A control theorem for the images of 7 0 Galois actions on certain infinite 0 2 families of modular forms n a J Luis Dieulefait 8 1 Dept. d’A´lgebra i Geometria, Universitat de Barcelona e-mail: [email protected] ] T N . h t 1. A letter with the result a m November 11, 2006 [ 1 Dear Colleagues, v 7 After reading a recent preprint by G. Wiese on the images of Galois re- 1 presentations attached to classical modular forms and applications to inverse 5 1 Galois theory I have been thinking again on the results of Ribet and Serre gi- 0 ving “image as large as possible for almost every prime”in the non-CM case. 7 TheresultofWieseisusedtorealizeasGaloisgroupoverQthegroupPGL2(ℓr) 0 or PSL2(ℓr) for a fixed ℓ and exponent r larger than any given exponent r0. I / h haven’treadhisproofindetail,butsinceituses“good-dihedral”primesandthe at result is for fixed ℓ, I can imagine that both the good-dihedral prime and the m modular form are constructed ad hoc to realize only the desired linear group in this specific characteristic (and large-but-unknown exponent). In any case, : v sincehismethodworksforeveryℓ,theresultthatheobtainsisquiteinteresting. i X r Ontheotherhand,ofcourse,youandIbelievethatastrongerresultistrue, a namely, a uniform result: so, assume for a moment that I just consider, for a fixed small prime t and every exponent n a modular form fn of level un ·tn and weight 2 (and trivial character) (*), where un is prime to t, without CM. Assume also that we know somehow that the images of the ℓ-adic and mod ℓ Galois representations attached to fn are “as large as possible”for every prime ℓ and for every fn. In other words, the special family fn has image large FOR EVERY PRIME instead of just for almost every prime (which is what Ribet’s result shows in general for any modular form without CM). If such a family fn exists, the well-known relation between the conductor and the minimal field of definition for a Galois representation with values on a finite field (as explained 1 in Serre’s Duke 1987 paper on modularity conjectures and exploited by Bru- mer to show that for a large power tn in the level the field of coefficients of the projective representations attached to fn must contain the real part of a cyclotomicfield oftm-rootsofunity,with m goingto infinity as n does)implies thatthe family fn gives anotherproofofWiese’s Galoisrealizationsresult,and in a uniform way: for any given ℓ and exponent r0 we know that taking any element fn with n sufficiently large in our family we will be realizing not only thedesiredprojectivelineargroupincharacteristicℓandwithexponentgreater thanr0 butalsoasimilarlineargroupforasetofprimesofdensityascloseto1 asdesired(butsmallerthan1),alwayswithexponentlargerthanr0.So,instead of realizing the desired “linear group over a finite field with large exponent”for an isolated prime our modular forms fn will do the job for a large density set of primes. As far as I know, it is not yet known how to construct an infinite family of modular forms with growing level fn as the one described above, having all of them large image for every prime. But I can construct a family with a slightly weaker property, that is still good enough to derive the above conclusion re- garding realizations of linear groups as Galois groups over Q. In particular the family fn that I have found with levels as in (*) has, of course, the property that the degree of the corresponding field of coefficients goes to infinity with n (because of Brumer result and the factor tn in the level), and concerning the images of the corresponding Galois representations it has the property that for eachfn withn≥4wecangiveanupper boundtothesetofexceptionalprimes computed as a function only of the level of fn (i.e., all the information we need is the value of the level, not a single eigenvalue is needed) and in particular we caneasily show that for any givenprimeℓ>3 there is a value n0 such that ℓ is notexceptionalforfn foranyexponentn>n0,wherehere“exceptional”means dihedral, reducible or (for ℓ = 5) some of the other cases of small image in Dickson’s result. Let us show one example of such a family fn: for any n ≥ 4 take fn to be ANY modular form of level 2·3n, weight 2, trivial character. Because of semistability at 2 none of them has CM. Since the large ramification at 3 for n sufficiently large makes easy to see (again, using the ideas of Serre and Brumer on conductors) that the small special groups in Dickson’s list can not occur for ℓ=5anditiswellknownthatthesegroupscannotoccurforlargerℓinweight 2, we can concentrate in the two problems that have to be solved to control the images of the representations attached to fn for any n: to control dihedral primes and to control reducible primes. In both cases we will use the large ra- mification at 3 and the semistability at 2 to do so. Dihedral primes: Let n≥4 and ℓ>3 be a dihedral prime for fn. Using the arguments created by Serre and Ribet, we see that the only possibility is that the mod ℓ representation is induced from the quadratic number field ramifying only at 3. Also, since a dihedral image does not contain unipotent elements, 2 this mod ℓ representationis unramifiedat2.Since 2 is a non-squaremod 3 this impliesthatthetracea2 oftheimageofFrob2inthismodℓrepresentationhas to be 0 (i.e.: as usual in the residually CM case, half of the traces have to be 0,and a2 is in that half). Onthe other hand the ℓ-adic representationattached to f has semistable ramificationat 2,so we are in a case of raising the level (or lowering the level, depending on the perspective), and as you know very well this can only happen if a2 = ±3 (these are numbers which only exist mod ℓ). Putting the two things together we conclude that 0 and ±3 are the same mod ℓ, and since ℓ>3 this gives us a contradiction. Reducible primes:this time assume ℓ>3 is a reducible prime for fn, n≥4. We anticipate that now such a prime can exist (for example 7 is reducible for some newforms of level 162) but we just want to bound the set of reducible primes in terms of the level 2·3n. Again, we will use the local information at 2 and 3 to do so. For simplicity of the exposition, we assume that n = 2·u is even. Since the mod ℓ representation is reducible (we semisimplify if necessary, so assume it is semisimple) and using the value of the level of fn (and, because ℓ>3,it is well-knownthat residuallythe conductorat 3 will be exactly 3n) we knowthatitis justthe directsumχ·ψ⊕ψ−1,whereχ isthe modℓ cyclotomic character and ψ is a character of conductor exactly 3u (remember that u is half of n, so it is at least 2). Computing the trace of the image of Frob 2 for the mod ℓ representation this time we obtain a2 = 2·ψ(2)+ψ−1(2). Here the importantthing to observeis thatthe orderofψ is φ(3u)=2·3u−1 (where φ is u Euler’sfunction),andthat2isprimitivemodulo3 ,sotheorderoftheelement ψ(2) is also 2·3u−1. On the other hand, using again raising the level since the ℓ-adic representation is semistable at 2 we must have a2 = ±3. Comparing the two formulas for a2 the first observation is that the roots of the characteristic polynomial of the image of Frob2 are 1,2 or −1,−2, in particular they belong to the prime field Fℓ, so ψ(2) must be in this field, and looking at the order of thiselementthismeansthat3u−1 dividesℓ−1(@).Thisalreadyshowsthatfor n sufficiently large any prime ℓ given a priori will not be reducible (thus, will not be exceptional), because the maximal power of 3 dividing ℓ−1 is finite. Just for fun, let us bound the set of possibly reducible primes for fn: Compa- ring the two formulas for a2 (comparing the roots of the polynomials deduced frombothformulas)andusingtheinformationontheorderofψ(2)weconclude that any reducible ℓ must satisfy, in addition to (@), the condition: ℓ divides 22·3(u−1)−1.So,thisisaboundforthesetofreducibleprimesforfn.Forexam- ple for n = 4 (thus u = 2) we conclude that ℓ has to be congruent to 1 mod 3 anddivides26−1=63,thustheprime7maybereducible(anditissoforsome newforms of level 2·34 =162),but it is the only possible reducible prime ℓ>3 (computing reducible primes for all newforms of this level using the method in my thesis confirms this fact). Conclusion: For any newform in the family fn described above, if n = 2·u (we assume it is even for simplicity) the residual image is “large”for any prime 3 which is not congruent to 1 modulo 3u−1. This, together with the fact proved bySerreandBrumerthatasn(the 3-partoftheconductor)goestoinfinity the exponents of the fields of coefficients of the projective residual representations also go to infinity, has as a corollary that with our family fn we are realizing, for any prime ℓ > 3, projective linear groups over the field of ℓr elements for r arbitrarilylargeas Galois groupsoverQ,and we arerealizingthese groupsina uniform way (i.e., for sufficiently large n we obtain these groups not only for a given ℓ but also for large density sets of primes, all with large exponent). Each of them is realized as an extension unramified outside 6·ℓ. Ofcoursewecanconstructothersimilarexamplestakingothersuitablepairs ofprimes insteadof 2 and 3,we canalso take more generallevels having semis- table ramificationat more primes,and other variations.The main point is that wecanboundthesetofexceptionalprimesforALLmodularformsinaninfinite family of increasing conductor, which is an interesting result that of course can not be obtained using just the computational method explained in my thesis years ago. It is interesting to observe how a very simple ramification condition (onesemistableprimeandotherdividingtheconductorwithalargepower)was enoughto obtain“uniformlylarge”images,to explainthat only primesthat are “very splitc¸an be exceptional, thus to generate a lot of large exponent linear groups as Galois groups. Maybe other combinations of ramification conditions can lead to similar, or even stronger, results. The idea used to control dihedral primes is an idea I had in Paris in 2002, when considering dihedral primes for the case of Q-curves coming from diop- hantineequations.Thenew ideaisthe ideatocontrolreducible primes,whichI had in Berkeley last week during the modularity conference (but I knew since I saw Wiese’s paper months ago that the arguments of Serre and Brumer should bekey:obtaininglineargroupsoverfieldswithlargeexponentsasGaloisgroups usingtheseresultsissomethingIwantedtodoalreadywhenstartingmythesis). Best regards, Luis Dieulefait 4