Towards a classification of entanglements of Galois representations attached to elliptic curves

• Harris B. Daniels ^[1] ; Alvaro Lozano-Robledo ^[2] ; Jackson S. Morrow ^[3]
  1. [1] Amherst College

     Amherst College

     Town of Amherst, Estados Unidos

     
  2. [2] University of Connecticut

     University of Connecticut

     Town of Mansfield, Estados Unidos

     
  3. [3] University of Montreal

     University of Montreal

     Canadá

     

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• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 39, Nº 3, 2023, págs. 803-844
• Idioma: inglés
• DOI: 10.4171/RMI/1424
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• Resumen

  • Let E=Q be an elliptic curve, let Q be a fixed algebraic closure of Q, and let GQ D Gal.Q=Q/ be the absolute Galois group of Q. The action of GQ on the adelic Tate module of E induces the adelic Galois representation E W GQ ! GL.2; Zy/: The goal of this paper is to explain how the image of E can be smaller than expected. To this end, we offer a group theoretic categorization of different ways in which an entanglement between division fields can be explained and prove several results on elliptic curves (and more generally, principally polarized abelian varieties) over Q where the entanglement occurs over an abelian extension