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Let K be a complete, algebraically closed nonarchimedean valued field, and let φ(z) ∈ K(z) have degree d ≥ 2. We study how the resultant of φ varies under changes of coordinates. For γ ∈ GL₂(K), we show that the map $γ \mapsto o r d (R e s (φ^{γ}))$ factors through a function $o r d R e s_{φ} (\cdot)$ on the Berkovich projective line, which is piecewise affine and convex up. The minimal resultant is achieved either at a single point in $P ¹_{K}$ , or on a segment, and the minimal resultant locus is contained in the tree in $P ¹_{K}$ spanned by the fixed points and poles...

The mininima of positive definite Heranmitian binary quadratic forms

A. Oppenheim (1934)

Mathematische Zeitschrift

The modified diagonal cycle on the triple product of a pointed curve

Benedict H. Gross, Chad Schoen (1995)

Annales de l'institut Fourier

Let $X$ be a curve over a field $k$ with a rational point $e$ . We define a canonical cycle $Δ_{e} \in Z^{2} (X^{3})_{hom}$ . Suppose that $k$ is a number field and that $X$ has semi-stable reduction over the integers of $k$ with fiber components non-singular. We construct a regular model of $X^{3}$ and show that the height pairing $⟨ τ_{*} (Δ_{e}), τ_{*}^{'} (Δ_{e}) ⟩$ is well defined where $τ$ and $τ^{'}$ are correspondences. The paper ends with a brief discussion of heights and $L$ -functions in the case that $X$ is a modular curve.

The modular degree and the congruence number of a weight 2 cusp form

Alina Carmen Cojocaru, Ernst Kani (2004)

Acta Arithmetica

The Modular Symbol and Continued Fractions in Higher Dimensions.

Avner Ash, Lee Rudolph (1979)

Inventiones mathematicae

The Modular Torus has Maximal Length Spectrum.

P. Schmutz Schaller (1996)

Geometric and functional analysis

The module of a 2-component link.

J. Levine (1982)

Commentarii mathematici Helvetici

The module of vector-valued modular forms is Cohen-Macaulay

Richard Gottesman (2020)

Czechoslovak Mathematical Journal

Let $H$ denote a finite index subgroup of the modular group $Γ$ and let $ρ$ denote a finite-dimensional complex representation of $H .$ Let $M (ρ)$ denote the collection of holomorphic vector-valued modular forms for $ρ$ and let $M (H)$ denote the collection of modular forms on $H$ . Then $M (ρ)$ is a $ℤ$ -graded $M (H)$ -module. It has been proven that $M (ρ)$ may not be projective as a $M (H)$ -module. We prove that $M (ρ)$ is Cohen-Macaulay as a $M (H)$ -module. We also explain how to apply this result to prove that if $M (H)$ is a polynomial ring, then $M (ρ)$ is a free...

The Moduli Space of (3,3,3) Trilinear forms.

Kok Onn Ng (1995)

Manuscripta mathematica

The moments of infinite series.

Ka-Lam Kueh (1988)

Journal für die reine und angewandte Mathematik

The moments of partitions, I

L. Richmond (1975)

Acta Arithmetica

The moments of partitions, II

L. Richmond (1975)

Acta Arithmetica

The monoid of ordered partitions of a natural number.

T.G. Lavers (1996)

Semigroup forum

The monotone Poisson process

Alexander C. R. Belton (2006)

Banach Center Publications

The coefficients of the moments of the monotone Poisson law are shown to be a type of Stirling number of the first kind; certain combinatorial identities relating to these numbers are proved and a new derivation of the Cauchy transform of this law is given. An investigation is begun into the classical Azéma-type martingale which corresponds to the compensated monotone Poisson process; it is shown to have the chaotic-representation property and its sample paths are described.

The Mordell conjecture revisited

Enrico Bombieri (1990)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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