# A description of an abelian variety as defined by equations having coefficients in any field

Only in did he prove that complete algebraic groups can be embedded into projective space. I[ edit ] Igusa zeta-function An Igusa zeta-functionnamed for Jun-ichi Igusais a generating function counting numbers of points on an algebraic variety modulo high powers pn of a fixed prime number p.

An abelian scheme over a base scheme S of relative dimension g is a propersmooth group scheme over S whose geometric fibers are connected and of dimension g. Given a curve with equation defined over the number field, we can apply this map to the coefficients to get a curve defined over some finite field, where the choices of finite field correspond to the finite primes of the number field.

It became a part of the Langlands programa list of important conjectures needing proof or disproof. Fermat[ edit ] Fermat's last theorem was first conjectured by Pierre de Fermat infamously in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin.

Hilbert[ edit ] David Hilbert unified the field of algebraic number theory with his treatise Zahlbericht literally "report on numbers".

An elliptic curve is an abelian variety of dimension 1. A polarisation induces a Rosati involution on the endomorphism ring E. This generalises the Weil pairing for elliptic curves. In the early s, Weil used the first definition over an arbitrary base field but could not at first prove that it implied the second.

When n and p are not coprime, the same result can be recovered provided one interprets it as saying that the n-torsion defines a finite flat group scheme of rank 2g. Only in did he prove that complete algebraic groups can be embedded into projective space.

Generally that is for a large number of variables, when the degree of an equation is held fixed. The word "Ring", introduced later by Hilbertdoes not appear in Dedekind's work. If factorizations into prime elements are permitted, then, even in the integers, there are alternative factorizations such as 6.

In the work of Niels Abel and Carl Jacobithe answer was formulated: However singularity theory enters: For later history see motive algebraic geometrymotivic cohomology.

An abelian variety can be defined by equations having coefficients in any field ; the variety is then said to be defined over that field. Applying this to the case when T is a point, we see that the points of Av correspond to line bundles of degree 0 on A, so there is a natural group operation on Av given by tensor product of line bundles, which makes it into an abelian variety.

Reduction modulo a prime number or ideal See good reduction. Abelian functions An abelian function is a meromorphic function on an abelian variety, which may be regarded therefore as a periodic function of n complex variables, having 2n independent periods; equivalently, it is a function in the function field of an abelian variety.

The concept underwent further development in the hands of Hilbert and, especially, of Emmy Noether. In the early s, Weil used the first definition over an arbitrary base field but could not at first prove that it implied the second.

See for example Birch—Tate conjectureLichtenbaum conjecture. So the isomorphism is actually defined over. Now a sophisticated equivariant Tamagawa number conjecture is a major research problem.

A principal polarisation is a polarisation that is an isomorphism. From toAndrew Wiles provided a proof of the modularity theorem for semistable elliptic curveswhich, together with Ribet's theoremprovided a proof for Fermat's Last Theorem. The extension of class field theory -style results on abelian coverings to varieties of dimension at least two is often called geometric class field theory.

The word "Ring", introduced later by Hilbertdoes not appear in Dedekind's work. An abelian variety is simple if it is not isogenous to a product of abelian varieties of lower dimension.

There are several equivalent formulations of this; all of them are known as the Riemann conditions. Applying this to the case when T is a point, we see that the points of Av correspond to line bundles of degree 0 on A, so there is a natural group operation on Av given by tensor product of line bundles, which makes it into an abelian variety.In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular ifongchenphoto.comn varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics.

Let S = spec(o) where 0 is a Dedekind ring, with quotient field F. Let AF be an abelian variety over F. By a Neron model of AF over 0 we mean a group scheme A satisfying the following properties. NM 1. A is smooth over S.

NM 2. The generic fiber AF is the given abelian variety AF • NM 3. by equations having coefficients in any field; of an abelian variety of dimension 2 (an abelian An abelian variety can be defined by equations having.

Type or paste a DOI name into the text box. Click Go. Your browser will take you to a Web page (URL) associated with that DOI name. Send questions or comments to doi. Sectional lectures are invited minute lectures. These are representative of the major scientific achievements in each field of mathematics. Start studying CRY- CHAP Learn use of an elliptic curve equation defined over an infinite field. in two variables with coefficients. A. abelian.

A description of an abelian variety as defined by equations having coefficients in any field
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