In this expository paper, we survey the various approaches to compactifying moduli stacks of polarized abelian varieties. To motivate the different approaches to compactifying, we first discuss three different points of view of the moduli stacks themselves. Then we explain how each point of view leads to a different compactification. Throughout we emphasize maximal degenerations which capture m...

Let X1, . . . , Xm denote smooth projective curves of genus gi ≥ 2 over an algebraically closed field of characteristic 0 and let n denote any integer at least equal to 1 + max i=1 gi. We show that the product JX1 × · · · × JXm of the corresponding Jacobian varieties admits the structure of a Prym-Tyurin variety of exponent n. This exponent is considerably smaller than the exponent of the struc...

For our study of Faltings’ heights on abelian varieties over number fields, it will be convenient if we can compactify the “moduli space” of g-dimensional abelian varieties (over Z) such that the universal abelian scheme extends to a semi-abelian scheme over the compactification. The reason for this wish is that ideally we’d like to set up a theory for which the Faltings height of an abelian va...

1 Algebraic Groups 7 1.1 Group Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Restriction of Scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Algebraic Groups over a Nonalgebraically Closed Field K . . . . . . . . . . . 13 1.4 Structure of Algebraic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5 Topologizing G(R...

Let k be a totally real field, and let A/k be an absolutely irreducible, polarized Abelian variety of odd, prime dimension whose endomorphisms are all defined over k. Then the only strictly compatible families of abstract, absolutely irreducible representations of Gal(k/k) coming from A are tensor products of Tate twists of symmetric powers of two-dimensional λ-adic representations plus field a...

Walnut (Juglans regia L.), which is an important nut fruit species in Türkiye, grown different ecological conditions. Economically, production possible by carrying out adaptation studies of varieties. This study was conducted between 2021-2022.It aimed to determine the performance Bursa-95, Chandler, Pedro, Şebin, Topak, Yalova-1 and Yavuz walnut varieties Kale (Denizli) ecology. Accordingly, a...

We give a group theoretic definition of “local models” as sought after in the theory of Shimura varieties. These are projective schemes over the integers of a p-adic local field that are expected to model the singularities of integral models of Shimura varieties with parahoric level structure. Our local models are certain mixed characteristic degenerations of Grassmannian varieties; they are ob...

Comparative political economy is often based on the premise that the latter is true. In particular, the now classic Varieties of Capitalism (VofC) comparative approach postulates two types of institutional frameworks. The general idea was that each institutional solution generates positive externalities which may or may not be captured by specific solutions in other institutional domains. There...

In this paper, we generalize to the “log regular case” a result of de Jong and Oort which states that any morphism (satisfying certain conditions) from the complement of a divisor with normal crossings in a regular scheme to a moduli stack of stable curves extends over the entire regular scheme. The proof uses the theory of “regular log schemes ” – i.e., schemes with singularities like those of...

Theorem 1.1 (Chudnovsky’s theorems). Let Λ = Zω+Zω′ ⊂ C be a lattice with invariants g2, g3, Weierstrass functions ℘, ζ and quasi-periods η, η′, all defined in the usual way (see [Sil94]). Each of the sets below contains at least two algebraically independent numbers : first (1) {g2, g3, η ω , π ω}, (2) {g2, g3, ω, η, ω′, η′}; if we assume g2, g3 ∈ Q̄ : (3) { η ω , π ω}, (4) {ω, ω′, η, η′}; and ...