Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userCounting points on superelliptic curves in average polynomial time
104
0
0.0
(
0
)
Authors
Andrew V. Sutherland
Ask ChatGPT about the research
No Arabic abstract
We describe the practical implementation of an average polynomial-time algorithm for counting points on superelliptic curves defined over $mathbb Q$ that is substantially faster than previous approaches. Our algorithm takes as input a superelliptic curves $y^m=f(x)$ with $mge 2$ and $fin mathbb Z[x]$ any squarefree polynomial of degree $dge 3$, along with a positive integer $N$. It can compute $#X(mathbb F_p)$ for all $ple N$ not dividing $mmathrm{lc}(f)mathrm{disc}(f)$ in time $O(md^3 Nlog^3 Nloglog N)$. It achieves this by computing the trace of the Cartier--Manin matrix of reductions of $X$. We can also compute the Cartier--Manin matrix itself, which determines the $p$-rank of the Jacobian of $X$ and the numerator of its zeta function modulo~$p$.
rate research
Read More
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 present an efficient algorithm to compute the Hasse-Witt matrix of a hyperelliptic curve C/Q modulo all primes of good reduction up to a given bound N, based on the average polynomial-time algorithm recently introduced by Harvey. An implementation for hyperelliptic curves of genus 2 and 3 is more than an order of magnitude faster than alternative methods for N = 2^26.
We give an explicit description of the stable reduction of superelliptic curves of the form $y^n=f(x)$ at primes $p$ whose residue characteristic is prime to the exponent $n$. We then use this description to compute the local $L$-factor of the curve and the exponent of conductor at $p$.
In this paper we study the Coleman-Oort conjecture for superelliptic curves, i.e., curves defined by affine equations $y^n=F(x)$ with $F$ a separable polynomial. We prove that up to isomorphism there are at most finitely many superelliptic curves of fixed genus $ggeq 8$ with CM Jacobians. The proof relies on the geometric structures of Shimura subvarieties in Siegel modular varieties and the stability properties of Higgs bundles associated to fibred surfaces.
We present an algorithm that computes the Hasse-Witt matrix of given hyperelliptic curve over Q at all primes of good reduction up to a given bound N. It is simpler and faster than the previous algorithm developed by the authors.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
