By Reinhard Schertz

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**Example text**

23) if c = 0, where ac and 0c = 1 denotes the Legendre symbol. To write (M ) in all three cases by one formula, we deﬁne c1 and λ ∈ Z by c = c1 2λ with c1 ≡ 1 mod 2 if c = 0, c1 = λ = 1 if c = 0. For c = 0 we have, according to the quadratic reciprocity law, c |a| = (−1) a−1 c1 −1 a2 −1 2 2 +λ 8 a c1 . 22). Then, with the above deﬁnition of c1 and λ we have ba+c(d(1−a2 )−a)+3(a−1)c1 +λ 32 (a2 −1) (M ) = ( ca1 )ζ24 . 1 from the formulae for (S) and (T ) by showing that the formula holds for (SM ) and (T M ) if it is valid for (M ).

Hence, all functions from CL and, in particular, x have a Laurent expansion of powers of W converging in a neighbourhood of 0. The coeﬃcients of this expansion are related to a1 , . . 2 In a neighbourhood of 0 we have Laurent expansions of the form − y1 = x = W3 + W −2 + An W n n≥4 Bn W n n≥−1 y = −W −3 + Cn W n n≥−2 with coeﬃcients An , Bn , Cn ∈ Z[a1 , . . , a6 ]. 10) with 1 s := − . 6 Weierstrass functions 19 Substituting this equation into itself recursively, s = f (W, f (W, f (W, . . , f (W, s)))), we obtain for every N ∈ N N −1 An W n + W N gN s= n=3 with an elliptic function gN holomorphic at 0 and coeﬃcients A3 = 1 and An ∈ Z[a1 , .

A6 ∈ C. So the map ⎧ ⎨ (1 : x(z) : y(z)) for z ∈ L, z + L → Q(z) := ⎩ 1 x(z) for y(z) = 0. 7) deﬁnes a bijection between C/L and the projective curve E deﬁned by y 2 t + a1 xyt + a3 yt2 = x3 + a2 x2 t + a4 xt2 + a6 t3 . Setting Q(z1 ) + Q(z2 ) := Q(z1 + z2 ) E is endowed with a group structure with the point Q(0) = (0 : 0 : 1) at inﬁnity as neutral element. Further, x and y being obtained from ℘ and ℘ by an aﬃne linear transformation, this group structure again allows the following geometric interpretation: for Q1 , Q2 , Q3 ∈ E with Q1 + Q2 + Q3 = 0, the point Q3 is the third intersection point of the line through Q1 and Q2 with the curve E.