Search results
Results from the WOW.Com Content Network
In mathematics, the Weil conjectures were highly influential proposals by André Weil . They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory .
In elliptic curve cryptography, the Weil descent attack uses the Weil restriction to transform a discrete logarithm problem on an elliptic curve over a finite extension field L/K, into a discrete log problem on the Jacobian variety of a hyperelliptic curve over the base field K, that is potentially easier to solve because of K's smaller size.
The function obtained in this way is, remarkably, a cusp form of weight two and level N and is also an eigenform (an eigenvector of all Hecke operators); this is the Hasse–Weil conjecture, which follows from the modularity theorem. Some modular forms of weight two, in turn, correspond to holomorphic differentials for an elliptic curve.
A Weil divisor D is effective if all the coefficients are non-negative. One writes D ≥ D′ if the difference D − D′ is effective. For example, a divisor on an algebraic curve over a field is a formal sum of finitely many closed points.
Instead of using the absolute Galois group of k, one can use the absolute Weil group, which has a natural map to the Galois group and therefore also acts on the root datum. The corresponding semidirect product is called the Weil form of the L-group. For algebraic groups G over finite fields, Deligne and Lusztig introduced a different dual group.
A Weil cohomology theory is a contravariant functor ... For -adic cohomology, for example, most of the above properties are deep theorems. The vanishing of ...
In mathematics, the Weil pairing is a pairing (bilinear form, though with multiplicative notation) on the points of order dividing n of an elliptic curve E, taking values in nth roots of unity. More generally there is a similar Weil pairing between points of order n of an abelian variety and its dual.
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures. Étale cohomology theory can be used to construct ℓ-adic cohomology, which is an example of a Weil cohomology theory in algebraic geometry.