Algorithmic Number Theory: 7th International Symposium, by Nigel Boston, Harris Nover (auth.), Florian Hess, Sebastian

By Nigel Boston, Harris Nover (auth.), Florian Hess, Sebastian Pauli, Michael Pohst (eds.)

This e-book constitutes the refereed lawsuits of the seventh overseas Algorithmic quantity concept Symposium, ANTS 2006, held in Berlin, Germany in July 2006.

The 37 revised complete papers provided including four invited papers have been rigorously reviewed and chosen for inclusion within the e-book. The papers are geared up in topical sections on algebraic quantity concept, analytic and simple quantity thought, lattices, curves and forms over fields of attribute 0, curves over finite fields and functions, and discrete logarithms.

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Extra info for Algorithmic Number Theory: 7th International Symposium, ANTS-VII, Berlin, Germany, July 23-28, 2006. Proceedings

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Since 2004, techniques for computing Heegner points of large height have improved very significantly, thanks to work of Delaunay and Watkins. The Magma implementation is now extremely fast and we have used it extensively for all the larger generators in the tables. 35. Here P = (a/c2 , b/c3 ) where h(P a= −13632833703140681033503023679128670529558218420063432397971439281876168936925608099278686103768271165751 437633556213041024136275990157472508801182302454436678900455860307034813576105868447511602833327656978462 242557413116494486538310447476190358439933060717111176029723557330999410077664104893597013481236052075987 42554713521099294186837422237009896297109549762937178684101535289410605736729335307780613198224770325365111 296070756137349249522158278253743039282375024853516001988744749085116423499171358836518920399114139315005, b= 776845386159678589635077615346492181601035042768002014396646962333772688446303892162606526955979081249211 185106671917236143678971202347339963247386055808925185619325909681380265508543158979491984235466881248491 978341526711100575326744746030922470291782156359389005809065313914236892470866399096616908015986267206085 816145609347461468770147859622405813347969542380216159923828490925517451952455079424426512616714569247069 065790676549942365146817589522964032348349807255751358289869629122053879780510640219504941970766697032823 589255263953926885142009701275092664710953135501372398976396568319085695054751879368605289437600720585853 465424006259176930980665902501637183477157293942231705607887213321716750749368884791336280387610317598902 0330254326477036682714837827401377115084796691, c= 113966855669333292896328833690552943933212422262287285858336471843279644076647486592460242089049033370292 485250756121056680073078113806049657487759641390843477809887412203584409641844116068236428572188929747 7694986150009319617653662693006650248126059704441347.

R. Acad. Sci. Paris S´er. I Math. 325 (1997), no. 8, 813–818. 13. B. Edixhoven , On the Manin constants of modular elliptic curves, in Arithmetic algebraic geometry (Texel, 1989), Progr. Math. 89, Birkh¨ auser, Boston (1991), 25–39. 14. de/TI/LiDIA/. 15. net/ntl/. 16. au/magma/. 17. fr/. 18. W. A. Stein and D. Joyner, SAGE: System for Algebra and Geometry Experimentation, Comm. Computer Algebra 39 (2005), 61–64. 19. W. A. org/sage. 20. W. A. Stein, G. Grigorov, A. Jorza, S. Patrikis and C. Patrascu, Verification of the Birch and Swinnerton-Dyer Conjecture for Specific Elliptic Curves, preprint 2006.

The Hasse bound |ap (E)| ≤ 2p1/2 implies that only primes l of size O(log p) are needed. Nowadays, since Wiles, modularity of elliptic curves over Q is known, and the ap (E) are indeed the coefficients at q p of the new-form fE associated to E. Hence Schoof’s algorithm shows that the coefficients ap (E) of fE can be computed in time polynomial in log p. The new-form fE has weight 2 and its level is the conductor of E. Pila’s generalisation of Schoof’s algorithm in [7] implies that for f a new-form of weight 2 the coefficients ap (f ) can be computed in time polynomial in log p (we stress that f is fixed here; the dependence of the running time on the degree over Q of the field generated by the coefficients of f is exponential).

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