mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or a family of abelian varieties. It goes back...

Historically the first abelian varieties to be studied were those defined over the field of complex numbers. Such abelian varieties turn out to be exactly...

Arithmetic dynamics Arithmetic of abelian varieties Birch and Swinnerton-Dyer conjecture Moduli of algebraic curves Siegel modular variety Siegel's theorem...

In arithmetic geometry, the Weil–Châtelet group or WC-group of an algebraic group such as an abelian variety A defined over a field K is the abelian group...

a foundational theorem of Diophantine geometry and the arithmetic of abelian varieties. The tangent-chord process (one form of addition theorem on a cubic...

of Mathematical Sciences. Vol. 49 (Second ed.). ISBN 978-3-540-20364-3. ISSN 0938-0396. Zbl 1079.11002. Arithmetic Chow groups Arithmetic of abelian varieties...

arithmetic geometry that explicitly includes the 'infinite primes'. Arithmetic of abelian varieties See main article arithmetic of abelian varieties Artin...

reductions of semi-stable varieties", Compositio Mathematica, 104 (1): 77–105. Zhang, Shou-Wu (1998), "Equidistribution of small points on abelian varieties",...

variety ResL/kX, defined over k. It is useful for reducing questions about varieties over large fields to questions about more complicated varieties over...

number Arithmetic of abelian varieties Elliptic divisibility sequences Mordell curve Fermat's Last Theorem Mordell conjecture Euler's sum of powers conjecture...

call the book "visionary". A larger field sometimes called arithmetic of abelian varieties now includes Diophantine geometry along with class field theory...

dynamics, the study of the iteration of self-maps of the complex plane or other complex algebraic varieties. Arithmetic dynamics is the study of the number-theoretic...

On the other hand, an abelian scheme may not be projective. Examples of abelian varieties are elliptic curves, Jacobian varieties and K3 surfaces. Let...

{A}}_{g}} of principally polarized complex abelian varieties of dimension g {\displaystyle g} (a principal polarization identifies an abelian variety with...

Rosati involution), leading to a classification of CM-type abelian varieties. To construct such varieties in the same style as for elliptic curves, starting...

Another class is formed by the abelian varieties, which are the algebraic groups whose underlying variety is a projective variety. Chevalley's structure theorem...

He was known for developing the theory of complex multiplication of abelian varieties and Shimura varieties, as well as posing the Taniyama–Shimura conjecture...

Abelian Varieties pp. 146–7); but if for example one looks at reduction mod p of a division equation, the number of solutions must drop. The rank of the...

asks which principally polarized abelian varieties are the Jacobians of curves. The Picard variety, the Albanese variety, generalized Jacobian, and intermediate...

Abelian varieties are a natural generalization of elliptic curves to higher dimensions. However, unlike the case of elliptic curves, there is no well-behaved...

In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements...

In arithmetic geometry, the Mordell–Weil group is an abelian group associated to any abelian variety A {\displaystyle A} defined over a number field K...

This is a timeline of the theory of abelian varieties in algebraic geometry, including elliptic curves. 3rd century AD Diophantus of Alexandria studies...

fiber variety, introduced by Kuga (1966), is a fiber space whose fibers are abelian varieties and whose base space is an arithmetic quotient of a Hermitian...

Shimura variety theory is related, by the idea that such varieties should parametrize not just good (arithmetically interesting) families of Hodge structures...

)_{\text{tors}}} involving the dual abelian variety A ^ {\displaystyle {\hat {A}}} . Elliptic curves as 1-dimensional abelian varieties are their own duals, i.e...

generalization of the Gross–Zagier theorem to modular abelian varieties of GL(2) type (Zhang 2001). In particular, the result led him to a proof of the rank...

In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is...

of abelian varieties. The Selmer group of an abelian variety A with respect to an isogeny f : A → B of abelian varieties can be defined in terms of Galois...

Joseph (1986). The Arithmetic of Elliptic Curves. New York: Springer-Verlag. ISBN 0-387-96203-4. James Milne, Abelian Varieties, available at www.jmilne...