Elliptic curve

In Chapter 29 we introduced the concept of the two dual data spaces. Each of the n rows of the data table X can be represented as a point in the p-dimensional column-space S . In Fig. 31.2a we have represented the n rows of X by means of the row-pattern F. The curved contour represents an equiprobability envelope, e.g. a curve that encloses 99% of the points. In the case of multinormally distributed data this envelope takes the form of an ellipsoid. For convenience we have only represented two of the p dimensions of SP which is in reality a multidimensional space rather than a two-dimensional one. One must also imagine the equiprobability envelope as an ellipsoidal (hyper)surface rather than the elliptical curve in the figure. The assumption that the data are distributed in a multinormal way is seldom fulfilled in practice, and the patterns of points often possess more complex structure than is shown in our illustrations. In Fig. 31.2a the centroid or center of mass of the pattern of points appears at the origin of the space, but in the general case this needs not to be so. 

For the next four lemmas let q be a prime power satisfying gcd(n,q) = 1 and let either S = A be an abelian surface over Fq or let S—>A be a geometrically ruled surface over an elliptic curve A over Fq. In this case we assume that there exist an open cover ([/ ),- of A and isomorphisms a 1( 7i) = /, x Pj over Fq. In both cases we assume that, for all l < n, all the /-division points of A are defined over Fq. All these conditions can be obtained by extending Fq if necessary. Let F be the geometric Frobenius over Fq. We put... 

Let S be a geometrically ruled surface over an elliptic curve over C. Then... 

Proof Let S be either a two dimensional abelian variety or a geometrically ruled surface over an elliptic curve over C. Let S be a good reduction of S modulo q, where gcd(q, n) = 1 such that the assumptions of lemma 2.4.7 hold. Then A 5n iis a good reduction of KSn- modulo q. (3) now follows by lemma 2.4.10 and remark 1.2.2. (1) and (2) follow from this by the formula of Macdonald for p(S n z) (see the proof of theorem 2.3.10). ... 

By applying the same argument to the case of a geometrically ruled surface over an elliptic curve we get that sign(KSn-i) = 0. This was however clear from the beginning as the dimension of KSn-i is not divisible by 4. It seems remarkable that in all cases the signatures and the Euler numbers can be expressed in terms of the coefficients of the (/-development of modular forms. For the first few of the X-y(KAn-i) we get ... 

These study will lead us to wonder the reason why we encounter objects, such as modular forms, affine Lie algebras, the conformal held theory etc. Usually we think these objects live with elliptic curves, not with surfaces. Formally, we have two possibilities one is that these objects are so universal (like Dynkin diagrams) that they appear everywhere. The other possiblity is that elliptic curves are hidden in the Hilbert schemes. We do not know which is correct at this moment, but we believe that the second one is correct. 

Silverman/Tate Rational Points on Elliptic Curves. 

Cotton. Cotton is furnished by the down surrounding the seeds of various species of Gossypium. This fibre, which is unicellular and closed at only one end, is always isolated, and appears under the microscope as a ribbon twisted at intervals on its own axis like a spiral (Fig. 68, Plate VI). The wall is comparatively thin and sometimes somewhat raised like a rim the lumen is wide—three or four times as wide as the walls. This lumen is mostly empty, but sometimes contains granulations representing the original protoplasm in a dried state. The cotton fibre, which consists solely of cellulose, is coated in the raw state with a very thin cuticle, which is readily seen in a dry microscopic preparation. When raw cotton is treated with ammoniacal cupric oxide solution, whilst the cellulose of the fibre first swells and then dissolves, the cuticle remains almost intact, so that the fibres assume characteristic microscopic forms. The section of the cotton fibre (see Fig. 69, Plate VI) is elliptical, curved or reniform, with a fissure-like lumen. 

If we consider the sun and one planet, namely a two-body case, the equations of motion for this case is solvable. We have the famous Kepler motion.lt is well known that there are four types of orbits, namely the circle (e = 0), the elliptic curve (0 < e < 1), the parabolic curve (e = 1), and the hyperbolic curve (e > 1), where e is the eccentricity. 

The only calculation in this direction we could make, is the following. Let p= 3i and let C be a hyperelliptic curve (i.e. a curve of genus 2), with Jacobian V2iriety J. Then o(J) <2 implies that J is isogeneous to a product of two elliptic curves (non--isomorphic curves have non-isomorphic polarized Jacobian varieties, by Torelli s theorem, but their Jacobians may be isomorphic). We use notations of IGUSA, cf. C20], page 6kk/Sk C can be of type n, with l[Pg.77]

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