This is TeX, Version 3.14159 (Web2C 7.3) (format=latex 1999.4.5) 7 JUN 1999 14:30 **\nonstopmode\input newidea.tex (newidea.tex (/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 ) (newidea.aux) \openout1 = `newidea.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 (11.99689pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 Hey! I just realised that by complete chance "type 17" is _a lso_ the[] [] Overfull \hbox (17.24684pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 type at 2 of Buhler's 800 repn!! So we can just look at what Buhler's[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 Do you happen to know how long it took Frey et al to do the one that[] [] Overfull \hbox (22.4968pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 we might be able to do?! Probably not as long. Aah well. I s till don't[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 know whether we were "unlucky" or whether their method is in herently[] [] Overfull \hbox (53.99652pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 I guess it's a shame that the computation of the new one...i s that 17051?...[] [] Overfull \hbox (27.74675pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 will take a year. I've never searched beyond 17051 so I don' t know what[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 Maybe it's time to re-trace our steps. Here's one random ide a that[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 Frey's list: it's 1376 (the first conductor is bad because I think[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 be possible, it's probably in Buhler, which I don't have to hand).[] [] Overfull \hbox (32.9967pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 Unfortunately we note that a character of order 3 won't ever take values[] [] Overfull \hbox (22.4968pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 in Z/5Z. So the coefficients of this form will probably take values in[] [] Overfull \hbox (17.24684pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 Unfortunately the level has now gone up! It's now 1376*43. P resumably[] [] Overfull \hbox (22.4968pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 this is too big to work with? I guess one would have to work at weight[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 in Z/5Z (even the troublesome a_5 !!). Is there a trick by w hich we[] [] Overfull \hbox (27.74675pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 can work at weight 5 and level 1376 (or even weight 2 and le vel 1376*5)[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 Galois is not in fact all of GL_2(F_25) but that it's just a twist[] [] [1 ] Overfull \hbox (6.74693pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 would maybe boil down to proving that a_p/chi(p) was always in F_5,[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 There should be a weight 1 form of level 1376 and character of order[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 3 at 43 and something funny at 2, whose Galois repn to GL_2( C) gives[] [] Overfull \hbox (17.24684pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 rise in the standard way to the second A_5 extension on p137 of Frey.[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 Say this form is sum a_nq^n. My belief is that you are now a ble to[] [] Overfull \hbox (48.74657pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 and level 1376, or weight 2 and level 1376*5 and character o f order 4 at 5.[] [] Overfull \hbox (17.24684pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 However you do it, I think you can find this one. The proble m is that[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 it won't be defined over Z/5Z, it'll probably be defined ove r F_25.[] [] Overfull \hbox (38.24666pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 Having found it though, you could now look for its twist by the character[] [] Overfull \hbox (17.24684pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 of conductor 43 and order 3 which was the local component of the form[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 you found. This form has got level equal to 43 * the old lev el, and[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 the twist by a character of a modular form is a modular form again,[] [] Overfull \hbox (27.74675pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 1) Find the F_25 form at level 1376 and weight 5 (or wt 2 le vel 1376*5)[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 3) Check that it lives in a Hecke-invariant 1-dimensional sp ace, and[] [] Overfull \hbox (48.74657pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 and then we'd probably be done? modulo a _really_ nasty calc ulation at 2 :)[] [] Overfull \hbox (59.24648pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 This might or not might work. Raw experience has shown that usually something[] [] [2] Overfull \hbox (64.49643pt too wide) in paragraph at lines 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\OT1/cmtt/m/n/10 so the conductor of the repn to GL_2 is 2. Well this is all well and[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 439--439 []\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 439--439 []\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 439--439 []\OT1/cmtt/m/n/10 But now you can do more; take the _determinant_ of this repn ; it will[] [] [3] Overfull \hbox (1.49698pt too wide) in paragraph at lines 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\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 439--439 []\OT1/cmtt/m/n/10 Ps I don't have time to proof-read this unfortunately. I hop e it's OK.[] [] [4] Overfull \hbox (43.49661pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 > Aargh!!! It knows the answer but it won't tell me! It w on't even tell[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 > I've been frustrated by this before. It you type "m" to t urn on[] [] Overfull \hbox (27.74675pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 > matrix display, then "c" to turn off charpoly display, bef ore running[] [] Overfull \hbox (22.4968pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 > I'll fix all of this when i get chance. Just tell me what you want.[] [] Overfull \hbox (48.74657pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 Rather interestingly, this comment really got me thinking: " well, what _do_[] [] Overfull \hbox (38.24666pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 level 2^N*43^2 which will be too big?". So I thought about w hat you might[] [] Overfull \hbox (22.4968pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 do anything at all. In fact we might have a lot of new examp les at our[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 fingertips. This is assuming that a certain calculation work s: but[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 calculations that I've been confident will work before have failed![] [] Overfull \hbox (32.9967pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 Anyway, here's the plan. There is this A_5 extension which i s unramified[] [] Overfull \hbox (17.24684pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 outside 2 and 43; at 43 it's of type...erm...can't remember, but it's[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 the type which has e=3 and f=1 at 43, so the inertia group a t 43 is[] [] Overfull \hbox (53.99652pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 cyclic of order 3 and equals the decomposition group. I thin k this is right;[] [] Overfull \hbox (22.4968pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 I'm at home and working from memory. Anyway, a minimal lifti ng of this[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 to GL_2(C) (minimal in the sense that the conductor is minim al) is[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 just 1+chi where chi is the character of order 3. This chara cter of[] [] Overfull \hbox (17.24684pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 order 3 doesn't reduce to Z/5Z; it forces us up to F_25 and initially[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 I just rejected these cases because I thought we wouldn't be able to[] [] Overfull \hbox (17.24684pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 deal with them. But seeing as we've got stuck with the F_5 e xtensions[] [] Overfull \hbox (22.4968pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 I've thought more about this F_25 one and I think we should be able to[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 I think I have checked that there are no problems at 5; so I think[] [] Overfull \hbox (32.9967pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 form sum a_n q^n, with all the a_n in F_25, giving us this m inimal lift.[] [] Overfull \hbox (27.74675pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 and (f-g)/(some appropriate elt of F_25) will both have coef fts in F_5;[] [] Overfull \hbox (38.24666pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 hence we *should* be able to find a 2-d space at level 1376 and weight 5,[] [] Overfull \hbox (43.49661pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 mod 5, which we can prove is Hecke equivariant, and which, w hen you tensor[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 What will hapen in practice is that we'll find a 2-d space w here we[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 can't diagonalise Hecke over F_5 but we can over F_25; the p roblem[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 will be to show that the two forms we have, which a priori g ive us[] [] Overfull \hbox (22.4968pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 repns to GL_2(F_25), are actually twists of repns to GL_2(F_ 5). Here's[] [] Overfull \hbox (17.24684pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 my plan. Firstly forget all that notation above; let's just work with[] [] Overfull \hbox (59.24648pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 what we have. We have a 2-d space defined over F_5 and it's Hecke equivariant[] [] [5] Overfull \hbox (11.99689pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 will be defined over F_25. Moreover, g will have to be the c onjugate[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 of f because that's the only way it can be; non-scalar eleme nts of[] [] Overfull \hbox (17.24684pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 Hecke will be forced to be diagonalisable with conjugate eig envalues.[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 Now the bit of luck: I reckon that U_43 will have eigenvalue s equal[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 to the 2 non-trivial cube roots of 1 in F_25. So by general theory[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 I think that one can deduce that the repn attached to f, res tricted[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 a character of order 3. Now twist f by chi^{-1} (not by chi) . We get[] [] Overfull \hbox (59.24648pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 some form, probably an oldform; but an eigenform. Take the a ssociated newform[] [] Overfull \hbox (43.49661pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 g'. Then my guess is that one will be able to show that the level of g' is[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 still 1376!! This is because the only thing that can go wron g will[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 be that the conductor at 43 goes up; but the conductor of th e repn[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 at 43 will still only be 1. It's hard to believe that we'll have to[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 use something like Ken's level-lowering theorem but we can d o this[] [] Overfull \hbox (27.74675pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 if necessary! The point is that the conductor of 1+chi^{-1} is still 1.[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 Anyway, I think we'll be able to check that g' is an eigenfo rm that[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 be f or g; we rule out f instantly by checking T_p for the f irst few[] [] Overfull \hbox (17.24684pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 primes p and deduce that f tensor chi^{-1} is hence g=f-bar, at least[] [] Overfull \hbox (32.9967pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 away from 43. In other words, if n is prime to 43, I think w e have shown[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 Now this means that a_n=a_n^5*chi(n) and let's set c_n=a_n*c hi(n).[] [] Overfull \hbox (17.24684pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 We see that (c_n)^5=a_n^5*chi(n)^2=a_n*chi(n)=c_n; so c_n is in Z/5Z.[] [] Overfull \hbox (53.99652pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 So, although we haven't ever computed f*chi, and indeed it's uncomputable---[] [] Overfull \hbox (32.9967pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 it's at level 1376*43---we have just shown that all its a_p are in Z/5Z,[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 And surely this is enough. We know exactly what's going on a t 43 and[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 And surely there will be other forms with this property: I r ejected[] [] Overfull \hbox (32.9967pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 a _lot_ of A_5 extns on the basis that the minimal lifting h ad character[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 not invariant" errors; I'll try and re-produce them. But if we get[] [] Overfull \hbox (48.74657pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 this is the computation we have to do (I'll try it Monday mo rning UK time);[] [] Overfull \hbox (27.74675pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 compute the space of forms that are weight 5 mod 5 level 137 6 and which[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 have T_5, T_19, T_31... acting as 0. The moment we get down to a 2-d[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 space I think we've won. We didn't win before when we were t rying to[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 get down to a 1-d space because a_5 really didn't live in Z/ 5Z; but[] [] Overfull \hbox (22.4968pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 this time it seems to me that a_2, a_5 and a_43 will all liv e in F_25;[] [] Overfull \hbox (22.4968pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 in fact I think a_2=0 (not 100% sure about this), a_5=0 and a_43=omega[] [] Overfull \hbox (53.99652pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 I might be wrong. If this works we get loads of new examples I think and you[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 don't even have to compute these twisting by a character map s that[] [] [6] Overfull \hbox (27.74675pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 can all be in hecke-v2.0, when there are routines for giving individual[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 forms names like Delta; and then being able to compute Delta (nz) or[] [] Overfull \hbox (38.24666pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 William: I think a lot of what I wrote yesterday was rubbish : not because[] [] Overfull \hbox (64.49643pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 it was wrong, but because it hadn't really dawned on me that Hecke was capable[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 of working over finite fields other than Z/pZ; now I've fina lly seen[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 another "f" the moment I decide I want to work out another k ernel![] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 can I make Hecke compute T_p on it in a sensible way? I thin k this[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 an f command with ker(T_2), ker(T_3-(x+2)), ker(T_19) and ke r(T_31);[] [] [7] Overfull \hbox (6.74693pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 > When copmputing the modular degree it is necessary to cut out the[] [] Overfull \hbox (48.74657pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 > eigenspaces in both hom<9m,c<0 and m and then compute the discriminant of[] [] Overfull \hbox (53.99652pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 > I can work on hecke when I get to Leiden in 1.5 weeks. Th e only draw back[] [] Overfull \hbox (43.49661pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 > is that my laptop compiles code at about 1/15th the speed of my desktop![] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 in CA? Of course, if you have nothing at all better to do, t hen feel[] [] Overfull \hbox (48.74657pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 free to continue; I'm just emphasizing that I'm not definite ly putting time[] [] Overfull \hbox (17.24684pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 I have been checking more of the details of the 1376 thing a nd things[] [] Overfull \hbox (32.9967pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 are still looking good. For all primes <=41, the trace of Fr ob_p divided[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 by the appropriate cube root of 1 is in Z/5Z, and I convince d myself[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 that Deligne-Langlands-Carayol could be used to prove that f -bar is[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 a twist of f, without having to do any computations at level 1376*43[] [] Overfull \hbox (27.74675pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 (Shimura's result only says that f*chi will be modular of le vel 1376*43[] [] Overfull \hbox (27.74675pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 but I think I can prove it's an oldform). Next the computati ons do give[] [] Overfull \hbox (22.4968pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 us a 2-dim space where T_2,3,19 and 31 all act diagonally by the right[] [] Overfull \hbox (53.99652pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 character, but T_5 doesn't. I think I once convinced myself that a companion[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 form argument showed that the repn was unramified at 5 but I can't[] [] Overfull \hbox (17.24684pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 reproduce it and I also can't find the email I sent you abou t it. I'm[] [] Overfull \hbox (43.49661pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 At 43 the repn is tame by Deligne-Langlands-Carayol. At 2 we have to check[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 we took a minimal lift maybe but this won't be too hard, it' s purely[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 local. If I manage to reconstruct the argument at 5 we get t o deduce[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 that there's an extension K/Q, Galois, unramified outside 2 and 43,[] [] Overfull \hbox (17.24684pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 discriminant of root field at 43 is just 43^2, disc at 2 wil l be some[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 439--439 []\OT1/cmtt/m/n/10 might well be done (assuming these arguments I never wrote d own but[] [] [8] (newidea.aux) ) Here is how much of TeX's memory you used: 196 strings out of 10898 1919 string characters out of 72048 63096 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,160b,300s stack positions out of 300i,100n,500p,30000b,4000s Output written on newidea.dvi (8 pages, 25868 bytes).