Diophantine approximation , irrationality and transcendence Michel
نویسنده
چکیده
1. Algebraic independence of the two functions ℘(z) and ez. Legendre’s relation η2ω1 − η1ω2 = 2iπ. Proof: integrate ζ(z)dz on a fundamental parallelogram. Application: algebraic independence of the two functions az + bζ(z) and ℘(z). 2. Section § 10.7.2: Morphisms between elliptic curves. The modular invariant. 3. Section § 10.7.3: Endomorphisms of an elliptic curve; complex multiplications. Algebraic independence of ℘ and ℘∗. Schneider’s Theorem on the transcendence of j(τ) (corollary 174).
منابع مشابه
Diophantine approximation , irrationality and transcendence Michel Waldschmidt
For proving linear independence of real numbers, Hermite [18] considered simultaneous approximation to these numbers by algebraic numbers. The point of view introduced by Siegel in 1929 [34] is dual (duality in the sense of convex bodies): he considers simultaneous approximation by means of independent linear forms. We define the height of a linear form L = a0X0 + · · · + amXm with complex coef...
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