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 \({\mathcal O}_K\).


14L15 Group schemes
11F80 Galois representations
14L05 Formal groups, \(p\)-divisible groups
11G05 Elliptic curves over global fields
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