### Algebra SeminarHow Often Is The Order Odd Modulo A Prime?Jeremy Rouse (Wake Forest University)

Wednesday, February 24, 2021
11:15am – 12:05pm

Storrs Campus
Online on WebEx

If you choose a prime number $$p$$ at random, what are the chances that it divides a number of the form $$2^{k} + 1$$? This question was answered by Hasse, and we will consider an elliptic curve analogue of this problem where we start with an elliptic curve $$E/\mathbb{Q}$$, a point $$\alpha \in E(\mathbb{Q})$$ with infinite order, and ask how often $$\overline{\alpha} \in E(\mathbb{F}_{p})$$ has odd order. We show that this is related to the image of a certain Galois representation, study its image, and use this to determine how large and how small the density of primes $$p$$ for which $$\alpha$$ has odd order can be (under natural restrictions on $$\alpha$$).