1Function: chardiv 2Section: number_theoretical 3C-Name: chardiv0 4Prototype: GGG 5Help: chardiv(cyc, a,b): given a finite abelian group (by its elementary 6 divisors cyc) and two characters a and b, return the character a/b. 7Doc: let \var{cyc} represent a finite abelian group by its elementary 8 divisors, i.e. $(d_j)$ represents $\sum_{j \leq k} \Z/d_j\Z$ with $d_k 9 \mid \dots \mid d_1$; any object which has a \kbd{.cyc} method is also 10 allowed, e.g.~the output of \kbd{znstar} or \kbd{bnrinit}. A character 11 on this group is given by a row vector $a = [a_1,\ldots,a_n]$ such that 12 $\chi(\prod g_j^{n_j}) = \exp(2\pi i\sum a_j n_j / d_j)$, where $g_j$ denotes 13 the generator (of order $d_j$) of the $j$-th cyclic component. 14 15 Given two characters $a$ and $b$, return the character 16 $a / b = a \overline{b}$. 17 \bprog 18 ? cyc = [15,5]; a = [1,1]; b = [2,4]; 19 ? chardiv(cyc, a,b) 20 %2 = [14, 2] 21 ? bnf = bnfinit(x^2+23); 22 ? bnf.cyc 23 %4 = [3] 24 ? chardiv(bnf, [1], [2]) 25 %5 = [2] 26 @eprog\noindent For Dirichlet characters on $(\Z/N\Z)^*$, additional 27 representations are available (Conrey labels, Conrey logarithm), 28 see \secref{se:dirichletchar} or \kbd{??character}. 29 If the two characters are in the same format, the 30 result is given in the same format, otherwise a Conrey logarithm is used. 31 \bprog 32 ? G = znstar(100, 1); 33 ? G.cyc 34 %2 = [20, 2] 35 ? a = [10, 1]; \\ usual representation for characters 36 ? b = 7; \\ Conrey label; 37 ? c = znconreylog(G, 11); \\ Conrey log 38 ? chardiv(G, b,b) 39 %6 = 1 \\ Conrey label 40 ? chardiv(G, a,b) 41 %7 = [0, 5]~ \\ Conrey log 42 ? chardiv(G, a,c) 43 %7 = [0, 14]~ \\ Conrey log 44 @eprog 45 46Variant: Also available is 47 \fun{GEN}{chardiv}{GEN cyc, GEN a, GEN b}, when \kbd{cyc} is known to 48 be a vector of elementary divisors and $a, b$ are compatible characters 49 (no checks). 50