Let $C\; : \;y^2=f(x)$ be a hyperelliptic curve of genus $g\geq 1$, defined over a complete discretely valued field $K$, with ring of integers $O_K$. Under certain conditions on $C$, mild when residue characteristic is not $2$, we explicitly construct the minimal regular model with normal crossings $\mathcal{C}/O_K$ of $C$. In the same setting we determine a basis of integral differentials of $C$, that is an $O_K$-basis for the global sections of the relative dualising sheaf $\omega _{\mathcal{C}/O_K}$.

This paper goes beyond Katz–Sarnak theory on the distribution of curves over finite fields according to their number of rational points, theoretically, experimentally, and conjecturally. In particular, we give a formula for the limits of the moments measuring the asymmetry of this distribution for (non-hyperelliptic) curves of genus $g\geq 3$. The experiments point to a stronger notion of convergence than the one provided by the Katz–Sarnak framework for all curves of genus $\geq 3$. However, for elliptic curves and for hyperelliptic curves of every genus, we prove that this stronger convergence cannot occur.

In this paper, we express the reduction types of Picard curves in terms of tropical invariants associated with binary quintics. We also give a general framework for tropical invariants associated with group actions on arbitrary varieties. The problem of finding tropical invariants for binary forms fits in this general framework by mapping the space of binary forms to symmetrized versions of the Deligne–Mumford compactification $\overline{M}_{0,n}$.

For a given genus $g \geq 1$, we give lower bounds for the maximal number of rational points on a smooth projective absolutely irreducible curve of genus g over $\mathbb{F}_q$. As a consequence of Katz–Sarnak theory, we first get for any given $g>0$, any $\varepsilon>0$ and all q large enough, the existence of a curve of genus g over $\mathbb{F}_q$ with at least $1+q+ (2g-\varepsilon) \sqrt{q}$ rational points. Then using sums of powers of traces of Frobenius of hyperelliptic curves, we get a lower bound of the form $1+q+1.71 \sqrt{q}$ valid for $g \geq 3$ and odd $q \geq 11$. Finally, explicit constructions of towers of curves improve this result: We show that the bound $1+q+4 \sqrt{q} -32$ is valid for all $g\ge 2$ and for all q.

We study plane curves over finite fields whose tangent lines at smooth $\mathbb {F}_q$-points together cover all the points of $\mathbb {P}^2(\mathbb {F}_q)$.

A superelliptic curve over a discrete valuation ring $\mathscr{O}$ of residual characteristic p is a curve given by an equation $\mathscr{C}\;:\; y^n=\,f(x)$, with $\textrm{Disc}(\,f)\neq 0$. The purpose of this article is to describe the Galois representation attached to such a curve under the hypothesis that f(x) has all its roots in the fraction field of $\mathscr{O}$ and that $p \nmid n$. Our results are inspired on the algorithm given in Bouw and WewersGlasg (Math. J. 59(1) (2017), 77–108.) but our description is given in terms of a cluster picture as defined in Dokchitser et al. (Algebraic curves and their applications, Contemporary Mathematics, vol. 724 (American Mathematical Society, Providence, RI, 2019), 73–135.).

We investigate a novel geometric Iwasawa theory for ${\mathbf Z}_p$ -extensions of function fields over a perfect field k of characteristic $p>0$ by replacing the usual study of p-torsion in class groups with the study of p-torsion class group schemes. That is, if $\cdots \to X_2 \to X_1 \to X_0$ is the tower of curves over k associated with a ${\mathbf Z}_p$ -extension of function fields totally ramified over a finite nonempty set of places, we investigate the growth of the p-torsion group scheme in the Jacobian of $X_n$ as $n\rightarrow \infty $ . By Dieudonné theory, this amounts to studying the first de Rham cohomology groups of $X_n$ equipped with natural actions of Frobenius and of the Cartier operator V. We formulate and test a number of conjectures which predict striking regularity in the $k[V]$ -module structure of the space $M_n:=H^0(X_n, \Omega ^1_{X_n/k})$ of global regular differential forms as $n\rightarrow \infty .$ For example, for each tower in a basic class of ${\mathbf Z}_p$ -towers, we conjecture that the dimension of the kernel of $V^r$ on $M_n$ is given by $a_r p^{2n} + \lambda _r n + c_r(n)$ for all n sufficiently large, where $a_r, \lambda _r$ are rational constants and $c_r : {\mathbf Z}/m_r {\mathbf Z} \to {\mathbf Q}$ is a periodic function, depending on r and the tower. To provide evidence for these conjectures, we collect extensive experimental data based on new and more efficient algorithms for working with differentials on ${\mathbf Z}_p$ -towers of curves, and we prove our conjectures in the case $p=2$ and $r=1$ .

We study the p-rank stratification of the moduli space of cyclic degree $\ell $ covers of the projective line in characteristic p for distinct primes p and $\ell $ . The main result is about the intersection of the p-rank $0$ stratum with the boundary of the moduli space of curves. When $\ell =3$ and $p \equiv 2 \bmod 3$ is an odd prime, we prove that there exists a smooth trielliptic curve in characteristic p, for every genus g, signature type $(r,s)$ , and p-rank f satisfying the clear necessary conditions.

Let $g \geq 1$ be an integer and let $A/\mathbb Q$ be an abelian variety that is isogenous over $\mathbb Q$ to a product of g elliptic curves defined over $\mathbb Q$ , pairwise non-isogenous over $\overline {\mathbb Q}$ and each without complex multiplication. For an integer t and a positive real number x, denote by $\pi _A(x, t)$ the number of primes $p \leq x$ , of good reduction for A, for which the Frobenius trace $a_{1, p}(A)$ associated to the reduction of A modulo p equals t. Assuming the Generalized Riemann Hypothesis for Dedekind zeta functions, we prove that $\pi _A(x, 0) \ll _A x^{1 - \frac {1}{3 g+1 }}/(\operatorname {log} x)^{1 - \frac {2}{3 g+1}}$ and $\pi _A(x, t) \ll _A x^{1 - \frac {1}{3 g + 2}}/(\operatorname {log} x)^{1 - \frac {2}{3 g + 2}}$ if $t \neq 0$ . These bounds largely improve upon recent ones obtained for $g = 2$ by Chen, Jones, and Serban, and may be viewed as generalizations to arbitrary g of the bounds obtained for $g=1$ by Murty, Murty, and Saradha, combined with a refinement in the power of $\operatorname {log} x$ by Zywina. Under the assumptions stated above, we also prove the existence of a density one set of primes p satisfying $|a_{1, p}(A)|>p^{\frac {1}{3 g + 1} - \varepsilon }$ for any fixed $\varepsilon>0$ .

Let $C$ be a smooth curve over a finite field of characteristic $p$ and let $M$ be an overconvergent $\mathbf {F}$-isocrystal over $C$. After replacing $C$ with a dense open subset, $M$ obtains a slope filtration. This is a purely $p$-adic phenomenon; there is no counterpart in the theory of lisse $\ell$-adic sheaves. The graded pieces of this slope filtration correspond to lisse $p$-adic sheaves, which we call geometric. Geometric lisse $p$-adic sheaves are mysterious, as there is no $\ell$-adic analogue. In this article, we study the monodromy of geometric lisse $p$-adic sheaves with rank one. More precisely, we prove exponential bounds on their ramification breaks. When the generic slopes of $M$ are integers, we show that the local ramification breaks satisfy a certain type of periodicity. The crux of the proof is the theory of $\mathbf {F}$-isocrystals with log-decay. We prove a monodromy theorem for these $\mathbf {F}$-isocrystals, as well as a theorem relating the slopes of $M$ to the rate of log-decay of the slope filtration. As a consequence of these methods, we provide a new proof of the Drinfeld–Kedlaya theorem for irreducible $\mathbf {F}$-isocrystals on curves.

In this work we generalise the main result of [1] to the family of hyperelliptic curves with potentially good reduction over a p-adic field which have genus $g=({p-1})/{2}$ and the largest possible image of inertia under the $\ell$ -adic Galois representation associated to its Jacobian. We will prove that this Galois representation factors as the tensor product of an unramified character and an irreducible representation of a finite group, which can be either equal to the inertia image (in which case the representation is easily determined) or a $C_2$ -extension of it. In this second case, there are two suitable representations and we will describe the Galois action explicitly in order to determine the correct one.

Let $E/\mathbb {Q}$ be an elliptic curve. For a prime p of good reduction, let $r(E,p)$ be the smallest non-negative integer that gives the x-coordinate of a point of maximal order in the group $E(\mathbb {F}_p)$ . We prove unconditionally that $r(E,p)> 0.72\log \log p$ for infinitely many p, and $r(E,p)> 0.36 \log p$ under the assumption of the Generalized Riemann Hypothesis. These can be viewed as elliptic curve analogues of classical lower bounds on the least primitive root of a prime.

We correct the proof of the main $\ell$-independence result of the above-mentioned paper by showing that for any smooth and proper variety over an equicharacteristic local field, there exists a globally defined such variety with the same ($p$-adic and $\ell$-adic) cohomology.

Recently E. Bombieri and N. M. Katz (2010) demonstrated that several well-known results about the distribution of values of linear recurrence sequences lead to interesting statements for Frobenius traces of algebraic curves. Here we continue this line of study and establish the Möbius randomness law quantitatively for the normalised form of Frobenius traces.

Given an elliptic curve E in Legendre form $y^2 = x(x - 1)(x - \lambda )$ over the fraction field of a Henselian ring R of mixed characteristic $(0, 2)$, we present an algorithm for determining a semistable model of E over R that depends only on the valuation of $\lambda $. We provide several examples along with an easy corollary concerning $2$-torsion.

Let $k$ be a finite field and $L$ be the function field of a curve $C/k$ of genus $g\geq 1$. In the first part of this note we show that the number of separable $S$-integral points on a constant elliptic curve $E/L$ is bounded solely in terms of $g$ and the size of $S$. In the second part we assume that $L$ is the function field of a hyperelliptic curve $C_{A}:s^{2}=A(t)$, where $A(t)$ is a square-free $k$-polynomial of odd degree. If $\infty$ is the place of $L$ associated to the point at infinity of $C_{A}$, then we prove that the set of separable $\{\infty \}$-points can be bounded solely in terms of $g$ and does not depend on the Mordell–Weil group $E(L)$. This is done by bounding the number of separable integral points over $k(t)$ on elliptic curves of the form $E_{A}:A(t)y^{2}=f(x)$, where $f(x)$ is a polynomial over $k$. Additionally, we show that, under an extra condition on $A(t)$, the existence of a separable integral point of ‘small’ height on the elliptic curve $E_{A}/k(t)$ determines the isomorphism class of the elliptic curve $y^{2}=f(x)$.

We prove an analogue of Belyi’s theorem in characteristic two. Our proof consists of the following three steps. We first introduce a new notion called pseudo-tameness for morphisms between curves over an algebraically closed field of characteristic two. Secondly, we prove the existence of a ‘pseudo-tame’ rational function by showing the vanishing of an obstruction class. Finally, we construct a tamely ramified rational function from the ‘pseudo-tame’ rational function.

We generalize the Cohen–Lenstra heuristics over function fields to étale group schemes $G$ (with the classical case of abelian groups corresponding to constant group schemes). By using the results of Ellenberg–Venkatesh–Westerland, we make progress towards the proof of these heuristics. Moreover, by keeping track of the image of the Weil-pairing as an element of $\wedge ^{2}G(1)$, we formulate more refined heuristics which nicely explain the deviation from the usual Cohen–Lenstra heuristics for abelian $\ell$-groups in cases where $\ell \mid q-1$; the nature of this failure was suggested already in the works of Malle, Garton, Ellenberg–Venkatesh–Westerland, and others. On the purely large random matrix side, we provide a natural model which has the correct moments, and we conjecture that these moments uniquely determine a limiting probability measure.

We prove that if $C$ is a reflexive smooth plane curve of degree $d$ defined over a finite field $\mathbb{F}_{q}$ with $d\leqslant q+1$, then there is an $\mathbb{F}_{q}$-line $L$ that intersects $C$ transversely. We also prove the same result for non-reflexive curves of degree $p+1$ and $2p+1$ when $q=p^{r}$.

In this paper we study the number of rational points on curves in an ensemble of abelian covers of the projective line: let $\ell$ be a prime, $q$ a prime power and consider the ensemble ${\mathcal{H}}_{g,\ell }$ of $\ell$-cyclic covers of $\mathbb{P}_{\mathbb{F}_{q}}^{1}$ of genus $g$. We assume that $q\not \equiv 0,1~\text{mod}~\ell$. If $2g+2\ell -2\not \equiv 0~\text{mod}~(\ell -1)\operatorname{ord}_{\ell }(q)$, then ${\mathcal{H}}_{g,\ell }$ is empty. Otherwise, the number of rational points on a random curve in ${\mathcal{H}}_{g,\ell }$ distributes as $\sum _{i=1}^{q+1}X_{i}$ as $g\rightarrow \infty$, where $X_{1},\ldots ,X_{q+1}$ are independent and identically distributed random variables taking the values $0$ and $\ell$ with probabilities $(\ell -1)/\ell$ and $1/\ell$, respectively. The novelty of our result is that it works in the absence of a primitive $\ell$th root of unity, the presence of which was crucial in previous studies.