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Let X and Y be curves over a finite field. In this article we explore methods to determine whether there is a rational map from Y to X by considering L-functions of certain covers of X and Y and propose a specific family of covers to address the special case of determining when X and Y are isomorphic. We also discuss an application to factoring polynomials over finite fields.
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This is an introduction to a probabilistic model for the arithmetic of elliptic curves, a model developed in a series of articles of the author with Bhargava, Kane, Lenstra, Park, Rains, Voight, and Wood. We discuss the theoretical evidence for the model, and we make predictions about elliptic curves based on corresponding theorems proved about the model. In particular, the model suggests that all but finitely many elliptic curves over $mathbb{Q}$ have rank $le 21$, which would imply that the rank is uniformly bounded.
We investigate the arithmetic of algebraic curves on coarse moduli spaces for special linear rank two local systems on surfaces with fixed boundary traces. We prove a structure theorem for morphisms from the affine line into the moduli space. We show that the set of integral points on any nondegenerate algebraic curve on the moduli space can be effectively determined.
Let $p$ be a prime, let $r$ and $q$ be powers of $p$, and let $a$ and $b$ be relatively prime integers not divisible by $p$. Let $C/mathbb F_{r}(t)$ be the superelliptic curve with affine equation $y^b+x^a=t^q-t$. Let $J$ be the Jacobian of $C$. By work of Pries--Ulmer, $J$ satisfies the Birch and Swinnerton-Dyer conjecture (BSD). Generalizing work of Griffon--Ulmer, we compute the $L$-function of $J$ in terms of certain Gauss sums. In addition, we estimate several arithmetic invariants of $J$ appearing in BSD, including the rank of the Mordell--Weil group $J(mathbb F_{r}(t))$, the Faltings height of $J$, and the Tamagawa numbers of $J$ in terms of the parameters $a,b,q$. For any $p$ and $r$, we show that for certain $a$ and $b$ depending only on $p$ and $r$, these Jacobians provide new examples of families of simple abelian varieties of fixed dimension and with unbounded analytic and algebraic rank as $q$ varies through powers of $p$. Under a different set of criteria on $a$ and $b$, we prove that the order of the Tate--Shafarevich group of $J$ grows quasilinearly in $q$ as $q to infty.$
We study the Jacobian $J$ of the smooth projective curve $C$ of genus $r-1$ with affine model $y^r = x^{r-1}(x + 1)(x + t)$ over the function field $mathbb{F}_p(t)$, when $p$ is prime and $rge 2$ is an integer prime to $p$. When $q$ is a power of $p$ and $d$ is a positive integer, we compute the $L$-function of $J$ over $mathbb{F}_q(t^{1/d})$ and show that the Birch and Swinnerton-Dyer conjecture holds for $J$ over $mathbb{F}_q(t^{1/d})$. When $d$ is divisible by $r$ and of the form $p^ u +1$, and $K_d := mathbb{F}_p(mu_d,t^{1/d})$, we write down explicit points in $J(K_d)$, show that they generate a subgroup $V$ of rank $(r-1)(d-2)$ whose index in $J(K_d)$ is finite and a power of $p$, and show that the order of the Tate-Shafarevich group of $J$ over $K_d$ is $[J(K_d):V]^2$. When $r>2$, we prove that the new part of $J$ is isogenous over $overline{mathbb{F}_p(t)}$ to the square of a simple abelian variety of dimension $phi(r)/2$ with endomorphism algebra $mathbb{Z}[mu_r]^+$. For a prime $ell$ with $ell mid pr$, we prove that $J[ell](L)={0}$ for any abelian extension $L$ of $overline{mathbb{F}}_p(t)$.
Arithmetic dynamics is the study of number theoretic properties of dynamical systems. A relatively new field, it draws inspiration partly from dynamical analogues of theorems and conjectures in classical arithmetic geometry, and partly from $p$-adic analogues of theorems and conjectures in classical complex dynamics. In this article we survey some of the motivating problems and some of the recent progress in the field of arithmetic dynamics.
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