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Moduli of finite flat group schemes, and modularity. (English) Zbl 1201.14034
The author proves:
Theorem. Let $$p>2$$ be a prime, and $$S$$ a finite set of primes containing $$p$$ and the infinite prime. We denote by $$G_{{\mathbb Q},S}$$ the Galois group of the maximal extension of $$\mathbb Q$$ unramified outside $$S$$. Let $$E/{\mathbb Q}_p$$ be a finite extension with ring of integers $$\mathcal O$$, and residue field $$\mathbb F$$. Let $\rho:G_{{\mathbb Q},S}\to\text{GL}_2({\mathcal O})$ be a continuous representation whose determinant is the cyclotomic character times a finite character. Suppose that
(1) The composite $$\overline{\rho}:G_{{\mathbb Q},S}\to\text{GL}_2({\mathcal O})\to\text{GL}_2({\mathbb F})$$ is absolutely irreducible when restricted to $${\mathbb Q}\left(\sqrt{(-1)^{\frac{p-1}{2}}p}\right)$$,
(2) $$\overline{\rho}$$ is modular,
(3) $$\rho$$ is potentially Barsotti-Tate at $$p$$.
Then $$\rho$$ is modular.
The main ingredient is a new technique for analysing flat deformation rings which involves the construction of certain auxiliary schemes named by the author moduli of finite flat group schemes. This provides a more conceptual way of establishing the Shimura-Taniyama-Weil conjecture, especially for elliptic curves which acquire good reduction over a wildly ramified extension of $${\mathbb Q}_3$$.
Suppose $$K/{\mathbb Q}_p$$ is a finite extension with absolute Galois group $$G_K$$, and that $$p>2$$. Let $${\mathbb F}/{\mathbb F}_p$$ be a finite extension, and $$V_{\mathbb F}$$ an $${\mathbb F}$$-vector space of finite dimension, equipped with a continuous $$G_K$$-action. Suppose that $$V_{\mathbb F}$$ arises as the generic fiber of a finite flat group scheme over $${\mathcal O}_K$$ By a finite flat model of $$V_{\mathbb F}$$ the author means a finite flat group scheme $$\mathcal G$$, equipped with an action of $$\mathbb F$$, and an isomorphism of $${\mathbb F}[G_K]$$-modules $${\mathcal G}(\overline{K})\to V_{\mathbb F}$$, where $$\overline{K}$$ is an algebraic closure of $$K$$. The author proves:
Theorem. There exists a projective $${\mathbb F}$$-scheme $${\mathcal G}{\mathcal R}_{V_{\mathbb F},0}$$ such that for any finite extension $${\mathbb F}'/{\mathbb F}$$ the set of finite flat models of $$V_{\mathbb F}\otimes_{\mathbb F}{\mathbb F}'$$ is in bijection with $${\mathcal G}{\mathcal R}_{V_{\mathbb F},0}({\mathbb F}')$$.
Let $$k$$ be the residue field of $$K$$, and let ${\mathfrak S}=W(k)[[u]],$ $$W$$ the Witt vectors. Fix an uniformiser of $${\mathcal O}_K$$ and let $$E(u)\in W(k)[u]$$ denote its Eisenstein polynomial. $${\mathfrak S}$$ is equipped with a Frobenius $$\phi$$ which is the canonical Frobenius on $$W(k)$$ and takes $$u$$ to $$u^p$$. We denote by $$(\text{Mod FI}/{\mathfrak S})_{{\mathbb Z}_p}$$ the category of finite free $${\mathfrak S}$$-modules $$\mathfrak M$$ equipped with a $$\phi$$-semilinear map $$\phi_{\mathfrak M}:{\mathfrak M}\to {\mathfrak M}$$ such that the image of $$1\otimes \phi_{\mathfrak M}:\phi^*{\mathfrak M}\to {\mathfrak M}$$ contains $$E(u){\mathfrak M}$$. The author proves:
Theorem. The category $$(\text{Mod FI}/{\mathfrak S})_{{\mathbb Z}_p}$$ is equivalent to the category of $$p$$-divisible groups over $${\mathcal O}_K$$.

##### MSC:
 14L15 Group schemes 11F80 Galois representations 14L05 Formal groups, $$p$$-divisible groups 11G05 Elliptic curves over global fields
##### Keywords:
flat group scheme
Full Text:
##### References:
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