinants of subquotients of these two representations. These two lists turn out to be remarkably similar.Applying our results in dimension $g= 1$, we recover a generalized version of a theorem of Momose on isogeny characters of elliptic curves over number fields, and obtain, conditionally on the Generalized Riemann Hypothesis, a generalization of Mazur’s bound on rational isogenies of prime degree to number fields.

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