By Manjul Bhargava (auth.), Claus Fieker, David R. Kohel (eds.)

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"The publication comprises 39 articles approximately computational algebraic quantity concept, mathematics geometry and cryptography. … The articles during this ebook mirror the large curiosity of the organizing committee and the individuals. The emphasis lies at the mathematical idea in addition to on computational effects. we suggest the publication to scholars and researchers who are looking to examine present study in quantity conception and mathematics geometry and its applications." (R. Carls, Nieuw Archief voor Wiskunde, Vol. 6 (3), 2005)

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**Additional info for Algorithmic Number Theory: 5th International Symposium, ANTS-V Sydney, Australia, July 7–12, 2002 Proceedings**

**Example text**

Is it true that for every number ﬁeld K, there exists an elliptic curve E over Q such that rk E(Q) = rk E(K) = 1? The author would conjecture so. If so, then Hilbert’s Tenth Problem over OK is undecidable for every number ﬁeld K. 2. Can one weaken the hypotheses of Theorem 1 and give a diophantine deﬁnition of OF over OK using any elliptic curve E over K with rk E(K) = 1, not necessarily deﬁned over F ? Such elliptic curves may be easier to ﬁnd. But our proof of Theorem 1 seems to require the fact that E is deﬁned over F and has rk E(F ) = 1, since Lemma 5 fails if the ideal I of OF is instead assumed to be an ideal of OK .

We ﬁrst show that for any ideal I ⊆ OK , the set GI := { Q ∈ rE(K) : I | den(x(Q)) } is a subgroup of rE(K). ) Since an intersection of subgroups is a subgroup, it suﬃces to prove this when I = pn for some prime p and some n ∈ Z≥1 . Let Op be the completion of OK at p. Let F ∈ OK [[z1 , z2 ]] denote the formal group of E with respect to the parameter z := −x/y, as in Chapter 4 of [Sil92]. Then there is an isomorphism E1 (Kp ), given by z → (x(z), y(z)) where x(z) = z −2 + . . and F(pOp ) −3 y(z) = −z + .

Yn ) in K m+n such that the fractional ideal (x1 , . . , xm ) divides the fractional ideal (y1 , . . , yn ) is diophantine over OK . 2. The set of (t, u) ∈ K × × K × such that den(t) | den(u) is diophantine over OK . 3. The set of (t, u) ∈ K × × K such that den(t) | num(u) is diophantine over OK . 4. The set of (t, u) ∈ OK × K × such that t | den(u) is diophantine over OK . Proof. Statement 1 is clear, since the condition is that there exist cij ∈ OK such that yj = i cij xi for each j. Statement 2 follows from statement 1, since den(t) | den(u) if and only if the fractional ideal (u, 1) divides (t, 1).