Canonical Models and Special Points
Each Shimura variety can be defined over a canonical number field E called the reflex field. This important result due to Shimura shows that Shimura varieties, which a priori are only complex manifolds, have an algebraic field of definition and, therefore, arithmetical significance. It forms the starting point in his formulation of the reciprocity law, where an important role is played by certain arithmetically defined special points.
The qualitative nature of the Zariski closure of sets of special points on a Shimura variety is described by the André-Oort conjecture. Conditional results have been obtained on this conjecture, assuming a Generalized Riemann Hypothesis.
Read more about this topic: Shimura Variety
Famous quotes containing the words canonical, models, special and/or points:
“If God bestowed immortality on every man then when he made him, and he made many to whom he never purposed to give his saving grace, what did his Lordship think that God gave any man immortality with purpose only to make him capable of immortal torments? It is a hard saying, and I think cannot piously be believed. I am sure it can never be proved by the canonical Scripture.”
—Thomas Hobbes (1579–1688)
“French rhetorical models are too narrow for the English tradition. Most pernicious of French imports is the notion that there is no person behind a text. Is there anything more affected, aggressive, and relentlessly concrete than a Parisan intellectual behind his/her turgid text? The Parisian is a provincial when he pretends to speak for the universe.”
—Camille Paglia (b. 1947)
“History repeats itself, but the special call of an art which has passed away is never reproduced. It is as utterly gone out of the world as the song of a destroyed wild bird.”
—Joseph Conrad (1857–1924)
“It’s my feeling that God lends you your children until they’re about eighteen years old. If you haven’t made your points with them by then, it’s too late.”
—Betty Ford (b. 1918)