In this paper we formulate a conjecture that partially generalizes the Gross-Kohnen-Zagier theorem to higher-weight modular forms. For $f \in S_{2k}(N)$ satisfying certain conditions, we construct a map from the Heegner points of level $N$ to a complex torus $\mathbb{C}/L_f$ defined by $f$. We define higher-weight analogues of Heegner divisors on $\mathbb{C}/L_f$.

We conjecture that they all lie on a line and that their positions are given by the coefficients of a certain Jacobi form corresponding to $f$. In weight 2, our map is the modular parameterization map (restricted to Heegner points), and our conjectures are implied by Gross-Kohnen-Zagier. For any weight, we expect that our map is the Abel-Jacobi map on a certain modular variety, and so our conjectures are consistent with the conjectures of Beilinson-Bloch. We have verified that our map is the Abel-Jacobi map for weight 4. We provide numerical evidence to support our conjecture for a variety of examples.

For a given integer $A$ and various families of elliptic curves over finite fields, we compare the number of occurrences of $A$ with the number of occurrences of $-A$ as the trace of Frobenius in the family.

We give explicit numerical values with 100 decimal digits for the Mertens constant involved in the asymptotic formula for $\sum_{\substack{p\le x\\ p\equiv a \operatorname{mod} q}} 1/p$ and, as a byproduct, for the Meissel-Mertens constant defined as $\sum_{p\equiv a \operatorname{mod}q} (\log(1-1/p) + 1/p)$, for $q \in \{3,\dots,100\}$ and $(q, a) = 1$. The complete set of results is available online (http://www.math.unipd.it/~languasc/Mertens-comput.html).

Normal and almost normal surfaces are essential tools for algorithmic 3-manifold topology, but to use them requires exponentially slow enumeration algorithms in a high-dimensional vector space. The quadrilateral coordinates of Tollefson alleviate this problem considerably for normal surfaces by reducing the dimension of this vector space from $7n$ to $3n$ (where $n$ is the complexity of the underlying triangulation). Here we develop an analogous theory for octagonal almost normal surfaces, using quadrilateral and octagon coordinates to reduce this dimension from $10n$ to $6n$. As an application, we show that quadrilateral– octagon coordinates can be used exclusively in the streamlined 3-sphere recognition algorithm of Jaco, Rubinstein, and Thompson, reducing experimental running times by factors of thousands. We also introduce joint coordinates, a system with only $3n$ dimensions for octagonal almost normal surfaces that has appealing geometric properties.

We develop algorithms for computations of Green’s function and its Fourier coefficients, $F_n(z;s)$, on Fuchsian groups with one cusp. An analogue of a Rankin-Selberg bound for $F_n(z;s)$ is presented.

We present some examples of numerical investigations of the value distribution of Green’s function and of its Fourier coefficients on the modular group $\mathrm{PSL}(2, \mathbb{Z})$. Our results indicate that both Green’s function $G_s(z;w)$ and its Fourier coefficients $F_n(z;s)$ have a Gaussian value distribution in the semiclassical limit when $\operatorname{Re} s = 1/2$.

This study is a follow-up to two recent works: [La Bretèche et Tenenbaum 05] and [Martin et Tenenbaum 10]. The former provides a friable (i.e., with respect to integers free of large prime factors) extension of the classical Turán–Kubilius inequality, while the latter furnishes a theoretical method for sharp evaluation of the involved constants. Here, we complement these investigations with a numerical study of the friable Turán–Kubilius constants, thereby supplying an effective, quantitative measure of the discrepancy between probabilistic number theory and its probabilistic model.

Cette étude fait suite à deux travaux récents : [La Bretèche et Tenenbaum 05] et [Martin et Tenenbaum 10]. Le premier fournit une extension friable (i.e., relative aux entiers sans grand facteur premier) de l’inégalité classique de Turán–Kubilius, tandis que le second propose une méthode théorique pour déterminer les constantes optimales impliquées. Ici, nous complétons ces recherches par une étude numérique des constantes friables de Turán–Kubilius, fournissant ainsi une mesure effective et quantitative de la discrépance entre la théorie probabiliste des nombres et son modèle probabiliste.

A matching on a set $X$ is a collection of pairwise disjoint subsets of $X$ of size two. Using computers, we analyze the integral homology of the matching complex ${\sf M}_n$, which is the simplicial complex of matchings on the set $\{1,\dots, n\}$. The main result is the detection of elements of order $p$ in the homology for $p \in \{5, 7, 11, 13\}$. Specifically, we show that there are elements of order $5$ in the homology of ${\sf M}_n$ for $n \ge 18$ and for $n \in \{14,16\}$. The only previously known value was $n = 14$, and in this particular case we have a new computer-free proof. Moreover, we show that there are elements of order 7 in the homology of ${\sf M}_n$ for all odd $n$ between $23$ and $41$ and for $n = 30$. In addition, there are elements of order $11$ in the homology of ${\sf M}_{47}$ and elements of order $13$ in the homology of ${\sf M}_{62}$. Finally, we compute the ranks of the Sylow 3- and 5-subgroups of the torsion part of $\tilde{H}_d({\sf M}_n; \mathbb{Z})$ for $13 \le n \le 16$; a complete description of the homology already exists for $n \le 12$. To prove the results, we use a representation-theoretic approach, examining subcomplexes of the chain complex of Mn obtained by letting certain groups act on the chain complex.