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We show how to efficiently evaluate functions on Jacobian varieties and their quotients. We deduce an algorithm to compute 
$(l,l)$ isogenies between Jacobians of genus two curves in quasi-linear time in the degree 
$l^{2}$.
$X_{0}(n)$ and isogenies of elliptic curves over quadratic fields
We study elliptic curves over quadratic fields with isogenies of certain degrees. Let 
$n$ be a positive integer such that the modular curve 
$X_{0}(n)$ is hyperelliptic of genus 
${\geqslant}2$ and such that its Jacobian has rank 
$0$ over 
$\mathbb{Q}$. We determine all points of 
$X_{0}(n)$ defined over quadratic fields, and we give a moduli interpretation of these points. We show that, with a finite number of exceptions up to 
$\overline{\mathbb{Q}}$-isomorphism, every elliptic curve over a quadratic field 
$K$ admitting an 
$n$-isogeny is 
$d$-isogenous, for some 
$d\mid n$, to the twist of its Galois conjugate by a quadratic extension 
$L$ of 
$K$. We determine 
$d$ and 
$L$ explicitly, and we list all exceptions. As a consequence, again with a finite number of exceptions up to 
$\overline{\mathbb{Q}}$-isomorphism, all elliptic curves with 
$n$-isogenies over quadratic fields are in fact 
$\mathbb{Q}$-curves.
We propose to generalize the work of Régis Dupont for computing modular polynomials in dimension 
$2$ to new invariants. We describe an algorithm to compute modular polynomials for invariants derived from theta constants and prove heuristically that this algorithm is quasi-linear in its output size. Some properties of the modular polynomials defined from quotients of theta constants are analyzed. We report on experiments with our implementation.
We solve the Diophantine equation 
$Y^{2}=X^{3}+k$ for all nonzero integers 
$k$ with 
$|k|\leqslant 10^{7}$. Our approach uses a classical connection between these equations and cubic Thue equations. The latter can be treated algorithmically via lower bounds for linear forms in logarithms in conjunction with lattice-basis reduction.
$\text{PSL}_{2}(13)$ in the Monster contains 
$13A$-elements
We prove the assertion in the title by conducting an exhaustive computational search for subgroups isomorphic to 
$\text{PSL}_{2}(13)$ and containing elements in class 
$13B$.
A prime sieve is an algorithm that finds the primes up to a bound 
$n$. We say that a prime sieve is incremental, if it can quickly determine if 
$n+1$ is prime after having found all primes up to 
$n$. We say a sieve is compact if it uses roughly 
$\sqrt{n}$ space or less. In this paper, we present two new results.
We describe the rolling sieve, a practical, incremental prime sieve that takes We also show how to modify the sieve of Atkin and Bernstein from 2004 to obtain a sieve that is simultaneously sublinear, compact, and incremental.
$O(n\log \log n)$ time and 
$O(\sqrt{n}\log n)$ bits of space.
The second result solves an open problem given by Paul Pritchard in 1994.
${\it\lambda}$-invariants attached to cyclic cubic number fields
We describe an algorithm for finding the coefficients of 
$F(X)$ modulo powers of 
$p$, where 
$p\neq 2$ is a prime number and 
$F(X)$ is the power series associated to the zeta function of Kubota and Leopoldt. We next calculate the 5-adic and 7-adic 
${\it\lambda}$-invariants attached to those cubic extensions 
$K/\mathbb{Q}$ with cyclic Galois group 
${\mathcal{A}}_{3}$ (up to field discriminant 
${<}10^{7}$), and also tabulate the class number of 
$K(e^{2{\it\pi}i/p})$ for 
$p=5$ and 
$p=7$. If the 
${\it\lambda}$-invariant is greater than zero, we then determine all the zeros for the corresponding branches of the 
$p$-adic 
$L$-function and deduce 
${\rm\Lambda}$-monogeneity for the class group tower over the cyclotomic 
$\mathbb{Z}_{p}$-extension of 
$K$.
We present a parallel algorithm to calculate a numerical approximation of a single, isolated root 
${\it\alpha}$ of a function 
$f:\mathbb{R}\rightarrow \mathbb{R}$ which is sufficiently regular at and around 
${\it\alpha}$. The algorithm is derivative free and performs one function evaluation on each processor per iteration. It requires at least three processors and can be scaled up to any number of these. The order with which the generated sequence of approximants converges to 
${\it\alpha}$ is equal to 
$(n+\sqrt{n^{2}+4})/2$ for 
$n+1$ processors with 
$n\geqslant 2$. This assumes that particular combinations of the derivatives of 
$f$ do not vanish at 
${\it\alpha}$.
We present a quasi-linear iterative method for solving a system of 
$m$-nonlinear coupled differential equations. We provide an error analysis of the method to study its convergence criteria. In order to show the efficiency of the method, we consider some computational examples of this class of problem. These examples validate the accuracy of the method and show that it gives results which are convergent to the exact solutions. We prove that the method is accurate, fast and has a reasonable rate of convergence by computing some local and global error indicators.