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Search: MSC category 11G30 ( Curves of arbitrary genus or genus $ )

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1. CMB 2009 (vol 53 pp. 58)

Dąbrowski, Andrzej; Jędrzejak, Tomasz
Ranks in Families of Jacobian Varieties of Twisted Fermat Curves
In this paper, we prove that the unboundedness of ranks in families of Jacobian varieties of twisted Fermat curves is equivalent to the divergence of certain infinite series.

Keywords:Fermat curve, Jacobian variety, elliptic curve, canonical height
Categories:11G10, 11G05, 11G50, 14G05, 11G30, 14H45, 14K15

2. CMB 2009 (vol 52 pp. 117)

Poulakis, Dimitrios
On the Rational Points of the Curve $f(X,Y)^q = h(X)g(X,Y)$
Let $q = 2,3$ and $f(X,Y)$, $g(X,Y)$, $h(X)$ be polynomials with integer coefficients. In this paper we deal with the curve $f(X,Y)^q = h(X)g(X,Y)$, and we show that under some favourable conditions it is possible to determine all of its rational points.

Categories:11G30, 14G05, 14G25

3. CMB 2007 (vol 50 pp. 313)

Tzermias, Pavlos
On Cauchy--Liouville--Mirimanoff Polynomials
Let $p$ be a prime greater than or equal to 17 and congruent to 2 modulo 3. We use results of Beukers and Helou on Cauchy--Liouville--Mirimanoff polynomials to show that the intersection of the Fermat curve of degree $p$ with the line $X+Y=Z$ in the projective plane contains no algebraic points of degree $d$ with $3 \leq d \leq 11$. We prove a result on the roots of these polynomials and show that, experimentally, they seem to satisfy the conditions of a mild extension of an irreducibility theorem of P\'{o}lya and Szeg\"{o}. These conditions are \emph{conjecturally} also necessary for irreducibility.

Categories:11G30, 11R09, 12D05, 12E10

4. CMB 2000 (vol 43 pp. 304)

Darmon, Henri; Mestre, Jean-François
Courbes hyperelliptiques à multiplications réelles et une construction de Shih
Soient $r$ et $p$ deux nombres premiers distincts, soit $K = \Q(\cos \frac{2\pi}{r})$, et soit $\F$ le corps r\'esiduel de $K$ en une place au-dessus de $p$. Lorsque l'image de $(2 - 2\cos \frac{2\pi}{r})$ dans $\F$ n'est pas un carr\'e, nous donnons une construction g\'eom\'etrique d'une extension r\'eguliere de $K(t)$ de groupe de Galois $\PSL_2 (\F)$. Cette extension correspond \`a un rev\^etement de $\PP^1_{/K}$ de \og{} signature $(r,p,p)$ \fg{} au sens de [3, sec.~6.3], et son existence est pr\'edite par le crit\`ere de rigidit\'e de Belyi, Fried, Thompson et Matzat. Sa construction s'obtient en tordant la representation galoisienne associ\'ee aux points d'ordre $p$ d'une famille de vari\'et\'es ab\'eliennes \`a multiplications r\'eelles par $K$ d\'ecouverte par Tautz, Top et Verberkmoes [6]. Ces vari\'et\'es ab\'eliennes sont d\'efinies sur un corps quadratique, et sont isog\`enes \`a leur conjugu\'e galoisien. Notre construction g\'en\'eralise une m\'ethode de Shih [4], [5], que l'on retrouve quand $r = 2$ et $r = 3$. Let $r$ and $p$ be distinct prime numbers, let $K = \Q(\cos \frac{2\pi}{r})$, and let $\F$ be the residue field of $K$ at a place above $p$. When the image of $(2 - 2\cos \frac{2\pi}{r})$ in $\F$ is not a square, we describe a geometric construction of a regular extension of $K(t)$ with Galois group $\PSL_2 (\F)$. This extension corresponds to a covering of $\PP^1_{/K}$ of ``signature $(r,p,p)$'' in the sense of [3, sec.~6.3], and its existence is predicted by the rigidity criterion of Belyi, Fried, Thompson and Matzat. Its construction is obtained by twisting the mod $p$ galois representation attached to a family of abelian varieties with real multiplications by $K$ discovered by Tautz, Top and Verberkmoes [6]. These abelian varieties are defined in general over a quadratic field, and are isogenous to their galois conjugate. Our construction generalises a method of Shih [4], [5], which one recovers when $r = 2$ and $r = 3$.

Categories:11G30, 14H25

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