Description
Dear Colleague,
It is our pleasure to invite you to the next talk in our Online Number Theory Seminar, on 6 December (Friday) at 17:00 CET (8:00 PST, 11:00 EST, 16:00 GMT, 18:00 IST, 21:30 India Standard Time; already Saturday 3:00 AEDT, 5:00 NZDT)
Speaker: J. Cremona (University of Warwick)
Title: On the equivalence of binary forms over a field
Abstract: We consider the question of determining whether two binary forms of small degree over a field K are equivalent under the actions of either GL(2,K) or SL(2,K). In the case of cubics, we give two necessary and sufficient criteria for such equivalence, one of which involves an algebraic invariant, the "Cardano invariant", which belongs to a quadratic extension of K; this is closely connected to classical formulas, and also similar to an invariant that appears in the work of Bhargava et al. The second criterion for cubics is in terms of the base field itself, and also gives explicit matrices in SL(2,K) or GL(2,K) transforming one cubic into the other, if any exist. These results may be also used to test equivalence of binary cubic forms over an integral domain such as Z. We will mention a connection between binary cubic forms and the arithmetic of elliptic curves, and discuss similar results for quartic forms, which have application to 2-descent on elliptic curves. The methods are elementary.
Meeting link: https://unideb.webex.com/unideb/j.php?MTID=md5836d028ef61310a7497e8e8f61288a
Meeting number: 2731 364 4264
Please, share the link with colleagues who are interested, tell us if you have a proposal for a talk, and
also inform us in case you are not willing to receive announcements of these events. The email address of the
seminar is ntrg@science.unideb.hu . We keep the program updated at https://ntrg.math.unideb.hu/seminar.html ,
where (upon the agreement of the Speaker), you can also find the video and the slides of the talk.
We are looking forward to meeting you, with kind regards,
the organizers (K.Győry, Á.Pintér, L.Hajdu, A.Bérczes, Sz.Tengely, I.Pink, Debrecen Number Theory Research Group , University of Debrecen)