Hasse Principle For Algebraic Groups
The Hasse principle for algebraic groups states that if G is a simply-connected algebraic group defined over the global field k then the map from
is injective, where the product is over all places s of k.
The Hasse principle for orthogonal groups is closely related to the Hasse principle for the corresponding quadratic forms.
Kneser (1966) and several others verified the Hasse principle by case-by-case proofs for each group. The last case was the group E8 which was only completed by Chernousov (1989) many years after the other cases.
The Hasse principle for algebraic groups was used in the proofs of the Weil conjecture for Tamagawa numbers and the strong approximation theorem.
Read more about this topic: Hasse Principle
Famous quotes containing the words principle, algebraic and/or groups:
“Well, you Yankees and your holy principle about savin’ the Union. You’re plunderin’ pirates that’s what. Well, you think there’s no Confederate army where you’re goin’. You think our boys are asleep down here. Well, they’ll catch up to you and they’ll cut you to pieces you, you nameless, fatherless scum. I wish I could be there to see it.”
—John Lee Mahin (1902–1984)
“I have no scheme about it,—no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?—and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?”
—Henry David Thoreau (1817–1862)
“Under weak government, in a wide, thinly populated country, in the struggle against the raw natural environment and with the free play of economic forces, unified social groups become the transmitters of culture.”
—Johan Huizinga (1872–1945)