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# Mathematical Research Letters

## Volume 20 (2013)

### Number 3

### Multivariable Lubin-Tate $(\varphi, \Gamma)$-modules and filtered $\varphi$-modules

Pages: 409 – 428

DOI: http://dx.doi.org/10.4310/MRL.2013.v20.n3.a1

#### Author

#### Abstract

We define some rings of power series in several variables, that are attached to a Lubin-Tate formal module. We then give some examples of $(\varphi, \Gamma)$-modules over those rings. They are the global sections of some reflexive sheaves on the $p$-adic open unit polydisk, that are constructed from a filtered $\varphi$-module using a modification process. We prove that we obtain every crystalline $(\varphi, \Gamma)$-module over those rings in this way.

#### Keywords

$(\varphi, \Gamma)$-module, Lubin-Tate group, filtered $\varphi$-module, crystalline representation, $p$-adic period, Fontaine theory, reflexive sheaf

#### 2010 Mathematics Subject Classification

11Fxx, 11Sxx, 14Gxx