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

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"The booklet includes 39 articles approximately computational algebraic quantity thought, mathematics geometry and cryptography. … The articles during this publication replicate the wide 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 learn in quantity thought and mathematics geometry and its applications." (R. Carls, Nieuw Archief voor Wiskunde, Vol. 6 (3), 2005)

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

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Field Curve Fp y 2 = x3 + ax Fp y 2 = x3 + a Fp 2 y 2 = x3 + a a ∈ Fp F3n y 2 = x3 + 2x + 1 F3n y 2 = x3 + 2x + 1 F3n y 2 = x3 + 2x − 1 F3n y 2 = x3 + 2x − 1 Distorsion (x, y) → (−x, iy) i2 = −1 (x, y) → (ζx, y) ζ3 = 1 yp xp (x, y) → (ω r(2p−1)/3 , rp−1 ) r2 = a, r ∈ Fp2 ω 3 = r, ω ∈ Fp6 (x, y) → (−x + r, uy) u2 = −1, u ∈ F32n 3 r + 2r + 2 = 0, r ∈ F33n (x, y) → (−x + r, uy) u2 = −1, u ∈ F32n r3 + 2r + 2 = 0, r ∈ F33n (x, y) → (−x + r, uy) u2 = −1, u ∈ F32n 3 r + 2r − 2 = 0, r ∈ F33n (x, y) → (−x + r, uy) u2 = −1, u ∈ F32n 3 r + 2r − 2 = 0, r ∈ F33n Conditions Group order Sec.

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 . 3. Can one prove an analogue of Theorem 1 in which the elliptic curve is replaced by an abelian variety? References CZ00. Gunther Cornelissen and Karim Zahidi, Topology of Diophantine sets: remarks on Mazur’s conjectures, Hilbert’s tenth problem: relations with arithmetic and algebraic geometry (Ghent, 1999), Amer. Math.

Kirsten Eisentr¨ ager, Ph. D. thesis, University of California, Berkeley, in preparation. KR92. K. H. Kim and F. W. Roush, Diophantine undecidability of C(t1 , t2 ), J. Algebra 150 (1992), no. 1, 35–44. Mat70. Ju. V. Matijaseviˇc, The Diophantineness of enumerable sets, Dokl. Akad. Nauk SSSR 191 (1970), 279–282. Maz94. B. Mazur, Questions of decidability and undecidability in number theory, J. Symbolic Logic 59 (1994), no. 2, 353–371. MB. Laurent Moret-Bailly, paper in preparation, extending results presented in a lecture 18 June 2001 at a conference in honor of Michel Raynaud in Orsay, France.

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