On the punctured torus the number of essential self-intersections of a homotopy class of closed curves is bounded (sharply) by a quadratic function of its combinatorial length (the number of letters required for its minimal description in terms of the two generators of the fundamental group and their inverses). We show that if a homotopy class has combinatorial length $L$, then its number of essential self-intersections is bounded by $(L-2)^{2}/4$ if $L$ is even, and $(L-1)(L-3)/4$ if $L$ is odd. The classes attaining this bound can be explicitly described in terms of the generators; there are $(L-2)^2+ 4$ of them if $L$ is even, and $2(L-1)(L-3)+8$ if $L$ is odd. Similar descriptions and counts are given for classes with self-intersection number equal to one less than the bound. Proofs use both combinatorial calculations and topological operations on representative curves.

Computer-generated data are tabulated by counting for each nonnegative integer how many length-$L$ classes have that self-intersection number, for each length $L$ less than or equal to $13$. Such experiments led to the results above. Experimental data are also presented for the pair-of-pants surface.

Let $L(E/\Q,s)$ be the $L$-function of an elliptic curve $E$ defined over the rational field $\Q$. We examine special values of the derivatives $L^{\prime}(E,1,\chi)$ of twists by Dirichlet characters of $L(E/\Q,s)$ when $L(E,1,\chi)=0$.

In families of Painlevé VI differential equations having common algebraic solutions we classify all the members that come from geometry, i.e., the corresponding linear differential equations that are Picard--Fuchs associated to families of algebraic varieties. In our case, we have one family with zero-dimensional fibers and all others are families of curves. We use the classification of families of elliptic curves with four singular fibers carried out by Herfurtner in 1991 and generalize the results of Doran in 2001 and Ben Hamed and Gavrilov in 2005.

The kissing number in $n$-dimensional Euclidean space is the maximal number of nonoverlapping unit spheres that simultaneously can touch a central unit sphere. Bachoc and Vallentin developed a method to find upper bounds for the kissing number based on semidefinite programming. This paper is a report on high-accuracy calculations of these upper bounds for $n \leq 24$. The bound for $n = 16$ implies a conjecture of Conway and Sloane: there is no $16$-dimensional periodic sphere packing with average theta series $1 + 7680q^3 + 4320q^4 + 276480q^5 + 61440q^6 + \cdots.$

For diagonal cubic surfaces $S$, we study the behavior of the height $\m(S)$ of the smallest rational point versus the Tamagawa-type number $\tau(S)$ introduced by E. Peyre. We determined both quantities for a sample of $849{,}781$ diagonal cubic surfaces. Our methods are explained in some detail. The results suggest an inequality of the type $\smash{\m(S) < C(\varepsilon)/\tau(S)^{1+\varepsilon}}$. We conclude the article with the construction of a sequence of diagonal cubic surfaces showing that the inequality $\m(S) < C/\tau(S)$ is false in general.

We prove weak forms of Kato's ${\rm K}_1$-congruences for elliptic curves with complex multiplication, subject to two technical hypotheses. We next use "Magma" to calculate the $\mu$-invariant measuring the discrepancy between the "motivic'' and "automorphic'' {$p$-adic} $L$-functions. Via the two-variable main conjecture, one can then estimate growth in this $\mu$-invariant using arithmetic of the $\Z_p^2$-extension.

In this paper we enumerate and classify the "simplest'' pairs $(M,G)$, where $M$ is a closed orientable $3$-manifold and $G$ is a trivalent graph embedded in $M$.

To enumerate the pairs we use a variation of Matveev's definition of complexity for $3$-manifolds, and we consider only $(0,1,2)$-irreducible pairs, namely pairs $(M,G)$ such that any $2$-sphere in $M$ intersecting $G$ transversely in at most two points bounds a ball in $M$ either disjoint from $G$ or intersecting $G$ in an unknotted arc. To classify the pairs, our main tools are geometric invariants defined using hyperbolic geometry. In most cases, the graph complement admits a unique hyperbolic structure with parabolic meridians; this structure was computed and studied using Heard's program "Orb" and Goodman's program "Snap".

We determine all $(0,1,2)$-irreducible pairs up to complexity $5$, allowing disconnected graphs but forbidding components without vertices in complexity $5$. The result is a list of $129$ pairs, of which $123$ are hyperbolic with parabolic meridians. For these pairs we give detailed information on hyperbolic invariants including volumes, symmetry groups, and arithmetic invariants. Pictures of all hyperbolic graphs up to complexity $4$ are provided. We also include a partial analysis of knots and links.

The theoretical framework underlying the paper is twofold, being based on Matveev's theory of spines and on Thurston's idea (later developed by several authors) of constructing hyperbolic structures via triangulations. Many of our results were obtained (or suggested) by computer investigations.

In this article we calculate explicitly the Ramanujan $(13,13)$-regular hypergraph introduced in Sarveniazi, "Explicit Construction of a Ramanujan $(n\sb1,n\sb 2,\dotsc,n\sb {d-1})$-Regular Hypergraph,'' Duke Math. J. 139:1 (2007), 141--171 using the computer algebra programs "Magma" and "Octave". This simple structure represents a nice and arithmetically very rich object in number theory, namely $\Gamma_f\setminus\Gamma(1)$.

In Brunier and Bundschuh, “On Borcherds Products Associated with Lattices of Prime Discriminant.” Ramanujan Journal 7 (2003), 49–61, the authors use Borcherds lifts to obtain Hilbert modular forms. Another approach is to calculate Hilbert modular forms using the Jacquet--Langlands correspondence, which was implemented by Lassina Dembele in "Magma". In Mayer, "Rings of Hilbert Modular Forms for the Fields $\Q(\sqrt{13})$ and $\Q(\sqrt{17})$,'' To appear, 2009, we use Brunier and Bundschuh to determine the rings of Hilbert modular forms for $\{Q}(\sqrt{13})$ and $\{Q}(\sqrt{17})$. In the present note we give the major calculational details and present some results for $\{K}=\{Q}(\sqrt{5})$, $\{K}=\{Q}(\sqrt{13})$, and $\K=\{Q}(\sqrt{17})$. For calculations in the ring $\{o}$ of integers of $\{K}$ we order $\{o}$ by the norm of its elements and get for fixed norm, modulo multiplication by $\pm \varepsilon_0^{2\Z}$, a finite set. We use this decomposition to describe Weyl chambers and their boundaries, to determine the Weyl vector of Borcherds products, and hence to calculate Borcherds products. As a further example we calculate Fourier expansions of Eisenstein series.