arXiv subjects

Simon Rubinstein-Salzedo

Publications and source records attributed to Simon Rubinstein-Salzedo.

23 records · Page 2Linked to original sources

arXiv2015

Identities for field extensions generalizing the Ohno-Nakagawa relations

In previous work, Ohno conjectured, and Nakagawa proved, relations between the counting functions of certain cubic fields. These relations may be viewed as complements to the Scholz reflection principle, and Ohno and Nakagawa deduced them as consequences of `extra functional equations' involving the Shintani zeta functions associated to the prehomogeneous vector space of binary cubic forms. In the present paper we generalize their result by proving a similar identity relating certain degree l fields with Galois groups D_l and F_l respectively, for any odd prime l, and in particular we give another proof of the Ohno-Nakagawa relation without appealing to binary cubic forms.

math.NT↗

arXiv2015

$N$-Division Points of Hypocycloids

We show that the $n$-division points of all rational hypocycloids are constructible with an unmarked straightedge and compass for all integers $n$, given a pre-drawn hypocycloid. We also consider the question of constructibility of $n$-division points of hypocycloids without a pre-drawn hypocycloid in the case of a tricuspoid, concluding that only the $1$, $2$, $3$, and $6$-division points of a tricuspoid are constructible in this manner.

math.NT↗

arXiv2014

Invariants for A4 fields and the Cohen-Lenstra heuristics

This article discusses deviations from the Cohen-Lenstra heuristics when roots of unity are present. In particular, we propose an explanation for the discrepancy between the observed number of cyclic cubic fields whose 2-class group is $C_2\times C_2$ and the number predicted by the Cohen-Lenstra heuristics, in terms of an invariant living in a quotient of the Schur multiplier group. We also show that, in some cases, the definition of the invariant can be simplified greatly, and we compute the invariant when the cubic field is ramified at exactly one prime, up to $10^8$.

math.NT↗

arXiv2014

Covers of Elliptic Curves with Unique, Totally Ramified Branch Points

A well-known and difficult problem in computational number theory and algebraic geometry is to write down equations for branched covers of algebraic curves with specified monodromy type. In this article, we present a technique for computing such covers in the case of totally ramified covers of elliptic curves branched at one point.

math.AG↗

arXiv2014

Period Computations for Covers of Elliptic Curves

In this article, we construct algebraic equations for a curve C and a map f to an elliptic curve E, with pre-specified branching data. We do this by determining certain relations that the periods of C and E must satisfy and use these relations to approximate the coefficients to high precision. We then conjecture which algebraic numbers the coefficients are, and then we prove this conjecture to be correct.

math.NT↗