Robert Rumely

Primes in arithmetic progressions

Strengthening work of Rosser, Schoenfeld, and McCurley, we establish explicit Chebyshev-type estimates in the prime number theorem for arithmetic progressions, for all moduli k ≤ 72 and other small moduli.

Harmonic Analysis on Metrized Graphs

This paper studies the Laplacian operator on a metrized graph, and its spectral theory.

Numerical computations concerning the ERH

This paper describes a computation which established the ERH to height 10000 for all primitive Dirichlet L-series with conductor Q < 13, and to height 2500 for all Q < 72, all composite Q < 112, and other moduli. The computations were based on Euler-Maclaurin summation. Care was taken to obtain mathematically rigorous results: the zeros were first located within 10-12, then rigorously separated using an interval arithmetic package. A generalized Turing Criterion was used to show there were no zeros off the critical line. Statistics about the spacings between zeros were compiled to test the Pair Correlation Conjecture and GUE hypothesis.

Transfinite diameter and the resultant

We prove a formula for the Fekete-Leja transfinite diameter of the pullback of a set E ⊂ℂ N by a regular polynomial map F, expressing it in terms of the resultant of the leading part of F and the transfinite diameter of E. We also establish the nonarchimedean analogue of this formula. A key step in the proof is a formula for the transfinite diameter of the filled Julia set of F.

A new equivariant in nonarchimedean dynamics

Let K be a complete, algebraically closed nonarchimedean valued field, and let φ(z) ∈ K(z) have degree d ≥ 2. We show there is a natural way to assign nonnegative integer weights wφ(P ) to points of the Berkovich projective line over K in such a way that ∑ P wφ(P ) = d− 1. When φ has bad reduction, the set of points with nonzero weight forms a distributed analogue of the unique point which occurs when φ has potential good reduction. Using this, we characterize the Minimal Resultant Locus of φ in dynamical and moduli-theoretic terms: dynamically, it is the barycenter of the weight-measure associated to φ; moduli-theoretically, it is the closure of the set of points where φ has semi-stable reduction, in the sense of Geometric Invariant Theory. Let K be a complete, algebraically closed nonarchimedean valued field with absolute value | · | and associated valuation ord(·). Write O for the ring of integers of K, M for its maximal ideal, and k̃ for its residue field. Let φ(z) ∈ K(z) be a function with degree d ≥ 2. Suppose (F,G) is a normalized representation for φ: a pair of homogeneous functions F (X, Y ), G(X, Y ) ∈ O[X, Y ] of degree d, such that φ(z) = F (z, 1)/G(z, 1) and some coefficient of F or G belongs to O. Such a pair (F,G) is unique up to scaling by a unit. Let Res(F,G) be the homogeneous resultant of F and G; then ordRes(φ) := ord(Res(F,G)) is well-defined and non-negative. Let P1K be the Berkovich projective line over K: a compact, uniquely path-connected Hausdorff space which contains P(K) as a dense subset. By Berkovich’s classification theorem, points of PK\P (K) correspond to discs D(a, r) ⊂ K, or to nested sequences of discs; points corresponding to discs with radius r ∈ |K| are said to be of type II. The point ζG corresponding to D(0, 1) is called the Gauss point. The natural action of GL2(K) on P(K) extends continuously to PK , and GL2(K) acts transitively on type II points. If γ ∈ GL2(K), we denote the conjugate γ ◦φ◦γ by φ. In [15] it is shown that the map γ 7→ ordRes(φ) factors through a function ordResφ(·) on P 1 K, given on type II points by ordResφ(γ(ζG)) = ordRes(φ ) . By ([15], Theorem 0.1) ordResφ(·) is piecewise affine and convex upward on paths, and takes the value ∞ on P(K). It achieves a minimum on PK . The set MinResLoc(φ) where the minimum occurs is called the Minimal Resultant Locus of φ. It is either a single type II point, or a closed segment joining two type II points. Date: February 23, 2014. 2000 Mathematics Subject Classification. Primary 37P50, 11S82; Secondary 37P05, 11Y40, 11U05.

Capacity theory and arithmetic intersection theory

We show that the sectional capacity of an adelic subset of a projective variety over a number field is a quasi-canonical limit of arithmetic top self-intersection numbers, and we establish the functorial properties of extremal plurisubharmonic Greens functions. We also present a conjecture that the sectional capacity should be a top self-intersection number in an appropriate adelic arithmetic intersection theory.

A formula for the grössencharacter of a parametrized elliptic curve

Abstract A formula for the grossencharacter of an elliptic curve with complex multiplication, in a family parametrized by modified Weierstrass functions or classical theta-functions, is given. The method is based on Shimuras Reciprocity Law for modular functions, and applies to Legendre, Jacobi, and Hesse curves. As an application, the conductors of the CM curves in these families are determined.

Well-Adjusted Models for Curves over Dedekind Rings

Let O be an excellent Dedekind ring with perfect residue fields, and let Y = Spec(O). Let C be a curve over Y. (For precise definitions, see §1; we assume C has a smooth geometrically irreducible general fibre, but we do not assume C is regular or complete). In this paper we will prove a relative minimal models Theorem, and a variant due to M. Artin of the Deligne-Mumford stable reduction Theorem. To state these results, let M(C) be the set of regular curves C′ for which there is a proper birational Y-morphism C′ → C. Let ≥ be the partial order on M(C) defined by C′ ≥ C″ if there is a proper morphism C′ → C″ over C.