In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or family of those. It goes back to the studies of Fermat on what are now recognised as elliptic curves; and has become a very substantial area both in terms of results and conjectures. Most of these can be posed for an abelian variety A over a number field K; or more generally (for global fields or more general finitely-generated rings or fields).
Read more about Arithmetic Of Abelian Varieties: Integer Points On Abelian Varieties, Rational Points On Abelian Varieties, Heights, Reduction Mod p, L-functions, Complex Multiplication, Manin–Mumford Conjecture
Famous quotes containing the words arithmetic of, arithmetic and/or varieties:
“Under the dominion of an idea, which possesses the minds of multitudes, as civil freedom, or the religious sentiment, the power of persons are no longer subjects of calculation. A nation of men unanimously bent on freedom, or conquest, can easily confound the arithmetic of statists, and achieve extravagant actions, out of all proportion to their means; as, the Greeks, the Saracens, the Swiss, the Americans, and the French have done.”
—Ralph Waldo Emerson (1803–1882)
“O! O! another stroke! that makes the third.
He stabs me to the heart against my wish.
If that be so, thy state of health is poor;
But thine arithmetic is quite correct.”
—A.E. (Alfred Edward)
“Now there are varieties of gifts, but the same Spirit; and there are varieties of services, but the same Lord; and there are varieties of activities, but it is the same God who activates all of them in everyone.”
—Bible: New Testament, 1 Corinthians 12:4-6.