By Johannes Buchmann, Michael J. Jacobson Jr., Stefan Neis, Patrick Theobald, Damian Weber (auth.), B. Heinrich Matzat, Gert-Martin Greuel, Gerhard Hiss (eds.)

ISBN-10: 3540646701

ISBN-13: 9783540646709

ISBN-10: 364259932X

ISBN-13: 9783642599323

This e-book includes 22 lectures provided on the ultimate convention of the German study application "Algorithmic quantity concept and Algebra 1991-1997", subsidized by means of the Deutsche Forschungsgemeinschaft. the aim of this learn application and the assembly used to be to compile builders of desktop algebra software program and researchers utilizing computational how you can achieve perception into experimental difficulties and theoretical questions in algebra and quantity idea. The booklet provides an outline on algorithmic equipment and effects got in this interval as a rule in algebraic quantity idea, commutative algebra and algebraic geometry, and team and illustration idea. a few of the articles illustrate the present country of the pc algebra structures constructed with aid from the study software, for instance KANT and LiDIA for algebraic quantity conception, SINGULAR, REDLOG and INVAR for commutative algebra and invariant idea respectively, and hole, SYSYPHOS and CHEVIE for staff and illustration theory.

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**Example text**

A point of Jc(IQ) corresponds to an ideal class of the integral closure of lQ[x] in F(x, y) := lQ[x, y] IUd which is a free module of rank 2 over lQ[x] with integral basis (1, Ll c ) (Lla is the discriminant of F). t. this basis. 42 G. Frey, M. 1. Splitting of SJ]ew(N) 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 1 0 0 1 1 0 1 0 1 1 1 2 2 1 0 2 1 2 2 3 2 1 3 3 3 1 2 4 3 3 3 5 3 4 3 1 0 0 1 1 0 1 0 1 1 1 0 2 1 0 2 1 0 2 1 2 1 1 1 3 1 2 2 3 1 3 1 3 1 1 1(1) 46 47 48 1(1) 49 1(1) 50 51 1(1) 52 53 54 1(1) 1(1) 55 1(1) 56 57 58 2(1) 1(1) 59 60 1(2) 61 62 1(1) 63 2(1) 64 65 1(1) 2(1) 66 1(1) 67 1(1) 68 69 1(1) 1(1), 2(1) 70 1(1) 71 1(2) 72 73 1(2) 1(1), 2(1) 74 1(1) 75 3(1) 76 1(1) 77 1(1), 2(1) 78 1(1) 79 1(1) 80 5 4 3 1 2 5 5 4 4 5 5 5 6 5 7 4 7 5 3 5 9 5 7 7 9 6 5 5 8 5 8 7 11 6 7 1 4 1 1 2 3 1 4 2 3 2 3 2 5 0 4 3 3 1 5 3 5 2 3 1 6 1 5 4 3 1 5 1 6 2 1(1) 4(1) 1(1) 1(1) 1(2) 1(1), 2(1) 1(1) 1(1), 3(1) 1(2) 1(1), 2(1) 1(2) 1(3) 1(2) 5(1) 1(1), 3(1) 1(1), 2(1) 1(1), 2(1) 1(1) 1(1), 2(2) 1(3) 1(1), 2(2) 2(1) 1(1), 2(1) 1(1) 3(2) 1(1) 1(1), 2(2) 2(2) 1(3) 1(1) 1(3), 2(1) 1(1) 1(1), 5(1) 1(2) Explanation of the fourth column: The notation a(b) means that SJ]ew(N) has b a-dimensional'll'N[GQ]-invariant subspaces.

S. 1. en does not depend on the level N and the weight k. 2. The 'claBsical' method of computing the Hecke operator on M is to use the isomorphism (17) to compute the Hecke operator on H1 (Xo(N), cusp, Z) and to map the result back to the space M. To do this explicitly one has to calculate a continued fraction expansion of some rational numbers (see [6] or [28]). ' 3. The same formula holds if we replaces M by M+ corresponding to the +l-eigenspace under complex conjugation (see below). 9. Take N = 31.

For the same reason as in characteristic two, we restrict our attention to the case whtlre v«(3) = -k with some kEN, k = 61 - m, m = {O, 1, ... , 5}. 7) where /3 := 11'61 (3 has v(/3) = m. 8 Proposition. , if m E {I, 2, 4, 5}. The conductors f(EI K) and Kodaira types are f = 6l + 1 = k + 2, type = II for m = 1, f = 6l = k + 2, type = IV for m = 2, f = 6l- 2 = k + 2, type = IV* for m = 4 and f = 6l- 3 = k + 2, type = II* for m = 5. Proof. 1 = 11'61/3. From this, the first two cases are straightforward.

### Algorithmic Algebra and Number Theory: Selected Papers From a Conference Held at the University of Heidelberg in October 1997 by Johannes Buchmann, Michael J. Jacobson Jr., Stefan Neis, Patrick Theobald, Damian Weber (auth.), B. Heinrich Matzat, Gert-Martin Greuel, Gerhard Hiss (eds.)

by Ronald

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