On uniform lower bound of the Galois images associated to elliptic curves

Keisuke Arai[1]

  • [1] Graduate School of Mathematical Sciences The University of Tokyo Tokyo 153-8914, Japan

Journal de Théorie des Nombres de Bordeaux (2008)

  • Volume: 20, Issue: 1, page 23-43
  • ISSN: 1246-7405

Abstract

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Let p be a prime and let K be a number field. Let ρ E , p : G K → Aut ( T p E ) ≅ GL 2 ( ℤ p ) be the Galois representation given by the Galois action on the p -adic Tate module of an elliptic curve E over K . Serre showed that the image of ρ E , p is open if E has no complex multiplication. For an elliptic curve E over K whose j -invariant does not appear in an exceptional finite set (which is non-explicit however), we give an explicit uniform lower bound of the size of the image of ρ E , p .

How to cite

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Arai, Keisuke. "On uniform lower bound of the Galois images associated to elliptic curves." Journal de Théorie des Nombres de Bordeaux 20.1 (2008): 23-43. <http://eudml.org/doc/10831>.

@article{Arai2008,
abstract = {Let $p$ be a prime and let $K$ be a number field. Let $\rho _\{E,p\}:\mathrm\{G\}_K \rightarrow \mathrm\{Aut\}(T_p E)\cong \mathrm\{GL\}_2(\mathbb\{Z\}_p)$ be the Galois representation given by the Galois action on the $p$-adic Tate module of an elliptic curve $E$ over $K$. Serre showed that the image of $\rho _\{E,p\}$ is open if $E$ has no complex multiplication. For an elliptic curve $E$ over $K$ whose $j$-invariant does not appear in an exceptional finite set (which is non-explicit however), we give an explicit uniform lower bound of the size of the image of $\rho _\{E,p\}$.},
affiliation = {Graduate School of Mathematical Sciences The University of Tokyo Tokyo 153-8914, Japan},
author = {Arai, Keisuke},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
number = {1},
pages = {23-43},
publisher = {Université Bordeaux 1},
title = {On uniform lower bound of the Galois images associated to elliptic curves},
url = {http://eudml.org/doc/10831},
volume = {20},
year = {2008},
}

TY - JOUR
AU - Arai, Keisuke
TI - On uniform lower bound of the Galois images associated to elliptic curves
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2008
PB - Université Bordeaux 1
VL - 20
IS - 1
SP - 23
EP - 43
AB - Let $p$ be a prime and let $K$ be a number field. Let $\rho _{E,p}:\mathrm{G}_K \rightarrow \mathrm{Aut}(T_p E)\cong \mathrm{GL}_2(\mathbb{Z}_p)$ be the Galois representation given by the Galois action on the $p$-adic Tate module of an elliptic curve $E$ over $K$. Serre showed that the image of $\rho _{E,p}$ is open if $E$ has no complex multiplication. For an elliptic curve $E$ over $K$ whose $j$-invariant does not appear in an exceptional finite set (which is non-explicit however), we give an explicit uniform lower bound of the size of the image of $\rho _{E,p}$.
LA - eng
UR - http://eudml.org/doc/10831
ER -

References

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