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SUMMARY:On the modularity of elliptic curves over imaginary quadratic fiel
 ds - Ana Caraiani (Imperial College London)
DTSTART:20220620T123000Z
DTEND:20220620T133000Z
UID:TALK174401@talks.cam.ac.uk
DESCRIPTION:I will survey how to prove the modularity of elliptic curves d
 efined over the rational numbers\, as pioneered by Wiles and Taylor-Wiles 
 and completed by Breuil\, Conrad\, Diamond and Taylor. I will also mention
  the case of elliptic curves defined over real quadratic fields\, more rec
 ently completed by Freitas\, Le Hung and Siksek. I will then explain why t
 he case of imaginary quadratic fields is qualitatively different from the 
 previous ones. Finally\, I will discuss joint work in progress with James 
 Newton\, where we prove a local-global compatibility result in the crystal
 line case for Galois representations attached to torsion classes occurring
  in the cohomology of locally symmetric spaces. This has an application to
  the modularity of elliptic curves over imaginary quadratic fields\, which
  also builds on recent work of Allen\, Khare and Thorne.&nbsp\;
LOCATION:Seminar Room 1\, Newton Institute
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