Number Theory

Elliptic Curves (student project) by Weng A.

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By Weng A.

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V n . Now given a lattice in some Euclidean space, there is a way to "roll up" the space around the lattice and get a quotient manifold. As an example, we take Z2 in the Euclidean space R 2 and identify two points of R 2 if their difference lies in the lattice Z2. So the quotient space R 2 / Z2 is a torus, as is probably familiar to the reader. In a similar way, we can construct higher-dimensional tori as the quotients of other Euclidean spaces by lattices contained in them. These tori are Riemannian manifolds since they "inherit" the Euclidean space metric.

So you won't get to another river. Integer-valued forms have periodic rivers For the form x 2 + 6xy - 3y 2, after moving along the river from the initial superbase Po surrounded by the numbers 1, 2, -3, we fmd another superbase PI surrounded in exactly the same way. If we move the same distance again, we shall see yet another such superbase P2 and so on; the surroundings of the river repeat periodically.. We shall Can You See the Values of 3:1: 2 + 6:1:Y - 21 5 y 2? 1 -7 --6 show that if the coefficients a, h, b, in the +- form ax 2 + hxy + by 2 are integers, then the river is necessarily periodic in this way.

If we read any vector of either tetralattice modulo 3 we get a tetracode word. Also, any nonzero codeword of the tetracode comes from reading one of the four basis vectors or their negatives modulo 3. For instance: + = [4,2,0, -2] = [1, -1,0,1] (mod 3), and this tetracode word arises from reading (mod 3). The kernel of this map from L + onto the tetracode is a sublattice M + of index 9. M + consists of vectors whose coordinates are all divisible by 3. For example, vi vt vi + vt - vt = vt [3,3, -3, -3] E M+ 44 THE SENSUAL (quadratic) FORM It is easy to check that M + is generated by the 8 vectors [±3, ±3, ±3, ±3] in which there are an even number of minus signs.

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