UNRAMIFIEDNESS OF GALOIS REPRESENTATIONS ARISING FROM HILBERT MODULAR SURFACES
Forum of Mathematics, Sigma, Tome 5 (2017)
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Let $p$ be a prime number and $F$ a totally real number field. For each prime $\mathfrak{p}$ of $F$ above $p$ we construct a Hecke operator $T_{\mathfrak{p}}$ acting on $(\text{mod}\,p^{m})$ Katz Hilbert modular classes which agrees with the classical Hecke operator at $\mathfrak{p}$ for global sections that lift to characteristic zero. Using these operators and the techniques of patching complexes of Calegari and Geraghty we prove that the Galois representations arising from torsion Hilbert modular classes of parallel weight $\mathbf{1}$ are unramified at $p$ when $[F:\mathbb{Q}]=2$ . Some partial and some conjectural results are obtained when $[F:\mathbb{Q}]>2$ .
@article{10_1017_fms_2017_26,
author = {MATTHEW EMERTON and DAVIDE REDUZZI and LIANG XIAO},
title = {UNRAMIFIEDNESS {OF} {GALOIS} {REPRESENTATIONS} {ARISING} {FROM} {HILBERT} {MODULAR} {SURFACES}},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {5},
year = {2017},
doi = {10.1017/fms.2017.26},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2017.26/}
}
TY - JOUR AU - MATTHEW EMERTON AU - DAVIDE REDUZZI AU - LIANG XIAO TI - UNRAMIFIEDNESS OF GALOIS REPRESENTATIONS ARISING FROM HILBERT MODULAR SURFACES JO - Forum of Mathematics, Sigma PY - 2017 VL - 5 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.1017/fms.2017.26/ DO - 10.1017/fms.2017.26 LA - en ID - 10_1017_fms_2017_26 ER -
%0 Journal Article %A MATTHEW EMERTON %A DAVIDE REDUZZI %A LIANG XIAO %T UNRAMIFIEDNESS OF GALOIS REPRESENTATIONS ARISING FROM HILBERT MODULAR SURFACES %J Forum of Mathematics, Sigma %D 2017 %V 5 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.1017/fms.2017.26/ %R 10.1017/fms.2017.26 %G en %F 10_1017_fms_2017_26
MATTHEW EMERTON; DAVIDE REDUZZI; LIANG XIAO. UNRAMIFIEDNESS OF GALOIS REPRESENTATIONS ARISING FROM HILBERT MODULAR SURFACES. Forum of Mathematics, Sigma, Tome 5 (2017). doi: 10.1017/fms.2017.26
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