Related papers: Genus-five hyperelliptic or trigonal curves with m…

Let $F(x_1,...,x_n)$ be a form of degree $d\geq 2$, which produces a geometrically irreducible hypersurface in $\mathbb{P}^{n-1}$. This paper is concerned with the number of rational points on F=0 which have height at most $B$. Whenever…

We determine the precise number of isomorphism classes of elliptic curves over $\mathbb{F}_q(t)$ with $\text{char}(\mathbb{F}_q) = 3,2$. The key idea is to obtain the exact unweighted number of rational points on the classifying stacks…

We compute rational points on genus $3$ odd degree hyperelliptic curves $C$ over $\mathbb{Q}$ that have Jacobians of Mordell-Weil rank $0$. The computation applies the Chabauty-Coleman method to find the zero set of a certain system of…

In this paper we enumerate nonhyperelliptic superspecial curves of genus $4$ over prime fields of characteristic $p\le 11$. Our algorithm works for nonhyperelliptic curves over an arbitrary finite field in characteristic $p \ge 5$. We…

We construct curves of each genus $g\geq 2$ for which Coleman's effective Chabauty bound is sharp and Coleman's theorem can be applied to determine rational points if the rank condition is satisfied. We give numerous examples of genus two…

Given a set $S$ of elements in a number field $k$, we discuss the existence of planar algebraic curves over $k$ which possess rational points whose $x$-coordinates are exactly the elements of $S$. If the size $|S|$ of $S$ is either $4,5$,…

We compute equations for the families of elliptic curves 9-congruent to a given elliptic curve. We use these to find infinitely many non-trivial pairs of 9-congruent elliptic curves over Q, i.e. pairs of non-isogenous elliptic curves over Q…

We establish the sharp estimate <<_d N^{2/d} for the number of rational points of height at most N on an irreducible projective curve of degree d. We deduce this from a result for general hypersurfaces that is sensitive to the coefficients…

According to a conjecture of H. Clemens, the dimension of the space of rational curves on a general projective hypersurface should equal the number predicted by a na\"ive dimension count. In the case of a general hypersurface of degree 7 in…

Combining $2$-descent techniques with Riemann-Roch and B\'ezout's theorems, we give an upper bound on the number of rational points of bounded height on elliptic and hyperelliptic curves over function fields of characteristic $\neq 2$. We…

We complete the computation of all $\mathbb{Q}$-rational points on all the $64$ maximal Atkin-Lehner quotients $X_0(N)^*$ such that the quotient is hyperelliptic. To achieve this, we use a combination of various methods, namely the…

We describe a practical algorithm for computing Brauer-Manin obstructions to the existence of rational points on hyperelliptic curves defined over number fields. This offers advantages over descent based methods in that its correctness does…

Koll\'ar proved that a very general $n$-dimensional complex hypersurface of degree at least $3\lceil (n+3)/4\rceil$ is not birational to a fibration in rational curves. This is most interesting when the hypersurface is Fano, in which case…

Currently, the best upper bounds on the number of rational points on an absolutely irreducible, smooth, projective algebraic curve of genus g defined over a finite field F_q come either from Serre's refinement of the Weil bound if the genus…

This note presents explicit equations (up to birational equivalence over $\mathbb{F}_2$) for a complete, smooth, absolutely irreducible curve $X$ over $\mathbb{F}_2$ of genus $50$ satisfying $#X(\mathbb{F}_2)=40$. In his 1985 Harvard…

A curve over a perfect field $K$ of characteristic $p > 0$ is said to be superspecial if its Jacobian is isomorphic to a product of supersingular elliptic curves over the algebraic closure $\overline{K}$. In recent years, isomorphism…

In the past several years, Howe curves have been studied actively in the field of algebraic curves over fields of positive characteristic. Here, a Howe curve is defined as the desingularization of the fiber product over a projective line of…

It has been conjectured that for $N$ sufficiently large, there are no quadratic polynomials in $\bold Q[z]$ with rational periodic points of period $N$. Morton proved there were none with $N=4$, by showing that the genus~$2$ algebraic curve…

We introduce and begin the topological study of real rational plane curves, all of whose inflection points are real. The existence of such curves is a corollary of results in the real Schubert calculus, and their study has consequences for…

In this paper the notion of rational simple connectedness for the quintic Fano threefold $V_5\subset \mathbb{P}^6$ is studied and unirationality of the moduli spaces $\overline{M}_{0,0}^{\text{bir}}(V_5,d)$, with $d \ge 1$, is proved. Many…