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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 ∈ 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 moduli space of commutative algebras of finite rank

Bjorn Poonen (2008)

Journal of the European Mathematical Society

The moduli space of rank- n commutative algebras equipped with an ordered basis is an affine scheme 𝔅 n of finite type over ℤ , with geometrically connected fibers. It is smooth if and only if n ≤ 3 . It is reducible if n ≥ 8 (and the converse holds, at least if we remove the fibers above 2 and 3 ). The relative dimension of 𝔅 n is 2 27 n 3 + O ( n 8 / 3 ) . The subscheme parameterizing étale algebras is isomorphic to GL n / S n , which is of dimension only n 2 . For n ≥ 8 , there exist algebras that are not limits of étale algebras. The dimension calculations...

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