On projectively rational lifts of mod 7 Galois representations Iftikhar Burhanuddin and Luis Dieulefait

Abstract We consider the problem of constructing “projectively rational” lifts of odd, two-dimensional Galois representations with values in F7 . Using modular forms, in particular the theory of congruences, we compute such lifts for many examples of mod 7 representations thus giving evidence that suggests that such lifts may always exist. We also consider the invariance after twist (weight change) of the existence of such lifts.

1

Introduction

In previous articles of D. Calegari and the second author (cf. [Ca] and [Di1]) examples of odd two-dimensional Galois representations with values in F7 were constructed such that because of a local obstruction (at the prime 2) the non-existence of lifts to a 7-adic representation corresponding to an elliptic curve was established. In this paper we examine the more general problem of lifting the projectivization of such mod 7 representations to 7-adic representations which are “projectively rational”, such as Galois representations attached to elliptic curves over Q, to Q-curves, or to rigid Calabi-Yau threefolds. Projectively rational means that the projective representation has Q as field of coefficients, and this is known to be equivalent to a condition on the traces ap of the images of Frobenius at unramified places (cf. [Ri1]): these traces must all satisfy a2p /ε(p) ∈ Z where ε is the Dirichlet character appearing in the determinant of the Galois representation (the “nebentypus” in the case of modular Galois representations).

1

Since for this projective version of the lifting problem no local obstruction shows up, it is reasonable to ask the following question: Does such a projectively rational 7-adic lift exist for every given odd two-dimensional Galois representation with values in F7 ? Using explicit computations with modular forms and the theory of congruences of modular forms, we have gathered experimental evidence that suggests that the answer to this question is affirmative. In particular, since Serre’s conjecture has recently been proved (cf. [KW1], [KW2] and [Di2]), using modular forms we can compute all irreducible Galois representations with coefficients in F7 , of Serre’s weight 2 and conductor up to a certain bound C and try to find a projectively rational lift for the projectivization of each of them. Taking C = 100 we have found for all such examples a modular form of weight 2 that solves the problem. We also have examples where we start from mod 7 representations obtained from weight 4 modular forms. We consider an example of Serre (given in his paper on Serre’s conjecture) where he takes a residual representation with values in F49 whose projectivization has values in F7 and he finds a weight 3 modular form that (now that Serre’s conjecture is a theorem we can easily deduce that) gives a 7-adic lift of it. We will show that this lift constructed by Serre is projectively rational. Finally, we will investigate the invariance after twist of the above question by computing lots of examples of mod 7 Galois representations ρ corresponding to weight 4 modular forms with integral coefficients and proving for most of them that there exists also an elliptic curve over Q such that the corresponding projective mod 7 representation is ρ. We note that reducible cases oftentimes occur but we do not present them in our tables. We would like to thank William Stein for providing us access to the meccah.math.harvard.edu and sage.math.washington.edu computers. The examples we present were computed using Magma and Sage mathematics software systems.

2

Weight k = 2 and level N ≤ 100

General Comment: The residual representations that we will consider arise from certain modular forms. Then to prove that a certain projectively rational Galois representation attached to another modular form gives a lift 2

amounts to proving a congruence between two modular forms. In general, to prove such a congruence one has to check the congruence between the eigenvalues ap up to a bound known as Sturm’s bound (cf. [St]) that depends on the weight and the level of the two modular forms (if the levels are different one takes the least common multiple of them). In the case of raising the level, i.e., if one of them has level N and the other has level N q where q satisfies the required condition for raising the level of the first modular form mod 7, since by Ribet’s results on raising the level it is known that the one of level N has to be congruent to some of the newforms of level N q, if after comparing a few eigenvalues mod 7 only one of them stands up, then by elimination the congruence must necessarily hold and there is no need to compare eigenvalues up to Sturm’s bound.

2.1

Weight 2 newforms that do not correspond to elliptic curves

We consider newforms f of weight 2 and squarefree levels from 1 to 100 that have non-rational coefficients (not elliptic curves) living in a number field, say, Kf . For each such form f we compute the reduction of f modulo prime ideals of Kf sitting above (7) with residue fields being F7 . We search for congruences between these reduced forms and forms corresponding to elliptic curves over Q with their coefficients reduced modulo 7. The elliptic curve can have the same level as f , its level can be a divisor of the level of f (lowering the level, as in the example of the level 55 newform that is congruent modulo 7 to the elliptic curve of conductor 11), or it can be a multiple of the level of f (raising the level, as in most examples). References for raising and lowering the level are [Ri2] and [Ri3] respectively. In the table below, let nf denote the case when the residue field associated to a form f and a prime ideal of Kf is not F7 and parentheses are used to enclose information about congruences or otherwise resulting from each prime ideal of Kf sitting above (7). As we have said in the introduction, we only consider irreducible residual representations. There are at most 2 newforms with non-rational coefficients in the levels we consider and these are labeled as f1 and f2 in the table. Levels with no newforms are omitted. In all these examples presented, we found a congruence with an elliptic curve.

3

N 23 29 31 35 39 41 43 47 51 53 55 59 61 62 65 67 69

f1 f2 nf 551(∗) nf nf 195(∗) nf 258(∗) 141 nf nf (11, 110) 3835 nf nf 195(∗) nf nf nf nf

N 71 73 74 77 79 82 83 85 86 87 89 91 93 94 95 97

f1 3337 nf nf nf nf 246(∗) nf (170, 595) 258 nf 178 182(∗) nf (470, 1786) nf 4171

f2 (∗) nf nf

nf nf 435 nf 186 nf nf

Table 1: Newforms of weight 2 and squarefree levels 1–100 with non-rational coefficients. The symbol nf denotes residual field is not F7 . The symbol (*) denotes that there is a residual reducible representation with values in F7 attached to this newform. In all these examples, we found a congruence with an elliptic curve.

3

Weight k = 4 and level N ≤ 70, 7 | N

We list the levels N ≤ 70 divisible by 7 such that we have found weight k = 4 newforms (with trivial nebentypus) whose field of coefficient is not Q but giving some irreducible mod 7 representation with values in F7 . In all cases we have found a projectively rational lift given either by: (a) a weight 4 modular form that is congruent mod 7 to the given one, or by (b) a weight 2 modular form that is congruent mod 7 to the given one after twisting by the mod 7 cyclotomic character.

4

In each case we indicate what is the level of the modular form that solves the problem, and if the modular lift found has coefficients in Z or squares of coefficients in Z (we only consider in this section newforms with trivial nebentypus). In all cases where we have added primes to the level we have used the recipe given by Ribet’s raising the level theorems to know which primes can be added. N = 21 : Qf quadratic, only one irreducible mod 7 representation (the other one is reducible). Congruent as in (b) to a newform of level 21 “in Z” (i.e., an elliptic curve). N = 35: Qf quadratic, 2 irreducible mod 7 representations. One is congruent as in (a) to a level 35 newform “in Z”, the other one is congruent as in (b) to a level 35 newform “in Z” (an elliptic curve). N = 35: Qf cubic, only one prime dividing 7 in Qf with inertial degree 1, so only one irreducible mod 7 representation with values in F7 . Congruent as in (a) to a level 70 newform “in Z”. N = 49: Qf quartic, but gives only three mod 7 irreducible representations (there are just 3 primes dividing 7 in Qf ). Two of them are congruent as in (a) to newforms of level 98 “squares in Z”. The third is congruent as in (a) to a newform of level 49 “in Z”. N = 56: Qf quadratic, 2 irreducible mod 7 representations. One is congruent as in (a) to a level 56 newform “in Z”, the other one is congruent as in (b) to a level 56 newform “in Z” (an elliptic curve). N = 63: Qf quadratic, 2 irreducible mod 7 representations. One is congruent as in (a) to a level 63 newform “in Z”, the other one is congruent as in (b) to a level 63 newform “in Z” (an elliptic curve).

4

The example of Serre

In his 1987 paper on Serre’s conjecture (cf. [Se], page 225), J.-P. Serre considers a mod 7 representation coming from a polynomial with Galois group PSL2 (F7 ) and he finds a modular form f of weight 3, level 27 and 5

nebentypus equal to the character θ corresponding to the quadratic extension √ Q( −3) such that it seems to give a modular lift of the Galois representation. More precisely, he first shows that the projective representation can be lifted to a Galois representation ρ¯ with values in the union of GL2 (F7 ) and wGL2 (F7 ) where w is a square root of −1 in F49 . Then, he checks numerically (i.e., he checks equality for several primes) that the traces of ρ¯ match with the mod 7 reduction of the eigenvalues of a modular form f of weight 3, level 27 and nebentypus θ. In the new century, since we know that Serre’s conjecture (both the weak and the strong form) is a theorem, we can easily prove (by elimination!) that f truly gives a lift of ρ¯, using the fact that the Serre’s weight of ρ¯ is 3 and its Serre’s level is 27 (as computed by Serre). The field generated by the eigenvalues of f is Q(i), thus we obtain an s-adic modular Galois representation lifting ρ¯ with values in GL2 (Z[i]s ) where s denotes the prime in Z[i] dividing 7. An easy computation shows that f has an inner twist, i.e., that it holds a2p /θ(p) ∈ Z for every p prime to 3. We conclude that the modular s-adic lift that Serre constructs in this example is also projectively rational.

5 5.1

Invariance after weight change? From weight 4 to weight 2 Levels co-prime to 7

Each (irreducible) newform f of weight 4 and level N (co-prime to 7) with rational coefficients is tested for congruences modulo 7 against elliptic curves over Q of level 49 ∗ N . More precisely, we consider the residual mod 7 representation attached to the weight 4 modular form, we twist it by the inverse of the mod 7 cyclotomic character, and we look for an elliptic curve whose corresponding mod 7 representation agrees with this twisted representation. If this fails to produce a congruence, the form f is tested for congruences withQelliptic curves of level N ∗ 49 ∗ r, where either r = q1 is a prime or r = qi is a product of primes, where in both cases the primes in question satisfy aqi ≡ ±qi (qi + 1) mod 7. In our program, newforms f with level N ≤ 100 are tested against elliptic curves from the Cremona database, which contains all elliptic curves up to 6

conductor 120, 000 and this limits the nature of our computation. If there does not exist a congruence at the minimal level of 49 ∗ N , f is compared against elliptic curves of higher level. Each “level-raising” prime should be less than 120, 000/N ∗49 due to the Cremona database limit. At most 15 such primes are considered in the program. If none of these result in a congruence, then a product of two such primes is used in “level-raising”. The only levels where congruences were not found using our program are 17, 53, 62, 73, 85, 89 and 95. Examples of congruences at nonminimal levels follow. There is no congruence minimally in an example with N = 15, though when 2 is added the level, congruences result. The first instance where a product of two primes appears in raising the level is at N = 71. The “level-raising” primes 2, 5, 13, 17, 31 do not give rise to a congruence but adding 2 ∗ 13 to the level does. Remark: In the example of level 44, as can be seen in the table that follows, we found a congruence with an elliptic curve of conductor 62524. There is also a projectively rational modular Galois representation attached to a modular form of weight 2 and level 44 ∗ 49 = 2156 (listed as 2156E1 in W. Stein’s Modular Forms Database) giving a mod 7 congruence for this example. The √ coefficient field of this modular form is Q( 2), and it is easy to see that it has an inner twist (this is why the attached Galois representations are projectively rational). Moreover, since the Galois representations attached to this modular form have semistable ramification at 11 this newform with inner twist is known (by a result of Ribet) to correspond to a Q-curve. So for the example of level 44 and weight 4 we have both an elliptic curve and a Q-curve giving a mod 7 congruence.

7

N 5 6 8 9 10 12 13 15 15 16 17 18 19 20 22 22 22 23 24 25 26 26 26 27 27 30 30 32 32 32 33 33 34 34 36 38

congruences 245b1 294e1, 294f 1 392b1 441a1, 441b1 490i1, 490j1 588d1, 588e1 637a1, 637c1 1470g1, 1470i1 735b1 784d1 882j1, 882l1 2793e1 980e1, 980f 1 5390r1 1078d1 1078m1 3381g1 1176h1 1225d1 1274b1 1274m1 1274l1 1323q1 1323j1 1470g1, 1470i1 1470o1, 1470q1 1568h1 1568b1 1568f 1 1617e1, 1617f 1 3234q1 1666h1 4998m1, 4998p1 1764h1, 1764i1 5586n1

N 39 40 40 40 44 45 45 45 45 45 46 46 48 48 48 50 50 50 51 51 51 52 53 54 54 54 54 55 57 58 58 60 60 62 62 64

congruences 1911a1 1960k1 1960l1, 1960m1 1960c1 62524d1 2205c1 2205l1, 2205m1 4410bc1, 4410bg1 2205l1, 2205m1 2205b1 2254d1 6762bj1 2352m1, 2352n1 2352b1 2352x1, 2352y1 2450r1 2450h1, 2450i1 2450bg1 4998bc1 4998bn1 4998i1 2548f 1 2646n1 2646c1, 2646d1 2646ba1, 2646z1 2646x1 5390h1 5586y1 31262l1 2940c1 2940e1 3038e1 3136h1 8

N 64 64 64 64 65 66 66 68 71 72 72 72 72 73 74 74 75 75 78 78 78 78 78 78 80 80 80 80 80 80 82 82 85 85 85 86

congruences 3136w1 3136u1 3136g1 3136l1 3185c1 3234m1 3234q1 3332c1 90454d1 3528o1 3528h1 3528w1 3528c1 54390e1 10878ba1 3675m1 7350bs1, 7350bx1 3822c1 3822k1 3822p1, 3822q1 3822s1 3822y1, 3822z1 3822bd1 3920e1 3920i1, 3920k1 3920bb1, 3920be1 3920y1 3920g1 3920bg1, 3920bh1 52234z1 4018r1 4165k1 4165p1

N 86 88 88 89 89 90 90 90 90

congruences 4214d1 4312i1 56056b1

N 90 93 95 95 26166r1 95 4410k1, 4410l1 95 4410a1 96 4410q1, 4410u1 96 4410bc1, 4410bg1 96

congruences 4410x1 4557h1 4655i1 4655n1 4655l1 4704d1 4704t1 4704e1

N 96 96 96 99 99 100 100 100

congruences 4704m1 4704bb1 4704bc1 9702s1 4851r1, 4851s1 4900e1, 4900h1 4900j1 4900u1

Table 2: Newforms of weight 4 and levels between 1–100 co-prime to 7. There is a congruence with an elliptic curve in most of these examples.

5.2

Levels divisible by 7

In this section we present results of modulo 7 congruences involving (irreducible) newforms of weight 4, level N ≤ 100 (divisible by 7) and rational coefficients. Congruences are searched for with elliptic curves of levels N, 7∗N and 49 ∗ N , raising the level if necessary. Elliptic curves are read from the Cremona database and raising the level is performed as stated in the previous section. The newforms which did not yield a congruence using the above procedure were compared against elliptic curves of conductor which were divisors of 7 ∗ N . Examples of this scenario are the newform at level 77 and one of the forms at level 98.

9

N 14 21 21 28 28 35 42 42

congruences 14a1 147c1 147c1 196a1 196a1 490b1 42a1 546f 1, 294b1

N 56 56 63 63 70 70 70 70

congruences 392f 1 56b1 441f 1 441f 1 490b1 490d1 490b1 490d1

N 70 77 84 84 98 98 98

congruences 70a1 11 84b1 84a1 98a1 98a1 14

Table 3: Newforms of weight 4 and levels between 1–100 divisible by 7. In all these examples, we found a congruence with an elliptic curve.

6

Final Comments

We should stress that the computations that we have performed in this paper are to be considered as part of the study of an interesting and obscure problem which is the control of coefficient fields of deformations of Galois representations. Since the only known examples of residual representations with values in a finite prime field (of characteristic greater or equal to 7) such that it can be proved that they do not have any lift corresponding to an elliptic curve have been constructed using local obstructions (cf. [Ca] and [Di1]), it is natural to investigate if in the absence of local obstructions a global lift with rational coefficients (at least projectively) must always exist or not. In general, the minimal degree among fields of coefficients of lifts or projectivizations of lifts (minimal among those lifts such that their traces of Frobenius elements generate a number field) is an elusive yet important invariant: a control of this invariant could have applications to proofs of modularity by induction on this degree via suitable congruences, in the spirit of the proof of Serre’s conjecture over F9 given in [Ell].

10

7

References

[Ca] Calegari, D., Mod p representations on Elliptic Curves, Pacific J. of Math. 225 (2006) 1-11. [Di1] Dieulefait, L., Existence of non-elliptic mod ` Galois representations for every ` > 5, Experiment. Math. 13 (2004) 327-329. [Di2] Dieulefait, L., Remarks on Serre’s modularity conjecture, preprint (2006), available at: http://www.arxiv.org [Ell] Ellenberg, J., Serre’s conjecture over F9 , Ann. Math. 161 (2005) 11111142. [KW1] Khare, C.; Wintenberger, J.-P., Serre’s modularity conjecture (I), preprint (2006), available at: http://www-irma.u-strasbg.fr/∼wintenb/ [KW2] Khare, C.; Wintenberger, J.-P., Serre’s modularity conjecture (II), preprint (2008), available at: http://www-irma.u-strasbg.fr/∼wintenb/ [Ri1] Ribet, K., Twists of modular forms and endomorphisms of abelian varieties, Math. Ann. 253 (1980) 43-62. [Ri2] Ribet, K., Raising the levels of modular representations, S´eminaire de Th´eorie des Nombres, Paris 1987–88, 259–271, Progr. Math., 81, Birkh¨auser Boston, Boston, MA, 1990. [Ri3] Ribet, K., On modular representations of Gal(Q/Q) arising from modular forms, Invent. Math. 100 (1990), no. 2, 431–476. [Se] Serre, J.-P., Sur les repr´esentations modulaires de degr´e 2 de Gal(Q/Q), Duke Math. J. 54 (1987) 179-230. [St] Sturm, J., On the Congruence of Modular Forms, Number theory (New York, 1984-1985), 275-280, Lecture Notes in Math., 1240, (1987) Springer.

11

On projectively rational lifts of mod 7 Galois ...

sider the invariance after twist (weight change) of the existence of such lifts. 1 Introduction. In previous articles of D. Calegari and the second author (cf. [Ca] and [Di1]) examples of odd two-dimensional Galois representations with values in F7 were constructed such that because of a local obstruction (at the prime 2).

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