By Noam D. Elkies (auth.), Joe P. Buhler (eds.)

This e-book constitutes the refereed court cases of the 3rd overseas Symposium on Algorithmic quantity conception, ANTS-III, held in Portland, Oregon, united states, in June 1998.
The quantity offers forty six revised complete papers including invited surveys. The papers are equipped in chapters on gcd algorithms, primality, factoring, sieving, analytic quantity thought, cryptography, linear algebra and lattices, sequence and sums, algebraic quantity fields, classification teams and fields, curves, and serve as fields.

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Extra info for Algorithmic Number Theory: Third International Symposiun, ANTS-III Portland, Oregon, USA, June 21–25, 1998 Proceedings

Example text

48) for some nonzero constant C. We evaluate C by taking t = 1 in (48). 122] F (a, b; c; 1) = Γ (c)Γ (c − a − b) Γ (c − a)Γ (c − b) (49) gives us the coefficient of C in the right-hand side in terms of gamma functions. 314837 . 472571 . 128545 . )i. Likewise we obtain convergent power series for computing z in neighborhoods of t = 1 and t = ∞. e. an embedding such Shimura Curve Computations 25 that OD = (OD ⊗ Q) ∩ O) and the embedding is unique up to conjugation in Γ ∗(1). Then there is a unique, and therefore rational, CM point on X ∗ (1) of discriminant D.

3 3 1 1 0 Y 4 4 1 1 1 Y 24 24 1 0 1 Y 84 12 7 −169 27 Y 40 8 5 2312 25 Y 51 3 17 −1377 1024 Y 19 1 19 3211 1024 Y 120 24 5 5776 3375 Y 52 4 13 6877 15625 Y 132 12 11 13689 15625 Y 75 3 52 152881 138240 Y 168 24 7 −701784 15625 Y 43 1 43 21250987 16000000 Y 228 12 19 66863329 11390625 N 88 8 11 15545888 20796875 Y 123 3 41 −296900721 16000000 N 100 4 52 421850521 1771561 Y 147 3 72 −1073152081 3024000000 Y 312 24 13 27008742384 27680640625 Y 67 1 67 77903700667 1024000000 N 148 4 37 69630712957 377149515625 N 372 12 31 −455413074649 747377296875 N 408 24 17 −32408609436736 55962140625 N 267 3 89 −5766681714488721 1814078464000000 N 232 8 29 66432278483452232 56413239012828125 N 708 12 59 71475755554842930369 224337327397603890625 N 163 1 163 699690239451360705067 684178814003344000000 N By (9), the curve X ∗ (1) has hyperbolic area 1/6.

51 − 19x8 . (82) w℘8 (x8 ) = 19 + 13x8 Note that all of these covers and involutions have rational coefficients even though a priori they are only known to be defined over K. This is possible because K is a normal extension of Q and the primes ℘7 , ℘8 used to define our curves and maps are Galois-invariant. To each of the three real places of K corresponds a quaternion algebra ramified only at the other two places, and thus a Shimura curve X (1) with three elliptic points P2 , P3 , P7 to which we may assign coordinates 0, 1, ∞.

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