arXiv subjects

Samuel Goodman

Publications and source records attributed to Samuel Goodman.

3 recordsLinked to original sources

arXiv2025

Regular Functions on Formal-Analytic Arithmetic Surfaces

In this paper, we show that for a broad class of pseudoconvex formal-analytic arithmetic surfaces over $\text{Spec}(\mathbb{Z})$, those which admit a nonconstant monic such regular function, that a conjecture of Bost-Charles that the ring of regular functions has continuum cardinality is implied by a purely complex-analytic conjecture. Under the conjecture, a Fekete-Szego-type approximation argument produces a polynomial "large" relative to the regular function, which in turn yields continuum many distinct regular functions. We also introduce a formula for the pushforward by a holomorphic function of the equilibrium Green's functions for our bordered Riemann surface with boundary, a formula which has constant term related to Arakelov degree.

math.CV↗

arXiv2024

On Galois Extensions of Local Fields with a Single Wild Ramification Jump

For a given positive integer $n$ and $K/\mathbb{Q}_p$ a finite extension of ramification degree $e$, we determine the number of finite Galois extensions $L/K$ with inertia degree $f$ and a single nonnegative ramification jump at $n$ as long as $(p,e)$ is outside of a finite set. This builds upon the tamely ramified case, which is a classical consequence of Serre's Mass Formula, exhibiting a more restrictive behavior than in the tamely ramified case because the degrees of such extensions are bounded. We do this by working in a fixed Lubin-Tate extension and exploiting the surjectivity of a map corresponding to the ramification jump to reconstruct the $U^1$ part of the norm subgroup (coming from local class field theory) from its fibers and then by understanding how the fibers interact by studying them in terms of properties of the formal logarithm and partitions.

math.NT↗

arXiv2023

On Power Sums and the Kummer Congruences

In this paper, we investigate the stabilizers of certain multisets $\mod p^k$ with respect to their natural multiplicative action, completely describing them for a certain family of polynomials whenever $p$ is an odd prime. This elucidates an underlying structure on the level of elements that yields a new proof of the classical Kummer Congruences. Hence the Kummer Congruences can be viewed as a consequence of putting together this local information, the interactions between which are understood. This gives a new perspective in that all known approaches are based directly on global considerations, namely either through global sum identities or $p$-adic integrals.

math.NT↗