By E.M. Chirka

One provider arithmetic has rendered the 'Et moi, .. " si j'avait so remark en revenir, human race. It has placed logic again je n'y semis aspect aile.' Jules Verne the place it belongs, at the topmost shelf subsequent to the dusty canister labelled 'discarded non The sequence is divergent; for that reason we can be sense'. capable of do whatever with it Eric T. Bell o. Heaviside arithmetic is a device for notion. A hugely precious software in a global the place either suggestions and non linearities abound. equally, all types of elements of arithmetic function instruments for different components and for different sciences. making use of an easy rewriting rule to the quote at the correct above one unearths such statements as: 'One provider topology has rendered mathematical physics .. .'; 'One provider common sense has rendered com puter technological know-how .. .'; 'One carrier type concept has rendered arithmetic .. .'. All arguably real. And all statements available this manner shape a part of the raison d'etre of this sequence.

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**Example text**

By a theorem whose origin may be traced back to Riemann (see [Hart77, App. B, Th. 1]), XΓ is a projective algebraic curve over C. We conclude from the compactness that XΓ contains only ﬁnitely many cusps and elliptic points. For example, XSL(2,ZZ) = SL(2, ZZ)\H∗ has exactly one cusp, represented by ∞, and one elliptic point of order 2 and 3, respectively, representatives being the complex numbers i and e2πi/3 , respectively. If we are given a subgroup Γ of ﬁnite index in SL(2, ZZ), the curve XΓ is a covering of XSL(2,ZZ) , and any possible elliptic point on it necessarily has order 2 or 3.

6, ex. 5]. We especially notice that all quantities aqn with n ∈ IN are uniquely determined as soon as aq is known. 12) ) in connection to the Frobenius endomorphism πq on E. it to globally deﬁned structures. Let p be a prime ideal of k with the norm q such that E has good reduction at p, and let be some prime number not divided by p. 1(b)]). Hence the -adic Tate˜ are isomorphic, and by conjugation with some such modules T (E) and T (E) isomorphism we get a representation ˜ . ˜ : Gal(k/k) → Aut T (E) → Aut T (E) If K is the ﬁxed ﬁeld in k corresponding to ker(˜ ), p is unramiﬁed in K /k.

Over κ(p) is given For any prime ideal p of k, the p-adic zeta function of E by ∞ n ˜ n (p))) T ˜ , #(E(κ Z(E/κ(p), T ) = exp n n=1 where κn (p) denotes the extension of κ(p) of degree n which is uniquely determined up to ﬁeld isomorphisms. 2. Let p = {0} be a prime ideal of Ok having the norm q = N (p) such that E has good reduction at p. Then the p-adic zeta function has the following properties: (a) It is a rational function in Q(T ). More precisely, we have ˜ Z(E/κ(p), T) = 1 − aq T + qT 2 .