By James R. Milgram

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1 − T )(1 − qT ) (b) It satisﬁes the functional equation ˜ Z(E/κ(p), 1 ˜ ) = Z(E/κ(p), T ). qT (c) The numerator polynomial in (a) has the following decomposition in C[T ] √ 1 − aq T + qT 2 = (1 − αT )(1 − α T ) with |α| = q. 38 3 Elliptic Curves Proof. These statements are a special case of the Weil conjectures on zeta functions of smooth projective varieties. See [Silv86, V. 4]. 2 we are able to estimate the size of aqn . 1. If p = {0} is a prime ideal of Ok having the norm q = N (p) such that E has good reduction at p, the numbers aqn satisfy the inequality √ |aqn | ≤ 2 q n .

6 Reduction Modulo p and L-Series Let k be a number ﬁeld and E an elliptic curve deﬁned over k. We establish a connection between locally and globally deﬁned structures related to E. At ﬁrst we describe the principle of reducing E modulo a prime ideal p. Within the entire following section, let p denote a prime ideal of Ok diﬀerent from {0} which has the absolute norm q = N (p). 1) of E modulo p we get the equation of a curve E deﬁned over the residue ﬁeld κ(p) = Ok /p. This curve, however, may possess singularities if the discriminant is reduced to 0.

5) 1N 01 for some Since Γ is a congruence subgroup, Γ contains the element N ∈ IN (for example N = (Γ (1) : Γ ) ). Then Condition 2 requires of f to have the real period N , and Condition 3 may be stated as follows. Let s be a cusp of XΓ and ρ ∈ SL(2, ZZ) any transformation with ρ(s) = ∞. Then f |[ρ−1 ]k has a Fourier expansion of the shape f |[ρ−1 ]k (τ ) = cn q n/N with q = e2πiτ . 6) n≥n0 The modular function f is called modular form, if it is holomorphic on H and at all cusps. 6) at any cusp satisﬁes n0 = 0, i.