# Is the weight in Serre's conjecture "minimal"?

Serre's conjecture says that given any odd, irreducible, continuous representation $$\rho:G_{\mathbb{Q}}\rightarrow GL_2(\overline{\mathbb{F}_p})$$ there is some eigenform $$f$$ of weight $$k(\rho)$$, level $$N(\rho)$$, and nebentype $$\epsilon(\rho)$$, such that $$\rho$$ is isomorphic to the mod $$p$$ representation $$\bar \rho_f$$ associated to $$f$$. Up until now, I have intuitively thought of the Serre weight $$k(\rho)$$ as being the minimal weight among all weights of eigenforms whose representations are isomorphic to $$\rho$$. I am no longer sure whether this is correct: I am currently reading Edixhoven's survey of Serre's conjecture, and it seems like one might be able to cook up an example where the Serre weight $$k(\bar\rho_f)$$ of some weight $$k$$ eigenform $$f$$ is actually greater than the weight of $$f$$, which seems strange to me... Can this happen? I.e., are there examples where the Serre weight of a representation coming from a weight $$k$$ eigenform is actually greater that $$k$$?

To give this a bit more context, fix $$p\nmid N$$ and let $$f \in S_{k}(\Gamma_0(N))$$ be an eigenform with mod $$p$$ representation $$\bar\rho_f$$. Fix an inertia subgroup $$I_p\subseteq G_{\mathbb{Q}}$$ at $$p$$ and write $$I_{p,w}\subseteq I_p$$ for its wild subgroup. If $$\bar\rho_f\mid_{I_{p,w}}$$ is nontrivial, we can uniquely write $$\bar\rho_f\mid_{I_{p}}=\begin{pmatrix}\chi^\beta &* \\ 0& \chi^{\alpha} \end{pmatrix}$$ where $$\chi$$ is the mod $$p$$ cyclotomic character and $$\alpha$$, $$\beta$$ are integers such that $$0\leq \alpha\leq p-2$$, $$1\leq\beta\leq p-1$$. As stated in the Edixhoven article, setting $$a=\min(\alpha,\beta)$$ and $$b=\max(\alpha,\beta)$$, we define $$k(\bar\rho_f)$$ to be $$1+pa+b+p-1$$ if $$\chi^{\beta-\alpha}=\chi$$ and $$\bar \rho_f\mid_{G_{\mathbb{Q}_p}} \otimes \chi^{-\alpha}$$ is not finite at $$p$$, otherwise we define it to be $$1+pa+b$$.

Now consider the case where $$p=3$$, and suppose that we have some weight 6 eigenform $$f$$ on $$\Gamma_0(N)$$ such that $$\bar\rho_f\mid_{I_{3}}=\begin{pmatrix}1 &* \\ 0& \chi \end{pmatrix}$$ and $$\rho_f\mid_{G_{\mathbb{Q}_3}} \otimes \chi^{-1}$$ is not finite at $$3$$. Then the formula above would say that $$f$$ has Serre weight $$k(\bar \rho_f)=8$$, which is larger than the weight of $$f$$. I am of course assuming that we can find such a form satisfying these conditions, but even so, I see no obvious reason why this shouldn't occur...

• The following article might be relevant (arxiv.org/abs/2004.07587) Aug 21 '20 at 13:01
• The weight in Serre's conjecture is minimal. The above example fails because in this example $\alpha = 0$ and $\beta = 1$, giving a minimal weight of $4$. Oct 25 '20 at 3:56
• Ah, sorry, there was a typo (now fixed): $\alpha$ and $\beta$ should be switched in the definition. (See Edixhoven's Defintion 1.7(b) in the FLT book.) Thus, in the example, we have $\beta=p-1$ and $\alpha=1$, in which case $a=1$ and $b=p-1$. So assuming $\rho_{G_{\mathbb{Q}_3}}\otimes \chi^{-1}$ is not finite, we get Serre weight $1+p+(p-1)+p-1=3p-1$. Oct 25 '20 at 13:11

I believe a modular form as you describe indeed cannot exist. I think it's easier to think about these issues if they're translated into the representation-theoretic language of Serre weights. Associated to $$\overline{\rho}$$ is a set of Serre weights, i.e., of irreducible mod $$3$$ representations of $$\mathrm{GL}(2,\mathbf{F}_3)$$. If $$\overline{\rho}|_{D_3}$$ is a tres ramifiee extension of $$\chi$$ by $$\chi^2 = 1$$, then this set is the singleton $$\{ \mathrm{det} \otimes \mathrm{Sym}^2\}$$, where $$\mathrm{Sym}^a$$ denotes the $$a$$th symmetric power of the standard representation. (Since we're in two dimensions you can think about this in terms of crystalline lifts: your $$\overline{\rho}|_{D_3}$$ will lift to a crystalline extension of cyclotomic by cyclotomic^4 (but not of cyclotomic by cyclotomic^2, because of the tres ramifiee condition). So it has a crystalline lift with Hodge-Tate weights $$(s,t) = (4,1)$$ [but not $$(2,1)$$] and the numerology is that the Serre weight associated to this lift is $$\mathrm{det}^t \otimes \mathrm{Sym}^{s-t-1}$$.)
Now the Breuil-Mézard conjecture for $$\mathrm{GL}(2,\mathbf{Q}_3)$$, which is known (and due to Shen-Ning Tung in this case), has the consequence that if $$\overline{\rho}|_{D_3}$$ has a crystalline lift of weights $$(0,k-1)$$ (in particular if $$\overline{\rho}$$ comes from a modular form of weight $$k$$ and level prime to $$p$$) then $$\{ \mathrm{det} \otimes \mathrm{Sym}^2\}$$ must be a Jordan-Hölder factor of $$\mathrm{Sym}^{k-2}$$.
If I haven't miscalculated, the Jordan-Hölder factors of $$\mathrm{Sym}^4$$ are $$\{\mathrm{Sym}^2, \mathrm{det}, \mathrm{det}^2\}$$, whereas the the Jordan-Hölder factors of $$\mathrm{Sym}^6$$ are $$\{\mathrm{Sym}^2, \mathrm{det} \otimes \mathrm{Sym}^2, \mathrm{det}^2\}$$, so indeed $$k=8$$ is the minimal weight where such $$\overline{\rho}$$ can be found.