This is TeX, Version 3.14159 (Web2C 7.3) (format=latex 1999.4.5) 7 JUN 1999 14:17 **kevin_1376.tex (kevin_1376.tex LaTeX2e <1998/12/01> Babel and hyphenation patterns for american, french, german, ngerman, n ohyphenation, loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 1999/01/07 v1.4a Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size10.clo File: size10.clo 1999/01/07 v1.4a Standard LaTeX file (size option) ) \c@part=\count79 \c@section=\count80 \c@subsection=\count81 \c@subsubsection=\count82 \c@paragraph=\count83 \c@subparagraph=\count84 \c@figure=\count85 \c@table=\count86 \abovecaptionskip=\skip41 \belowcaptionskip=\skip42 \bibindent=\dimen102 ) (kevin_1376.aux) \openout1 = `kevin_1376.aux'. LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 4. LaTeX Font Info: ... okay on input line 4. LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 4. LaTeX Font Info: ... okay on input line 4. LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 4. LaTeX Font Info: ... okay on input line 4. LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 4. LaTeX Font Info: ... okay on input line 4. LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 4. LaTeX Font Info: ... okay on input line 4. LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 4. LaTeX Font Info: ... okay on input line 4. Overfull \hbox (64.49643pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 This is 1376 we're talking about here. This might not be suc h a great example,[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 seeing as 2 is so horrible. One thing that _must_ work is th at once[] [] Overfull \hbox (22.4968pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 we've found the form and proved there's a repn of A_5 type, we have to[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 be able to work out what the A_5 extension is. The proof sho uld be[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 The step I'm worried about is (1). To do this we have to fin d a good[] [] Overfull \hbox (17.24684pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 The plan is as follows: the field will be unramified outside 5N where[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 in the sense of p46 of Buhler, using Local Langlands. [By th e way,[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 about is doing this calculation at 2. Actually though... hmm ... :)[] [] Overfull \hbox (17.24684pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 that's pretty cool. I'm even more optimistic now. The proof goes like[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 this: the extension is an A_5 extn unramified outside 2,5,43 and we[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 can also check it's unram at 5 I think; at 43 Local Langland s says[] [] Overfull \hbox (43.49661pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 that disc=43^2; this leaves 2 but disc at 2 is <=2^8 so sqrt (disc)<=2^4*43[] [] Overfull \hbox (38.24666pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 GL_2(C) from PGL_2(C). Well...erm...hmm. I've forgotten how to do this :)[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 How does this all work? Oh yeah. I remember. Here's an examp le which[] [] Overfull \hbox (69.74638pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 OK here's the basic principle: for any "type" there is a Dir ichlet character of[] [] Overfull \hbox (53.99652pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 (Z/p^nZ) associated to it, where n="cond" in his table. They 're not too hard[] [] Overfull \hbox (27.74675pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 consider A_5 as living in PSL_2(C) (it does). We have a C_3 in this A_5[] [] Overfull \hbox (38.24666pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 and we can think of the C_3 as being represented by the elem ent (1 0;0 z)[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 where z is a primitive cube root of 1. Now we lift back to S L_2(C);[] [] Overfull \hbox (27.74675pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 repn Gal(Q_p-bar/Q_p)-->GL_2(C) with image C_3 generated by (z 0;0 z^2)[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 and it's ramified and the image of inertia is this too, but p isn't[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 What's the conductor of this repn? Well, it's abelian and co nductors[] [] Overfull \hbox (32.9967pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 add; both characters are tame so the conductor is 1 (or p^1 if you like)[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 so the conductor of the repn to GL_2 is 2. Well this is all well and[] [] [1 ] Overfull \hbox (1.49698pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 the form (1 0;0 *) where * is a tamely ramified char and thi s repn[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 has conductor 1. That's why "1" is the entry for "cond" in t ype 2.[] [] Overfull \hbox (17.24684pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 But now you can do more; take the _determinant_ of this repn ; it will[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 image will have order 3 (which is great because p=1 mod 3 so there[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 are some of these) and this is the character associated to t ype 2.[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 I guess in the input to your program this would just be deno ted "3".[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 Now I'm brave enough to try 17. IT looks nasty though, doesn 't it.[] [] Overfull \hbox (48.74657pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 First we lift to SL_2(C). I reckon this A_4 in PSL_2(C) must become a group[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 of order 24 with a centre of order 2; there _is_ one of thes e, it's[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 just the auts of the elliptic curve over F_2-bar with j-inva riant 0.[] [] Overfull \hbox (17.24684pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 I think it's something like SL_2(Z/3Z). The D_2 in here will probably[] [] Overfull \hbox (38.24666pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 become a Sylow 2-subgroup of this which is probably a strang e non-abelian[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 thing...erm..it's (1 0;0 1) (-1 0;0 -1) +/-(0 1;-1 0) +/-(1 1;1 -1)[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 and +/-(-1 1;1 1 ) mod 3. OK so now we have to explicitly fi nd a 2-d[] [] Overfull \hbox (59.24648pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 faithful complex repn of this group; ugh. Well I guess +/-(1 0;0 1) can go to[] [] Overfull \hbox (17.24684pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 OK now what's the conductor of this representation? Well...h mm...this[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 is subtle. I don't know what's going on here. Is there now a G_2 or[] [] Overfull \hbox (32.9967pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 something? I'm a bit confused now. I think what must be goin g on is that[] [] Overfull \hbox (38.24666pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 this extra +-1 which we've introduced might make the thing m ore ramified,[] [] Overfull \hbox (27.74675pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 i.e....yeah I guess it must. I think G_2 is now +-1, whereas in the A_5[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 conductors?* Is it in his book? I think that what must have happened[] [] Overfull \hbox (32.9967pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 It's really hard to believe that twisting can make this repn any better.[] [] Overfull \hbox (32.9967pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 But if that were the case then we'd end up showing that the lift to SL_2[] [] Overfull \hbox (48.74657pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 was minimal, so the character would be trivial because we're going to SL_2.[] [] Overfull \hbox (38.24666pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 And this can't be right because chi(-1) has to be not 1 for _some_ prime![] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 I have to go out. I think something weird is happening. Mayb e G_1 is[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 still cyclic of order 4, say generated by (1 1;1 -1). Then t wisting[] [] Overfull \hbox (17.24684pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 by a character of order 4 will do wonders, it will make the conductor[] [] Overfull \hbox (27.74675pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 (or weight 2 character of order 4 at 5 and the same for the other ones)[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 and then see if there's anything there which looks like it's giving[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 and a linear; then T_p will be 0 for those p. Alternatively try and[] [] Overfull \hbox (22.4968pt too wide) in paragraph at lines 120--120 []\OT1/cmtt/m/n/10 Ps I don't have time to proof-read this unfortunately. I hop e it's OK.[] [] [2] [3] (kevin_1376.aux) ) Here is how much of TeX's memory you used: 196 strings out of 10898 1934 string characters out of 72048 50156 words of memory out of 263001 3199 multiletter control sequences out of 10000+0 3808 words of font info for 15 fonts, out of 200000 for 1000 15 hyphenation exceptions out of 1000 23i,4n,17p,132b,300s stack positions out of 300i,100n,500p,30000b,4000s Output written on kevin_1376.dvi (3 pages, 7372 bytes).