The weight in Serre’s conjectures on modular forms. (English) Zbl 0777.11013

Let \(p\) be a prime number. In [J.-P. Serre, Duke Math. J. 54, 179- 230 (1987; Zbl 0641.10026)], to a continuous, irreducible, odd representation \(\rho: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \text{GL}_ 2(\overline{\mathbb{F}}_ p)\), Serre attached a triple \((N(\rho),k_ \rho,\varepsilon(\rho))\), where \(N(\rho)\) is a positive integer prime to \(p\) (the prime to \(p\) part of the Artin conductor of \(\rho\)), \(\varepsilon(\rho): (\mathbb{Z}/N(\rho)\mathbb{Z})^*\to \overline{\mathbb{F}}^*_ p\) is a character, and \(k_ \rho\) is a well defined positive integer called the weight. It depends only on the restriction of \(\rho\) to the (tame) ramification group at \(p\). Serre conjectured that for such a \(\rho\) there exists a cusp form \(f\) of type \((N(\rho),k_ \rho,\varepsilon(\rho))\), which is an eigenform of all Hecke operators \(T^*_ \ell\), \(\ell\) prime, such that \(\rho\) is isomorphic to the modular representation \(\rho_ f\) determined by \(f\). \(N(\rho)\) and \(k_ \rho\) should be as small as possible. Here it is shown that if \(\rho\) comes from a modular form at all, say of type \((N,k,\varepsilon)\), then it also comes from a modular form of type \((N,k_ \rho,\varepsilon)\) with \(k_ \rho\) (almost) minimal.
Write \(\rho_ p\) for the restriction of \(\rho\) to the decomposition group \(G_ p\subset \text{Gal} (\overline{\mathbb{Q}}/\mathbb{Q})\) at \(p\). \(G_ p\) can be identified with \(\text{Gal} (\overline{\mathbb{Q}}_ p/\mathbb{Q}_ p)\). Parallel to the definition of \(k_ \rho\) one defines an integer \(k(\rho)\). One always has \(k(\rho)\leq k_ \rho\), and, as a matter of fact there are only two cases where \(k(\rho)<k_ \rho\). These occur when the restriction of \(\rho\) to the wild ramification group \(I_ p\) at \(p\) is trivial: then \(k(\rho)=1\) and \(k_ \rho=p\). In the second case \(p=2\), the restriction of \(\rho\) to \(I_ p\) has a particular (non-trivial) form, and one has: \(k(\rho)=3\) and \(k_ \rho=4\). \(\rho\) is called exceptional if \(\rho_ p\) is isomorphic to an extension of two copies of an unramified character \(\varepsilon\) of \(G_ p\). Then the precise statement of the main result with respect to Serre’s conjecture is:
Let \(\rho: \text{Gal} (\overline{\mathbb{Q}}/\mathbb{Q})\to\text{GL}_ 2 (\overline{\mathbb{F}}_ p)\) be a continuous irreducible and odd representation. Suppose there exists a cusp form \(g\) of some type \((N,k,\varepsilon)\) with \(p\nmid N\), which is an eigenform for all \(T^*_ \ell\), such that \(\rho\cong\rho_ g\). Then there exists a cuspidal eigenform \(f\) of type \((N,k_ \rho,\varepsilon)\) which has the same eigenvalues for \(T^*_ \ell\) (\(\ell\neq p\)) as \(g\) has, such that \(\rho\cong\rho_ f\). If \(\rho\) is not exceptional then there exists an eigenform \(f\) of type \((N,k(\rho),\varepsilon)\) with the same eigenvalues for \(T^*_ \ell\) (\(\ell\neq p\)) as \(g\) has, such that \(\rho\cong\rho_ f\). If \(\rho\) is not exceptional then there is no eigenform of level prime to \(p\) and of weight less than \(k(\rho)\) whose associated Galois representation is isomorphic to \(\rho\).
For the proof one is led to construct an eigenform \(f_ 1\) of weight \(k_ 1\leq p+1\), such that \(\rho_ g\simeq\rho_{f_ 1}\otimes \chi^ a\), where \(\chi\) is the \(p\)-cyclotomic character, and then ‘untwist’ \(f_ 1\) by applying \(a\) times Tate’s \(\theta\)-operator, and finally divide as many times as possible by the Hasse invariant to obtain the desired form \(f\) of minimal weight \(k_ \rho\). To fill in the details one needs several side results. The proofs of these are technical and consume a great part of the article.
The paper closes with a multiplicity one result: Let \(f\) be a cuspidal eigenform of type \((N,k,\varepsilon)\), defined over \(\overline{\mathbb{F}}_ p\), with \(p\nmid N\) and \(2\leq k\leq p+1\). Let \(J_ \mathbb{Q}\) be the Jacobian of the curve \(X_ 1(pN)_ \mathbb{Q}\) if \(k>2\) and let \(J_ \mathbb{Q}\) be the Jacobian of \(X_ 1(N)\) if \(k=2\). Let \(H\subset\text{End}(J_ \mathbb{Q})\) be the subring generated by all \(T_ \ell\) and \(\langle a\rangle_ N\) and \(\langle b\rangle_ p\) if \(k>2\), and write \(m\) for the maximal ideal of \(H\) corresponding to \(f\). Also, let \(\mathbb{F}=H/m\subset\overline{\mathbb{F}}_ p\). Suppose that the representation \(\rho_ f: G_ \mathbb{Q}\to\text{GL}_ 2(\overline{\mathbb{F}}_ p)\) is irreducible. Then \(J_ \mathbb{Q} (\overline{\mathbb{Q}})[m]\) is an \(\mathbb{F}\)-vector space of dimension two in each of the following cases: (i) \(2\leq k<p\); (ii) \(k=p\) and \(a^ 2_ p\neq \varepsilon(p)\), where \(T^*_ p f=a_ p f\); (iii) \(k=p\) and \(\rho_ f\) is ramified at \(p\); (iv) \(k=p+1\) and there is no form \(g\) of type \((N,2,\varepsilon)\) with \(\rho_ g\cong \rho_ f\).


11F11 Holomorphic modular forms of integral weight
11F80 Galois representations


Zbl 0641.10026
Full Text: DOI EuDML


[1] Ash, A., Stevens, G.: Modular forms in characteristicl and special values of theirL-functions. Duke Math. J.53, No. 3 (1986)
[2] Boston, N., Lenstra, H.W., Ribet, K.A.: Quotients of group rings arising from two-dimensional representations. C.R. Acad. Sci. Paris, S?r.I 312, 323-328 (1991) · Zbl 0718.16018
[3] Carayol, H.: Sur les repr?sentationsl-adiques associ?es aux formes modulaires de Hilbert. Ann. Sci. Ec. Norm. Super., IV. S?r.19, 409-468 (1986) · Zbl 0616.10025
[4] Coleman, R.F.: Ap-adic Shimura isomorphism andp-adic periods of modular forms. (Preprint)
[5] Coleman, R.F., Voloch, J.F.: Companion forms and Kodaira-Spencer theory (to appear) · Zbl 0770.11024
[6] Cornell, G., Silverman, J.H.: Arithmetic geometry. Berlin Heidelberg New York: Springer 1986 · Zbl 0596.00007
[7] Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. Publ. Math., Inst. Hautes Etud. Sci.36, 75-125 (1969) · Zbl 0181.48803
[8] Deligne, P., Rapoport, M.: Les sch?mas de modules des courbes elliptiques. In: Deligne, P., Kuyk, W. (eds.) Modular Functions of One Variable II. (Lect. Notes Math., vol. 349, pp. 143-316) Berlin Heidelberg New York: Springer 1973
[9] Deligne, P., Serre, J.-P.: Formes modulaires de poids 1. Ann. Sci. Ec. Norm. Super., IV. S?r.7, 507-530 (1974) · Zbl 0321.10026
[10] Gross, B.H.: A tameness criterion for Galois representations associated to modular forms (mod p). Duke Math. J.61, No. 2, (1990) · Zbl 0743.11030
[11] Jordan, B., Livn?, R.: Conjecture ?epsilon? for weightk>2. Bull. Am. Math. Soc.21, 51-56 (1989) · Zbl 0675.10020
[12] Grothendieck, A.: S?minaire de g?om?trie alg?brique. (Lect. Notes Math., vols. 151, 152, 153, 224, 225, 269, 270, 288, 305, 340, 589) Berlin Heidelberg New York: Springer 1970-1977
[13] Jochnowitz, N.: The local components of the Hecke algebra modl. Trans. Am. Math. Soc.270, 253-267 (1982) · Zbl 0536.10021
[14] Katz, N.M.:p-adic properties of modular schemes and modular forms. In: Kuyk, W., Serre J.-P. (eds.) Modular Functions of One Variable III. (Lect. Notes Math., vol. 350, pp. 69-190) Berlin Heidelberg New York: Springer 1973
[15] Katz, N.M.: A result on modular forms in characteristicp. In: Serre, J.-P., Zagier, D.B. (eds.) Modular Functions of One Variable V. (Lect. Notes Math., vol. 601, pp. 53-61) Berlin Heidelberg New York: Springer 1976
[16] Katz, N.M., Mazur, B.: Arithmetic moduli of elliptic curves. (Ann. Math. Stud., vol. 108) Princeton: Princeton University Press 1985 · Zbl 0576.14026
[17] Mazur, B.: Modular curves and the Eisenstein ideal. Publ. Math., Inst. Hautes Etud. Sci. 47 (1977) · Zbl 0394.14008
[18] Mazur, B., Ribet, K.A.: Two-dimensional representations in the arithmetic of modular curves Ast?risque (to appear) · Zbl 0780.14015
[19] Oda, T.: The first De Rham cohomology group and Dieudonn? modules. Ann. Sci. Ec. Norm. Super., IV. S?r.2, 63-135 (1969) · Zbl 0175.47901
[20] Raynaud, M.: Sp?cialisation du foncteur de Picard. Publ. Math. Inst. Hautes Etud. Sci. 38 (1970) · Zbl 0207.51602
[21] Raynaud, M.: Sch?mas en groupes de type (p,...,p). Bull. Soc. Math. Fr.102, 241-280 (1974) · Zbl 0325.14020
[22] Ribet, K.A.: On modular representations of Gal (Q/?) arising from modular forms. Invent. Math.100, 431-476 (1990) · Zbl 0773.11039
[23] Robert, G.: Congruences entres s?ries d’Eisenstein, dans le cas supersingulier. Invent. Math.61, 103-158 (1980) · Zbl 0442.10020
[24] Serre, J.-P.: Une interpr?tation des congruences relatives ? la fonction ? de Ramanujan. S?minaire Delange-Pisot-Poitou 1967/68, 14, Oeuvres 80
[25] Serre, J.-P.: Valeurs propres des op?rateurs de Hecke modulol. Journ?es arithm?tiques Bordeaux. Ast?risque 24-25, 109-117 (1975), Oeuvres 104
[26] Serre, J.-P., Sur les repr?sentations de degr? 2 de Gal (594-1). Duke Math. J. 54, No. 1, (1987)
[27] Serre, J.-P., R?sum? des cours au Coll?ge de France, 1987-1988
[28] Szpiro, L.: S?minaire sur les pinceaux arithm?tiques: la conjecture de Mordell. Ast?risque 127 (1985)
[29] Tate, J.: ?p-divisible groups.? In: Proceedings of a conference on local fields at Driebergen, pp. 158-184. Berlin Heidelberg New York: Springer 1967 · Zbl 0157.27601
[30] Ulmer, D.L.:L-functions of universal curves over Igusa curves. Am. J. Math.112, 687-712 (1990) · Zbl 0731.14013
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