Melanie Matchett Wood

On the probabilities of local behaviors in abelian field extensions

For a number field K and a finite abelian group G , we determine the probabilities of various local completions of a random G -extension of K when extensions are ordered by conductor. In particular, for a fixed prime ℘ of K , we determine the probability that ℘ splits into r primes in a random G -extension of K that is unramified at ℘ . We find that these probabilities are nicely behaved and mostly independent. This is in analogy to Chebotarev’s density theorem, which gives the probability that in a fixed extension a random prime of K splits into r primes in the extension. We also give the asymptotics for the number of G -extensions with bounded conductor. In fact, we give a class of extension invariants, including conductor, for which we obtain the same counting and probabilistic results. In contrast, we prove that neither the analogy with the Chebotarev probabilities nor the independence of probabilities holds when extensions are ordered by discriminant.

P-orderings: a metric viewpoint and the non-existence of simultaneous orderings

Abstract For a prime ideal ℘ and a subset S of a Dedekind ring R, a ℘ -ordering of S is a sequence of elements of S with a certain minimizing property. These ℘ -orderings were introduced in Bhargava (J. Reine Angew. Math., 490 (1997) 101) to generalize the usual factorial function and many classical results were thereby extended, including results about integer-valued polynomials. We consider ℘ -orderings from the viewpoint of the ℘ -adic metric on R. We find that the ℘ -sequences of S depend only on the closure of S in R ℘ . When R is a Dedekind domain and R′ is the integral closure of R in a finite extension of the fraction field of R, we relate the ℘ -sequences of R and R′. Lastly, we investigate orderings that are simultaneously ℘ -orderings for all prime ideals ℘⊂R , and show that such simultaneous orderings do not exist for imaginary quadratic number rings.

Rings and ideals parameterized by binary n-ic forms

The association of algebraic objects to forms has had many important applications in number theory. Gauss, over two centuries ago, studied quadratic rings and ideals associated to binary quadratic forms, and found that ideal classes of quadratic rings are exactly parametrized by equivalence classes of integral binary quadratic forms. Delone and Faddeev, in 1940, showed that cubic rings are parametrized by equivalence classes of integral binary cubic forms. Birch, Merriman, Nakagawa, Corso, Dvornicich, and Simon have all studied rings associated to binary forms of degree n for any n, but it has not previously been known which rings, and with what additional structure, are associated to binary forms. In this paper, we show exactly what algebraic structures are parametrized by binary n-ic forms, for all n. The algebraic data associated to an integral binary n-ic form includes a ring isomorphic to

Gauss composition over an arbitrary base

\mathbb{Z}^n

Belyi-Extending Maps and the Galois Action on Dessins d'Enfants

as a

The distribution of points on superelliptic curves over finite fields

\mathbb{Z}

Quartic Rings Associated to Binary Quartic Forms

-module, an ideal class for that ring, and a condition on the ring and ideal class that comes naturally from geometry. In fact, we prove these parametrizations when any base scheme replaces the integers, and show that the correspondences between forms and the algebraic data are functorial in the base scheme. We give geometric constructions of the rings and ideals from the forms that parametrize them and a simple construction of the form from an appropriate ring and ideal.

The density of discriminants of ₃-sextic number fields

Abstract The classical theorems relating integral binary quadratic forms and ideal classes of quadratic orders have had important applications in number theory, and many authors have given extensions of these theorems to rings other than the integers. However, such extensions have always included hypotheses on the rings, and the theorems involve only binary quadratic forms satisfying further hypotheses. We give a complete statement of the relationship between binary quadratic forms and modules for quadratic algebras over any base ring, or in fact base scheme. The result includes all binary quadratic forms, and commutes with base change. We give global geometric as well as local explicit descriptions of the relationship between forms and modules.

Semiample Bertini theorems over finite fields

We study the absolute Galois group by looking for invariants and orbits of its faithful action on Grothendieck’s dessins d’enfants. We define a class of functions called Belyi-extending maps, which we use to construct new Galois invariants of dessins from previously known invariants. Belyi-extending maps are the source of “new-type” relations on the injection of the absolute Galois group into the ∆

A heuristic for the distribution of point counts for random curves over a finite field

We give the distribution of points on smooth superelliptic curves over a fixed finite field, as their degree goes to infinity. We also give the distribution of points on smooth m-fold cyclic covers of the line, for any m, as the degree of their superelliptic model goes to infinity. This builds on previous work of Kurlberg, Rudnick, Bucur, David, Feigon, and Lalin for p-fold cyclic covers, but the limits taken differ slightly and the resulting distributions are interestingly different.