# Tag Info

28

To expand my comment, there are at least 3 subtle ways to get BSD wrong: 1) The BSD period over ${\mathbb Q}$ is the real period $\Omega_\infty$ when $E({\mathbb R})$ is connected ($\Delta(E)<0$) and $2\Omega_\infty$ when it has two connected components ($\Delta(E)>0$). The same thing happens over number fields, at every real place. So in your example ...

17

This curve has rank 0 over $\mathbb{Q}$. The 2-descent fails to determine this, because the $2$-torsion subgroup of the Tate-Shafarevich group is non-trivial. Instead, one can compute the $L$-value. One can prove that $L(E,1) = 16 \Omega_{+}$. By Kolyvagin, this implies that the rank is $0$. Now one just needs to compute the torsion order and since there ...

16

Magma is computing the automorphism group of the associated projective curve $E$ defined over the base field (according to the link you gave). You are thinking of the automorphism group of $E$ as an elliptic curve i.e. with $\infty$ fixed. The group magma is computing will also include the translations which are defined over the base, so there is a short ...

16

Tim's answer is great, but I want to mention one other place where people have lost a power of 2. The canonical height is often defined relative to the divisor (O), as I do in my books. So it is given by $$\hat h(P) = \frac12 \lim_{n\to\infty} 4^{-n} h\bigl(x([2^n]P)\bigr).$$ Here the $\frac12$ is inserted because $x$ has a double pole at $\infty$. ...

8

In theory, Groebner basis software can do this. I don't know whether there is a standard package for this purpose, or what practical issues may arise. Here is the theory: Your goal is the following. You are given rational functions $r_1$, $r_2$, ..., $r_n$ in $x$ and $y$. You would like know whether there are polynomials $f(t_1, \ldots, t_n)$ and $g(t_1,\... 7 The point is that in GAP, a^b means b^-1*a*b rather than b*a*b^-1. If you adjust your relations accordingly, you get what you expect: gap> rels:=[F.1^2*F.2^(-1),F.2^2,F.3^5,F.1^-1*F.3*F.1*F.3^(-2), > F.2^-1*F.3*F.2*F.3^(-4)];; gap> G:=F/rels;; S:=[G.2,G.2*G.3,G.1*G.3^2,G.1*G.2*G.3^4];; gap> S^2 = One(G); true gap> S^2 = One(G);... 5 AutomorphismGroup computes the automorphism group over the base field, and it only works with certain types of base fields - in particular, it won't work over the reals, complexes, and the "algebraic closure" (none of which are really suited to geometric computations). But you can get what you want by working over a number field, and in your example it's ... 5 Paragraph 3.2.1 of On Certain Subgroups of$E_8(2)$and their Brauer Character Tables explains precisely how the Brauer characters are defined in MAGMA. The ambiguity in the definition refers to the choice of a definite root of unity. MAGMA uses Conway polynomials for that purpose and from page 306 of Lux and Pahlings I gather that they use the same ... 5 I just ran this calculation on a machine with lots of memory, and it completed in just over two hours using about 85GB of memory. There are 59 classes of maximal subgroups. You can follow the progress of the computation by turning on verbosity. For this one I would recommend SetVerbose("Subgroups",3); The calculation proceeds by finding the maximal ... 4 The problem with inconsistent results in the final part of your calculation is not just due to different choices of random elements, as I said in my comment, but probably due to a confusion about the meaning of$S^t$, which is defined to be$t^{-1}St$in Magma. Think of the group$S$as acting by right multiplication on the cosets$Tt$of$Tt$of$T$. Then ... 4 This question is not about research level mathematics, so it is really not suitable for mathoverflow. But I'll answer it anyway. The code below works. > G := Sym(6); > H := sub< G | (1,2,3) >; > N := Normalizer(G,H); > M := PermutationModule(G, GF(7)); > MH := Restriction(M,H); > MN := Restriction(M,N); > FMH := Fix(MH); &... 3 I am not sure if I understand completely what you are trying to do, but I get the impression that the heart of the problem is that you are given$kH$-module homomorphism$M \to N$, and you want to compute the induce homomorphism$M_H^G \to N_H^G$. I think the following code does that. InducedHom := function(phi, G) //phi:M->N is a KH-module M->N ... 3 I cannot answer 100%, but I can tell you what I know is there, and maybe its enough with some tweaking. AR-sequences are not something I've needed to implement in Magma yet, so I've not grappled with this one. Magma can compute projective covers and a syzygy, first off. Then it can compute$\texttt{AHom(A,B)}$, which is simply$\mathrm{Hom}_{kG}(A,B)$. It ... 3 (I meant to post this as a comment, but that doesn't seem to allow me to format code, so I'll post it as an answer instead) Note that GAP also has the database of small groups built in, so you can also directly access the group in question and obtain a presentation for it: gap> G:=SmallGroup(20,3); <pc group of size 20 with 3 generators> gap> F:... 3 Between David's answer and much staring at the paper I linked to above, I think I've mostly figured out what's going on there so I thought I'd post a summary of their method here as well in case anyone else had a similar problem. I'm focusing here on$how$to use their algorithm, not$why$it works: I found it was a lot easier to follow the theory once I ... 3 I take it you want birational invariants of projective desingularisations of a surface. Magma can't do very much at the moment for surfaces in weighted projective space but if you can find an embedding into ordinary projective space with singularities that aren't too bad (only simple singularities), it can practically handle surfaces in P^n for n reasonable ... 2 I once used the following MAGMA-code (just for finite groups, I don't know if it works for general reductive groups). intrinsic Presentation(R::RngInvar) -> SeqEnum {A presentation of the invariant ring R.} fund := FundamentalInvariants(R); prim := PrimaryInvariants(R); sec := IrreducibleSecondaryInvariants(R); invar := prim cat sec; P ... 2 Probably your program received a "kill" signal (they are numbered, SIGKILL is number 9), typically these signals are sent by programs run by your OS, "OOM-killers" (out of memory). TL;DNR, you use too much memory. 2 My eventual solution was as follows: Use Magma to compute a Groebner basis of$A$, then compute a matrix representation of$A$. Export this representation to Sage,* then compute the matrices of the linear maps over the vector space of matrices with coefficients in the symbolic ring (which was suitably vectorized.) Compute the intersections of the kernels in ... 1 Given an order$O$in a quaternion algebra$B$over a number field$K$, I believe you want to construct an integral ideal$I$in$O$with some given norm$n$(an ideal in$K$). I don't believe Magma has any built-in function to do this directly, but you should be able to do this in one of the following ways using various built-in Magma commands: (1) If$n$... 1 In Magma, the unit group is returned as an abstract group together with a map from that group into the field. So you would do something like the following: > Zx<x> := PolynomialRing(Integers()); > K<w> := NumberField(x^3 + x^2 - 2*x - 1); > G,phi := UnitGroup(K); > G; Abelian Group isomorphic to Z/2 + Z + Z Defined on 3 generators ... 1 You can also do this in Macaulay2 http://www.math.uiuc.edu/Macaulay2/ Here's the commands: A = ZZ[a,b,c]/ideal(a*b-c^2); I = ideal(x,y); (ker(matrix{{h}}**(A/I))) == 0 To briefly explain: you're constructing the multiplication by$h$map over$A/I$and computing its kernel and testing if it's the 0 module. So you'll get true if$h$is a nonzerodivisor. 1 It sounds like Magma does not currently include the exact functionality that you want to use. An alternative that you could try is manually building the$L$-function you wish to evaluate numerically (see the section of the Magma documentation "Arithmetic Geometry -$L\$-functions - Constructing a general L-series"). This accesses the algorithms (primarily ...

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