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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 \(\rho_ g\cong \rho_ f\).

MSC:

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

Citations:

Zbl 0641.10026
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References:

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