Explicit Resolutions of Cubic Cusp Singularities

Resolutions of cusp singularities are crucial to many techniques in computational number theory, and therefore finding explicit resolutions of these singularities has been the focus of a great deal of research. This paper presents an implementation of a sequence of algorithms leading to explicit resolutions of cusp singularities arising from totally real cubic number fields. As an example, the implementation is used to compute values of partial seta functions associated to these cusps

Computing the Arithmetic Genus of Hilbert Modular Fourfolds

The Hilbert modular fourfold determined by the totally real quartic number field k is a desingularization of a natural compactification of the quotient space Gamma(k)\H-4, where Gamma(k) = PSL2(O-k) acts on H-4 by fractional linear transformations via the four embeddings of k into R. The arithmetic genus, equal to one plus the dimension of the space of Hilbert modular cusp forms of weight (2, 2, 2, 2), is a birational invariant useful in the classification of these varieties. In this work, we describe an algorithm allowing for the automated computation of the arithmetic genus and give sample results

Computing the Arithmetic Genus of Hilbert Modular Fourfolds

The Hilbert modular fourfold determined by the totally real quartic number field k is a desingularization of a natural compactification of the quotient space Gamma(k)\H-4, where Gamma(k) = PSL2(O-k) acts on H-4 by fractional linear transformations via the four embeddings of k into R. The arithmetic genus, equal to one plus the dimension of the space of Hilbert modular cusp forms of weight (2, 2, 2, 2), is a birational invariant useful in the classification of these varieties. In this work, we describe an algorithm allowing for the automated computation of the arithmetic genus and give sample results

Automatic Realizability of Galois Groups of Order 16

This article examines the realizability of small groups of order 2(k), k less than or equal to 4, as Galois groups over arbitrary fields of characteristic not 2. In particular we consider automatic realizability of certain groups given the realizability of others

Automatic Realizability of Galois Groups of Order 16

This article examines the realizability of small groups of order 2(k), k less than or equal to 4, as Galois groups over arbitrary fields of characteristic not 2. In particular we consider automatic realizability of certain groups given the realizability of others

Augmented generalized happy functions

An augmented happy function, $S_{[c,b]}$ maps a positive integer to the sum of the squares of its base-$b$ digits and a non-negative integer $c$. A positive integer $u$ is in a cycle of $S_{[c,b]}$ if, for some positive integer $k$, $S_{[c,b]}^k(u) = u$ and for positive integers $v$ and $w$, $v$ is $w$-attracted for $S_{[c,b]}$ if, for some non-negative integer $\ell$, $S_{[c,b]}^\ell(v) = w$. In this paper, we prove that for each $c\geq 0$ and $b \geq 2$, and for any $u$ in a cycle of $S_{[c,b]}$, (1) if $b$ is even, then there exist arbitrarily long sequences of consecutive $u$-attracted integers and (2) if $b$ is odd, then there exist arbitrarily long sequences of 2-consecutive $u$-attracted integers