## 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

Zbl 0641.10026
Full Text:

### References:

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