IISER Pune
INDIAN INSTITUTE OF SCIENCE EDUCATION AND RESEARCH (IISER) PUNE
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An Autonomous Institution, Ministry of Human Resource Development, Govt. of India
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Seminars and Colloquia

Mathematics

Tetrahedral elliptic curves and the local-global principle for isogenies 
 
Mon, Jan 06, 2020,   11:00 AM to 12:00 PM at Madhava Hall

Dr. Barinder Banwait
Cambridge University

Let E/K be an elliptic curve over a number field. If E admits a K-rational l-isogeny for some prime l, then the reduction of E modulo almost all primes of K admits an l-isogeny rational over the residue field. Sutherland showed that the converse of this - the "local-global principle for isogenies" - is not necessarily true, and gave a classification of such failures when K does not contain the quadratic subfield of the l-th cyclotomic field. In this talk we will review Sutherland's work, and give a classification of failures in the case when K does contain this quadratic field. In attempting to provide explicit examples of such failures, we are led to constructing an explicit model of - and determining the rational and quadratic points on - the genus-3 level-13 modular curve X_{S_4}(13) corresponding to the pullback to GL(2,13) of S_4 \\in PGL(2,13). This is joint work with John Cremona.
 

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