Growth of points on hyperelliptic curves over number fields

Abstract: Fix a hyperelliptic curve

C / m a t h b b Q

of genus

g

, and consider the number fields

K / m a t h b b Q

generated by the algebraic points of

C

. In this paper, we study the number of such extensions with fixed degree

n

and discriminant bounded by

X

. We show that when

g g e q 1

and

n

is sufficiently large relative to the degree of

C

, with

n

even if the degree of the defining polynomial of

C

is even, there are

g g X^{c_{n}}

such extensions, where

c_{n}

is a positive constant depending on

g

which tends to

1 / 4

as

n o i n f t y

. This result builds on work of Lemke Oliver and Thorne who, in the case where

C

is an elliptic curve, put lower bounds on the number of extensions with fixed degree and bounded discriminant over which the rank of

C

grows with specified root number.

Cites Work