Related papers: Computing binary curves of genus five

We construct families of hyperelliptic curves over Q of arbitrary genus g with (at least) g integral elements in K_2. We also verify the Beilinson conjectures about K_2 numerically for several curves with g=2, 3, 4 and 5. The paper is…

The genus g of an F_{q^2}-maximal curve satisfies g=g_1:=q(q-1)/2 or g\le g_2:= [(q-1)^2/4]. Previously, such curves with g=g_1 or g=g_2, q odd, have been characterized up to isomorphism. Here it is shown that an F_{q^2}-maximal curve with…

A projective, smooth, absolutely irreducible algebraic curve X of genus g defined over a finite field F_q is called optimal if for every other such genus g curve Y over F_q one has $\#Y(F_q)\le \#X(F_q)$. In this paper we show that for…

We study the moduli space of genus 3 hyperelliptic curves via the weighted projective space of binary octavics. This enables us to create a database of all genus 3 hyperelliptic curves defined over $\mathbb Q$, of weighted moduli height…

We study mapping class group orbits of homotopy and isotopy classes of curves with self-intersections. We exhibit the asymptotics of the number of such orbits of curves with a bounded number of self-intersections, as the complexity of the…

We describe a method to determine all the isomorphism classes of principal polarizations of the modular abelian surfaces $A_f$ with quaternionic multiplication attached to a normalized newform $f$ without complex multiplication. We include…

We give explicit formulas for the number of distinct elliptic curves over a finite field, up to isomorphism, in the families of Legendre, Jacobi, Hessian and generalized Hessian curves.

We present algorithms to compute the topology of 2D and 3D hyperelliptic curves. The algorithms are based on the fact that 2D and 3D hyperelliptic curves can be seen as the image of a planar curve (the Weierstrass form of the curve), whose…

We consider systems of simple closed curves on surfaces and their total number of intersection points, their so-called crossing number. For a fixed number of curves, we aim to minimise the crossing number. We determine the minimal crossing…

The well known Hurwitz upper bound states that a closed Riemann surface $S$ of genus $g \geq 2$ has at most $84(g-1)$ conformal automorphisms. If $S$ has exactly $84(g-1)$ conformal automorphisms, then it is called a Hurwitz curve. The…

For every finite collection C of abelian varieties over F_q, we produce an explicit upper bound on the genus of curves over F_q whose Jacobians are isogenous to a product of powers of elements of C.

We classify, up to isomorphism, maximal curves covered by the Hermitian curve \mathcal H by a prime degree Galois covering. We also compute the genus of maximal curves obtained by the quotient of \mathcal H by several automorphisms groups.…

We compile a complete list of isomorphism class representatives of curves of genus 6 over $\mathbb{F}_2$. We use explicit descriptions of canonical curves in each stratum of the Brill--Noether stratification of the moduli space…

In algebraic geometry, superspecial curves are important research objects. While the number of superspecial genus-3 curves in characteristic $p$ is known, the number of hyperelliptic ones among them has not been determined even for small…

In this paper the number of $\bar{\mathbb{F}}_q$-isomorphism classes of general Jacobi quartic curves, i.e., the number of general Jacobi quartic curves with distinct $j$-invariants, over the finite field $\mathbb{F}_q$ is enumerated.

We exploit three classical characterizations of smooth genus two curves to study their tropical and analytic counterparts. First, we provide a combinatorial rule to determine the dual graph of each algebraic curve and the metric structure…

The genus of a maximal curve over a finite field with r^2 elements is either g_0=r(r-1)/2 or less than or equal to g_1=(r-1)^2/4. Maximal curves with genus g_0 or g_1 have been characterized up to isomorphism. A natural genus to be studied…

We continue our study of genus 2 curves $C$ that admit a cover $ C \to E$ to a genus 1 curve $E$ of prime degree $n$. These curves $C$ form an irreducible 2-dimensional subvariety $\L_n$ of the moduli space $\M_2$ of genus 2 curves. Here we…

We prove the existence of curves of genus $7$ and $12$ over the field with $11^5$ elements, reaching the Hasse-Weil-Serre upper bound. These curves are quotients of modular curves and we give explicit equations. We compute the number of…

We describe the invariants of plane quartic curves -- nonhyperelliptic genus 3 curves in their canonical model -- as determined by Dixmier and Ohno, with application to the classification of curves with given structure. In particular, we…