×

On the ramification of Hecke algebras at Eisenstein primes. (English) Zbl 1145.11314

From the introduction: Fix a prime \(p\) and a modular residual representation \(\overline\rho: G_\mathbb Q\to\text{GL}_2(\overline{\mathbb F_p})\). Suppose \(f\) is a normalized cuspidal Hecke eigenform of some level \(N\) and weight \(k\) that gives rise to \(\overline\rho\), and let \(K_f\) denote the extension of \(\mathbb Q_p\) generated by the \(q\)-expansion coefficients \(a_n( f )\) of \(f\). The field \(K_f\) is a finite extension of \(\mathbb Q_p\). What can one say about the extension \(K_f/\mathbb Q_p\)? K. Buzzard [Astérisque 298, 1–15 (2005; Zbl 1122.11025)] has made the following conjecture: if \(N\) is fixed, and \(k\) is allowed to vary, then the degree \([K_f: \mathbb Q_p]\) is bounded independently of \(k\).
Little progress has been made on this conjecture so far; indeed, very little seems to have been proved at all regarding the degrees \([K_f: \mathbb Q_p]\). The goal of this paper is to consider a question somewhat orthogonal to that of Buzzard [loc. cit.], namely, to fix the weight and vary the level. Moreover, we only consider certain reducible representations \(\overline\rho\) that arise in B. Mazur’s study of the Eisenstein ideal [Publ. Math., Inst. Hautes Étud. Sci. 47, 33–186 (1977; Zbl 0394.14008)]. Our results suggest that the degrees \([K_f: \mathbb Q_p]\) are, in fact, arithmetically significant.
Suppose that \(N\geq 5\) is prime and that \(p\) is a prime which exactly divides the numerator of \((N-1)/12\). Mazur [loc. cit. (Prop. 9.6, p. 96, and Prop. 19.1, p. 140)] has shown that there is a weight two normalized cuspidal Hecke eigenform defined over \(\overline{\mathbb Q}_p\), unique up to conjugation by \(G_{\mathbb Q_p}\) (the Galois group of \(\overline{\mathbb Q}_p\) over \(\mathbb Q_p\)), satisfying the congruence \[ a_l(f)\equiv 1+l\bmod{\mathfrak p}\tag{1} \] (where \(\mathfrak p\) is the maximal ideal in the ring of integers of \(K_f\), and \(l\) ranges over primes distinct from \(N\)). It follows moreover from [B. Mazur, loc. cit. (Prop. 19.1, p. 140)] that \(K_f\) is a totally ramified extension of \(\mathbb Q_p\), and thus that the degree \([K_f: \mathbb Q_p]\) is equal to the (absolute) ramification degree of \(K_f\). Denote this ramification degree by \(e_p\).
In this paper we prove the following theorem in the case when \(p=2\):
Theorem 1.1. Suppose that \(p=2\) and that \(N\equiv 9\bmod{16}\), and let \(f\) be a weight two eigenform on \(\Gamma_0(N)\) satisfying the congruence (1). If \(2^m\) is the largest power of 2 dividing the class number of the field \(\mathbb Q(\sqrt{-N})\), then \(e_2=2^{m-1}-1\).
When \(p\) is odd, we establish the following less definitive result:
Theorem 1.2. Suppose that \(p\) is an odd prime exactly dividing the numerator of \((N - 1)/12\). Let \(f\) be a weight two eigenform on \(\Gamma_0(N)\) satisfying the congruence (1).
(i) Suppose that \(p=3\). (Our hypothesis on \(N\) thus becomes \(N \equiv 10\) or \(19 \bmod 27\).) Then \(e_3=1\) if and only if the 3-part of the class group of \(\mathbb Q(\sqrt{-3}, N^{1/3})\) is cyclic.
(ii) Suppose that \(p\geq5\). (Our hypothesis on \(N\) thus becomes \(p\parallel N-1\).) Then \(e_p=1\) if the \(p\)-part of the class group of \(\mathbb{Q}(N^{1/p})\) is cyclic.
The question of computing \(e_p\) has been addressed previously, in the paper of L. Merel [J. Reine Angew. Math. 477, 71–115 (1996; Zbl 0859.11036)]. In this work, Merel established a necessary and sufficient criterion for \(e_p=1\). Merel’s criterion for \(e_p=1\) is not expressed in terms of class groups; rather, it is expressed in terms of whether or not the congruence class modulo \(N\) of a certain explicit expression is a \(p\)th power.
When \(p=2\), Merel, using classical results from algebraic number theory, was able to reinterpret his explicit criterion for \(e_2=1\) so as to prove that \(e_2=1\) if and only if \(m=2\). (It is known that \(m\geq 2\) if and only if \(N\equiv 1 \bmod 8\); see Proposition 4.1.) Theorem 1.1 strengthens this result by relating the value of \(e_2\) in all cases to the order of the 2-part of the class group of \(Bbb Q(\sqrt{-N})\).
When \(p\) is odd, Merel was not able to reinterpret his explicit criterion in algebraic number theoretic terms. However, combining Merel’s result with Theorem 1.2 (and the analogue of this theorem for more general primes \(N\), i.e., those for which \(p\) divides \(N-1\), but not necessarily exactly) yields the following result:
Theorem 1.3. Let \(N \geq 5\) be prime.
(i) Let \(N \equiv 1 \bmod 9\). The 3-part of the class group of \(\mathbb Q(\sqrt{-3}, N^{1/3})\) is cyclic if and only if \((\frac{N-1}{3})!\) is not a cube modulo \(N\). Equivalently, if we let \(N =\pi\overline{\pi}\) denote the factorization of \(N\) in \(\mathbb Q(\sqrt{-3})\), then the 3-part of the class group of \(\mathbb Q(N^{1/3},\sqrt{-3})\) is cyclic if and only if the 9th power residue symbol \((\frac{\pi} {\overline\pi})_9\) is non-trivial. Furthermore, if these equivalent conditions hold, then the 3-part of the class group of \(Bbb Q(N^{1/3})\) (which a fortiori is cyclic of order divisible by three) has order exactly three.
(ii) Let \(p\geq 5\), and let \(N \equiv 1 \bmod p\). If the \(p\)-part of the class group of \(\mathbb Q(N^{1/p})\) is cyclic then \[ \prod^{(N-1)/2}_{l=1}l^l \] is not a \(p\)th power modulo \(N\).
The proofs of Theorems 1.1 and 1.2 depend on arguments using deformations of Galois representations. Briefly, if \(T\) denotes the completion at its \(p\)-Eisenstein ideal of the Hecke algebra acting on weight two modular forms on \(\Gamma_0(N)\), then we identify \(T\) with the universal deformation ring for a certain deformation problem. The theorems are then proved by an explicit analysis of this deformation problem over Artinian \(F_p\)-algebras.
It may be of independent interest to note that our identification of \(T\) as a universal deformation ring also allows us to recover all the results of Mazur proved in [loc. cit.] regarding the structure of \(T\) and the Eisenstein ideal: for example, that \(T\) is monogenic over \(Bbb Z_p\) (and hence Gorenstein); that the Eisenstein ideal is principal, and is generated by \(T_l-(1+l)\) if and only if \(l\neq N\) is a good prime; and also that \(T_N=1\) in \(T\).

MSC:

11F80 Galois representations
11F33 Congruences for modular and \(p\)-adic modular forms
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Buzzard, K.: Questions about slopes of modular forms. To appear in Astérisque · Zbl 1122.11025
[2] Fontaine, No article title, C. R. Acad. Sci., Paris, Sér. A, 280, 1423 (1975)
[3] Fontaine, No article title, Invent. Math., 81, 515 (1985) · Zbl 0612.14043
[4] III, No article title, Acta Arith., 30, 307 (1976)
[5] Halter-Koch, No article title, Acta Arith., 33, 355 (1977)
[6] Lenstra Jr., H.W.: Complete intersections and Gorenstein rings. In: Elliptic curves, modular forms and Fermat’s Last Theorem, ed. by J. Coates, S.T. Yau. Cambridge: International Press 1995 · Zbl 0860.13012
[7] Mazur, No article title, Publ. Math., Inst. Hautes Étud. Sci., 47, 33 (1977) · Zbl 0394.14008
[8] Mazur, B.: An introduction to the deformation theory of Galois representations. In: Modular forms and Fermat’s last theorem (Boston, MA, 1995), pp. 243-311. New York: Springer 1997 · Zbl 0901.11015
[9] Merel, No article title, J. Reine Angew. Math., 477, 71 (1996)
[10] Oort, F.: Commutative group schemes. Lect. Notes Math., vol. 15. Springer 1966 · Zbl 0216.05603
[11] Oort, No article title, Ann. Sci. Éc. Norm. Supér., IV. Sér., 3, 1 (1970)
[12] Quillen, D.: Higher algebraic K-theory. I. In: Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972). Lect. Notes Math., vol. 341, pp. 85-147. Berlin: Springer 1973 · Zbl 0292.18004
[13] Ramakrishna, No article title, Compos. Math., 87, 269 (1993)
[14] Raynaud, No article title, ..,p). Bull. Soc. Math. Fr., 102, 241 (1974)
[15] Scholz, No article title, Monatsh. Math. Phys., 40, 211 (1933) · Zbl 0007.00301
[16] Skinner, No article title, Proc. Natl. Acad. Sci. USA, 94, 10520 (1997) · Zbl 0924.11044
[17] Wiles, No article title, Ann. Math., 141, 443 (1995) · Zbl 0823.11029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.