The Analytic Rank of a Family of Jacobians of Fermat Curves

Tomasz Jędrzejak

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Frobenius nonclassicality with respect to linear systems of curves of arbitrary degree

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For each integer s ≥ 1, we present a family of curves that are $_{q}$ -Frobenius nonclassical with respect to the linear system of plane curves of degree s. In the case s=2, we give necessary and sufficient conditions for such curves to be $_{q}$ -Frobenius nonclassical with respect to the linear system of conics. In the $_{q}$ -Frobenius nonclassical cases, we determine the exact number of $_{q}$ -rational points. In the remaining cases, an upper bound for the number of $_{q}$ -rational points will follow from Stöhr-Voloch...

An iterative construction for ordinary and very special hyperelliptic curves

Francis J. Sullivan (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Si costruiscono famiglie di curve iperellittiche col $p$ —rango della varietà jacobiana uguale a zero. La costruzione sfrutta le proprietà elementari dell’operatore di Cartier e delle estensioni $p$ -cicliche dei corpi con la caratteristica $p$ maggiore di zero.

Some examples of 5 and 7 descent for elliptic curves over $Q$

Tom Fisher (2001)

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We perform descent calculations for the families of elliptic curves over $Q$ with a rational point of order $n = 5$ or 7. These calculations give an estimate for the Mordell-Weil rank which we relate to the parity conjecture. We exhibit explicit elements of the Tate-Shafarevich group of order 5 and 7, and show that the 5-torsion of the Tate-Shafarevich group of an elliptic curve over $Q$ may become arbitrarily large.

Complete pluripolar curves and graphs

Tomas Edlund (2004)

Annales Polonici Mathematici

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It is shown that there exist $C^{\infty}$ functions on the boundary of the unit disk whose graphs are complete pluripolar. Moreover, for any natural number k, such functions are dense in the space of $C^{k}$ functions on the boundary of the unit disk. We show that this result implies that the complete pluripolar closed $C^{\infty}$ curves are dense in the space of closed $C^{k}$ curves in ℂⁿ. We also show that on each closed subset of the complex plane there is a continuous function whose graph is complete pluripolar. ...

The automorphism group of ${\bar{M}}_{0, n}$

Andrea Bruno, Massimiliano Mella (2013)

Journal of the European Mathematical Society

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The paper studies fiber type morphisms between moduli spaces of pointed rational curves. Via Kapranov’s description we are able to prove that the only such morphisms are forgetful maps. This allows us to show that the automorphism group of ${\bar{M}}_{0, n}$ is the permutation group on $n$ elements as soon as $n \geq 5$ .

An index inequality for embedded pseudoholomorphic curves in symplectizations

Michael Hutchings (2002)

Journal of the European Mathematical Society

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Let $Σ$ be a surface with a symplectic form, let $φ$ be a symplectomorphism of $Σ$ , and let $Y$ be the mapping torus of $φ$ . We show that the dimensions of moduli spaces of embedded pseudoholomorphic curves in $ℝ \times 𝕐$ , with cylindrical ends asymptotic to periodic orbits of $φ$ or multiple covers thereof, are bounded from above by an additive relative index. We deduce some compactness results for these moduli spaces. This paper establishes some of the foundations for a program with Michael Thaddeus, to...

Euler characteristics of moduli spaces of curves

Gilberto Bini, John Harer (2011)

Journal of the European Mathematical Society

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Let $M_{g}^{n}$ be the moduli space of $n$ -pointed Riemann surfaces of genus $g$ . Denote by $\bar{M_{g}^{n}}$ the Deligne-Mumford compactification of $M_{g}^{n}$ . In the present paper, we calculate the orbifold and the ordinary Euler characteristic of $\bar{M_{g}^{n}}$ for any $g$ and $n$ such that $n > 2 - 2 g$ .

On Zariski's theorem in positive characteristic

Ilya Tyomkin (2013)

Journal of the European Mathematical Society

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In the current paper we show that the dimension of a family $V$ of irreducible reduced curves in a given ample linear system on a toric surface $S$ over an algebraically closed field is bounded from above by $- K_{S} . C + p_{g} (C) - 1$ , where $C$ denotes a general curve in the family. This result generalizes a famous theorem of Zariski to the case of positive characteristic. We also explore new phenomena that occur in positive characteristic: We show that the equality $𝚍𝚒𝚖 (V) = - K_{S} . C + p_{g} (C) - 1$ does not imply the nodality of $C$ even if $C$ belongs...

Ch flat curves in Euclidean space $E^{2 n + 1}$

Lj. Protić, N. Bokan (1980)

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Infinite rank of elliptic curves over $ℚ^{a b}$

Bo-Hae Im, Michael Larsen (2013)

Acta Arithmetica

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If E is an elliptic curve defined over a quadratic field K, and the j-invariant of E is not 0 or 1728, then $E (ℚ^{a b})$ has infinite rank. If E is an elliptic curve in Legendre form, y² = x(x-1)(x-λ), where ℚ(λ) is a cubic field, then $E (K ℚ^{a b})$ has infinite rank. If λ ∈ K has a minimal polynomial P(x) of degree 4 and v² = P(u) is an elliptic curve of positive rank over ℚ, we prove that y² = x(x-1)(x-λ) has infinite rank over $K ℚ^{a b}$ .

On ramified covers of the projective plane II: Generalizing Segre’s theory

Michael Friedman, Rebecca Lehman, Maxim Leyenson, Mina Teicher (2012)

Journal of the European Mathematical Society

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The classical Segre theory gives a necessary and sufficient condition for a plane curve to be a branch curve of a (generic) projection of a smooth surface in $ℙ^{3}$ . We generalize this result for smooth surfaces in a projective space of any dimension in the following way: given two plane curves, $B$ and $E$ , we give a necessary and sufficient condition for $B$ to be the branch curve of a surface $X$ in $ℙ^{N}$ and $E$ to be the image of the double curve of a $ℙ^{3}$ -model of $X$ . In the classical Segre theory, a...

Explicit Teichmüller curves with complementary series

Carlos Matheus, Gabriela Weitze-Schmithüsen (2013)

Bulletin de la Société Mathématique de France

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We construct an explicit family of arithmetic Teichmüller curves $𝒞_{2 k}$ , $k \in ℕ$ , supporting $SL (2, ℝ)$ -invariant probabilities $μ_{2 k}$ such that the associated $SL (2, ℝ)$ -representation on $L^{2} (𝒞_{2 k}, μ_{2 k})$ has complementary series for every $k \geq 3$ . Actually, the size of the spectral gap along this family goes to zero. In particular, the Teichmüller geodesic flow restricted to these explicit arithmetic Teichmüller curves $𝒞_{2 k}$ has arbitrarily slow rate of exponential mixing.

On invariants of elliptic curves on average

Amir Akbary, Adam Tyler Felix (2015)

Acta Arithmetica

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We prove several results regarding some invariants of elliptic curves on average over the family of all elliptic curves inside a box of sides A and B. As an example, let E be an elliptic curve defined over ℚ and p be a prime of good reduction for E. Let $e_{E} (p)$ be the exponent of the group of rational points of the reduction modulo p of E over the finite field $_{p}$ . Let be the family of elliptic curves $E_{a, b} : y^{2} = x^{3} + a x + b$ , where |a| ≤ A and |b| ≤ B. We prove that, for any c > 1 and k∈ ℕ, $1 / | | \sum_{E \in} \sum_{p \leq x} e_{E}^{k} (p) = C_{k} l i (x^{k + 1}) + O ((x^{k + 1}) / {(l o g x)}^{c}$ )as x → ∞, as long...

On covering and quasi-unsplit families of curves

Laurent Bonavero, Cinzia Casagrande, Stéphane Druel (2007)

Journal of the European Mathematical Society

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Given a covering family $V$ of effective 1-cycles on a complex projective variety $X$ , we find conditions allowing one to construct a geometric quotient $q : X \to Y$ , with $q$ regular on the whole of $X$ , such that every fiber of $q$ is an equivalence class for the equivalence relation naturally defined by $V$ . Among other results, we show that on a normal and $ℚ$ -factorial projective variety $X$ with canonical singularities and $dim X \leq 4$ , every covering and quasi-unsplit family $V$ of rational curves generates a geometric...

On linked Jordan curves in $R^{3}$

M. Mateljević (1975)

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Rational points on curves

Michael Stoll (2011)

Journal de Théorie des Nombres de Bordeaux

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This is an extended version of an invited lecture I gave at the Journées Arithmétiques in St. Étienne in July 2009. We discuss the state of the art regarding the problem of finding the set of rational points on a (smooth projective) geometrically integral curve $C$ over $ℚ$ . The focus is on practical aspects of this problem in the case that the genus of $C$ is at least $2$ , and therefore the set of rational points is finite.