Exact formulas for the number of partitions with parts in a finite set are proved without the use of partial fractions. Corresponding formulas with parts outside of a finite set are found. Several numerical examples are given.

We consider the Bloch-Kato conjecture applied to the symmetric square L-function of an elliptic curve over $$, at $$. In particular, we use a construction of elements of order l in a generalised Shafarevich-Tate group, which works when E has a rational point of infinite order and a rational point of order l. The existence of the latter places us in a situation where the recent theorem of Diamond, Flach, and Guo does not apply, but we find that the numerical evidence is quite convincing.

As a complement to a recent paper by Jade Vinson we study the distribution of the sequence $$ modulo 1 with the aim of explaining its different behaviour when $$ and when $$. We tackle this question from a different point of view using the theory of uniformly distributed sequences.

Let p be an odd prime number which splits into two distinct primes in an imaginary quadratic field K. Then K has certain kinds of noncyclotomic $$-extensions which are constructed through ray class fields with respect to a prime ideal lying above p. We try to show that Iwasawa invariants $$ and $$ both vanish for these specfic noncyclotomic $$-extensions.

In order to deal with nonhyperbolic fixed points of a given real iteration function g, we construct new iteration functions C which will be called combined. When a nonhyperbolic fixed point of g becomes a super attractor fixed point of C, the iteration function C is called flat.

Some flat iteration functions are constructed based on Newton's iteration function. Several numerical examples illustrating the good properties of flat iteration functions are presented.

We review previous methods of computing the modular degree of an elliptic curve, and present a new method (conditional in some cases), which is based upon the computation of a special value of the symmetric square L-function of the elliptic curve. Our method is sufficiently fast to allow large-scale experiments to be done. The data thus obtained on the arithmetic character of the modular degree show two interesting phenomena. First, in analogy with the class number in the number field case, there seems to be a Cohen--Lenstra heuristic for the probability that an odd prime divides the modular degree. Secondly, the experiments indicate that {\small $$} should always divide the modular degree, where r is the Mordell--Weil rank of the elliptic curve. We also discuss the size distribution of the modular degree, or more exactly of the special L-value which we compute, again relating it to the number field case.

We give an effective version of a result reported by Serre asserting that the images of the Galois representations attached to an abelian surface with $$ are as large as possible for almost every prime. Our algorithm depends on the truth of Serre's conjecture for two-dimensional odd irreducible Galois representations. Assuming this conjecture, we determine the finite set of primes with exceptional image. We also give infinite sets of primes for which we can prove (unconditionally) that the images of the corresponding Galois representations are large. We apply the results to a few examples of abelian surfaces.

We study computational aspects of amoebas associated with varieties in $$, both from an exact and from an experimental point of view. In particular, we give explicit characterizations for the amoebas of classes of linear and nonlinear varieties and present homotopy-based techniques to compute the boundary of two-dimensional amoebas.

Pursuant to the authors' previous chaotic-dynamical model for random digits of fundamental constants, we investigate a complementary, statistical picture in which pseudorandom number generators (PRNGs) are central. Some rigorous results are achieved: We establish b-normality for constants of the form $$ for certain sequences $$ of integers. This work unifies and extends previously known classes of explicit normals. We prove that for coprime $$ the constant $$ is b-normal, thus generalizing the Stoneham class of normals. Our approach also reproves b-normality for the Korobov class $$, for which the summation index n above runs instead over powers $$ with $$. Eventually we describe an uncountable class of explicit normals that succumb to the PRNG approach. Numbers of the $$ classes share with fundamental constants such as $$ the property that isolated digits can be directly calculated, but for these new classes such computation tends to be surprisingly rapid. For example, we find that the googol-th (i.e., $$-th) binary bit of $$ is 0. We also present a collection of other results---such as digit-density results and irrationality proofs based on PRNG ideas---for various special numbers.

All plane curves of degree less than 7 with coefficients in $$ are examined for curves with a large number of $$ rational points on their smooth model, for $$. Known lower bounds are improved, and new curves are found meeting or close to Serre's, Lauter's, and Ihara's upper bounds for the maximal number of $$ rational points on a curve of genus g.

We give a global description of the Frobenius for the division fields of an elliptic curve E that is strictly analogous to the cyclotomic case. This is then applied to determine the splitting of a prime p in a subfield of such a division field. These subfields include a large class of nonsolvable quintic extensions and our application provides an arithmetic counterpart to Klein's "solution" of quintic equations using elliptic functions. A central role is played by the discriminant of the ring of endomorphisms of the reduced curve modulo p.

In this paper, we study simple modules for a reduced enveloping algebra $$ in the critical case when $$ is "nilpotent.'' Some dimension formulas computed by Jantzen suggest modified versions of Weyl's dimension formula, based on certain reflecting hyperplanes for the affine Weyl group which might be associated to Kazhdan--Lusztig cells.