In this paper, we discuss the properties of curves of the form y3 = f(x) over a given field K of characteristic different from 3. If f(x) satisfies certain properties, then the Jacobian of such a curve is isomorphic to the ideal class group of the maximal order in the corresponding function field. We seek to make this connection concrete and then use it to develop an explicit arithmetic for the...

Good, but does an inverse g of f have to exist? A necessary condition is that f(1) 6= 0. Indeed, if g is the inverse of f , then 1 = I(1) = (f ∗ g)(1) = f(1)g(1). We now show that this necessary condition is also sufficient. We assume that f(1) 6= 0 and we try and solve for g. What this means is that we are solving for infinitely many unknowns: g(1), g(2), . . . . From the above, we see that th...

The purpose of this paper is to study the arithmetic function f : Z+ → Q ∗ + defined by f(2l) = l (∀k, l ∈ N, l odd). We have, for example, f(1) = 1, f(2) = 1, f(3) = 3, f(12) = 1 3 , f(40) = 1 25 , . . . , so it is clear that f(n) is not always an integer. However, we will show in what follows that f satisfies the property that the product of the f(r) for 1 ≤ r ≤ n is always an integer, and it...

In celebration of G.E. Andrews' 60 th birthday.

Let φ denote Euler’s totient function. A century-old conjecture of Carmichael asserts that for every n, the equation φ(n) = φ(m) has a solution m 6= n. This suggests defining F (n) as the number of solutions m to the equation φ(n) = φ(m). (So Carmichael’s conjecture asserts that F (n) ≥ 2 always.) Results on F are scattered throughout the literature. For example, Sierpiński conjectured, and For...

provided x is sufficiently large. An asymptotic formula of type (1) Df (x; q, a) = (1 +O((log x)))Df (x; q) , in which the error term is smaller than the main term by a suitable power of log x, is good enough for basic applications. More important than the size of the error term is the range where (1) holds uniformly with respect to the modulus q. In this paper we consider the problem for the d...

— We introduce the positive intersection product in Arakelov geometry and prove that the arithmetic volume function is continuously differentiable. As applications, we compute the distribution function of the asymptotic measure of a Hermitian line bundle and several other arithmetic invariants. Résumé. — On introduit le produit d’intersection positive en géométrie d’Arakelov et on démontre que ...

Although there is a vast literature on the properties of p(n), typically motivated by work of Ramanujan, some of the simplest questions remain open. For example, little is known about p(n) modulo 2 and 3. Most results concerning the congruence properties of p(n) have been proved using properties of (1.1). Theorems typically depend on q-series identities, the theory of modular equations, or the ...

let $f$ be a finite extension of $bbb q$‎, ‎${bbb q}_p$ or a global‎ ‎field of positive characteristic‎, ‎and let $e/f$ be a galois extension‎. ‎we study the realization fields of‎ ‎finite subgroups $g$ of $gl_n(e)$ stable under the natural‎ ‎operation of the galois group of $e/f$‎. ‎though for sufficiently large $n$ and a fixed‎ algebraic number field $f$ every its finite extension $e$ is‎ ‎re...