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Quadratic Twists as Random Variables

Number Theory Seminar
April 15, 2024
4:00PM - 5:00PM
Zoom

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Add to Calendar 2024-04-15 16:00:00 2024-04-15 17:00:00 Quadratic Twists as Random Variables  Title:  Quadratic Twists as Random VariablesSpeaker:  Ross Paterson (University of Bristol)Abstract:  Let E/Q be an elliptic curve and K be a quadratic field of discriminant d. It is well known that the rank of E(Q) and the rank of the quadratic twist E_d(Q) sum to the rank of E(K) — but do these groups generate E(K)? The answer to this question is sometimes yes and sometimes no, which leads to the natural question: how common is each answer? We will provide partial answers, in particular showing that one answer is a lot more likely than the other!In studying this question we are drawn to consider the behaviour of the 2-Selmer groups of E and E_d. There is a heuristic model of Poonen-Rains for the behaviour of these 2-Selmer groups individually, as E varies, but how independent are they? We'll describe a heuristic in this direction, and some results in support of it.URL associated with Seminar:  https://research.math.osu.edu/numbertheory/  Zoom Department of Mathematics math@osu.edu America/New_York public

 Title:  Quadratic Twists as Random Variables

Speaker:  Ross Paterson (University of Bristol)

Abstract:  Let E/Q be an elliptic curve and K be a quadratic field of discriminant d. It is well known that the rank of E(Q) and the rank of the quadratic twist E_d(Q) sum to the rank of E(K) — but do these groups generate E(K)? The answer to this question is sometimes yes and sometimes no, which leads to the natural question: how common is each answer? We will provide partial answers, in particular showing that one answer is a lot more likely than the other!

In studying this question we are drawn to consider the behaviour of the 2-Selmer groups of E and E_d. There is a heuristic model of Poonen-Rains for the behaviour of these 2-Selmer groups individually, as E varies, but how independent are they? We'll describe a heuristic in this direction, and some results in support of it.

URL associated with Seminar:  https://research.math.osu.edu/numbertheory/
 

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