Related papers: On the genera of X_0(N)

A seminal theorem due to Weyl states that if (a_n) is any sequence of distinct integers, then, for almost every real number x, the sequence (a_n x) is uniformly distributed modulo one. In particular, for almost every x in the unit interval,…

We consider sets of positive integers containing no sum of two elements in the set and also no product of two elements. We show that the upper density of such a set is strictly smaller than 1/2 and that this is best possible. Further, we…

We show that the Galois group of a random monic polynomial %of degree $d>12$ with integer coefficients between $-N$ and $N$ is NOT $S_d$ with probability $\ll \frac{\log^{\Omega(d)}N}{N}.$ Conditionally on NOTbeing the full symmetric group,…

We study the probabilistic properties of the Greatest Increase Grid (GIG) digraph. We compute the probability of a particular sequence of directed edges connecting two random vertices. We compute the joint probability that a set of vertices…

Suppose l=2m+1, m>0. We introduce m "theta-series", [1],...,[m], in Z/2[[x]]. It has been conjectured that the n for which the coefficient of x^n in 1/[i] is 1 form a set of density 0. This is probably always false, but in certain cases,…

Let $N$ be a non-squarefree integer such that the quotient $X_0(N)^*$ of the modular curve $X_0(N)$ by the full group of Atkin-Lehner involutions has positive genus. Elkies conjectures that the rational points on $X_0(N)^*$ are only cusps…

Let $g(n)$ be the largest positive integer $k$ such that there are distinct primes $p_i$ for $1\leq i\leq k$ so that $p_i |n+i$. This function is related to a celebrated conjecture of C.A. Grimm. We establish upper and lower bounds for…

We define a sequence of positive integers recursively, where each term is determined as follows: starting with a given positive integer, if the term is odd, the next is the sum of its positive divisors; if the term is even, the subsequent…

An integer partition of $n$ is called graphical if its parts form a degree sequence of a simple graph. While unrestricted graphical partitions have been extensively studied, much less is known when the parts are restricted to a prescribed…

Let $X$ be a scheme of finite type over $\mathbf{Z}$. For $p \in \mathcal{P}$ the set of prime numbers, let $N_{X}(p)$ be the number of $\mathbf{F}_{p}$-points of $X/\mathbf{F}_{p}$. For fixed $n\geq 1$ and $a_{1}, \ldots, a_{n} \in…

Consider a random multigraph G* with given vertex degrees d_1,...,d_n, contructed by the configuration model. We show that, asymptotically for a sequence of such multigraphs with the number of edges (d_1+...+d_n)/2 tending to infinity, the…

For a given sequence of weights (non-negative numbers), we consider partitions of the positive integer n. Each n-partition is selected uniformly at random from the set of all such partitions. Under a classical scheme of assumptions on the…

Integer sequences where each element is determined by a previous randomly chosen element are investigated analytically. In particular, the random geometric series x_n=2x_p with 0<=p<=n-1 is studied. At large n, the moments grow…

By some extremely simple arguments, we point out the following: (i) If n is the least positive k-th power non-residue modulo a positive integer m, then the greatest number of consecutive k-th power residues mod m is smaller than m/n. (ii)…

We resolve a 1983 question of Serre by constructing curves with many points of every genus over every finite field. More precisely, we show that for every prime power q there is a positive constant c_q with the following property: for every…

We describe the birational geometry of the moduli space S_g^{-} of odd spin curves (theta-characteristics) for all genera g. The odd spin moduli space is a uniruled variety for g<12, and of general type for g at least 12. Furthermore, for…

Suppose that we are given an infinite binary sequence which is random for a Bernoulli measure of parameter $p$. By the law of large numbers, the frequency of zeros in the sequence tends to~$p$, and thus we can get better and better…

A theorem of Green says that a line bundle of degree at least $2g+1+p$ on a smooth curve $X$ of genus $g$ has property $N_p$. We prove a similar conclusion for certain singular, reducible curves $X$ under suitable degree bounds over all…

In this paper we compute the degree of a curve which is the image of a mapping $z\longmapsto (f(z): g(z): h(z))$ constructed out of three linearly independent modular forms of the same even weight $\ge 4$ into $\mathbb P^2$. We prove that…

Let A be a matrix whose entries are real i.i.d. centered random variables with unit variance and suitable moment assumptions. Then the smallest singular value of A is of order n^{-1/2} with high probability. The lower estimate of this type…