Consider a hypereliptic curve $ C_F $ defined about $ mathbb {P} (1.1, g + 1) $ by the equation

$$ displaystyle C_F: z ^ 2 = F (x, y), $$

where $ F in mathbb {Z} (x, y) $ it is a non-singular binary form of degree $ 2g + $ 2.

It is conjectured (and demonstrated by Caporaso, Harris and Mazur under the conjecture of Bombieri-Lang) that the number of rational points in $ C_F $ is *uniformly bounded* when $ g geq 2 $ in the sense that there is a number $ N (g) $ depending only on $ g geq 2 $ such that for all $ F $, we have $ | C_F ( mathbb {Q}) | leq N (g) $. Stoll proved to be so tied with the additional condition that the range $ r $ of the Mordell-Weil group of the Jacobins of $ C_F $ satisfies $ r leq g-3 $.

My question is about restricting $ (x, y) $ to an orbit of a small (but infinite) subgroup of $ text {SL} _2 ( mathbb {Z}) $. In fact, the simplest example would be the subgroup $ G_M $ generated by some $ M in text {SL} _2 ( mathbb {Z}) $ of infinite order.

For such a group $ G $, put $ G_0 $ for the orbit of $ (1.0) $ below $ G $. Then the counting function can be formed

$$ S (F, G) = # {(x, y) in mathbb {Z} ^ 2 cap G_0: text {exists}} mathbb {Z} text {s.t. } z ^ 2 = F (x, y) }. $$

Clearly, $ S (F, G) leq | C_F ( mathbb {Q}) | $, therefore finite when $ g geq 2 $.

My question is when $ G = G_M $ for some $ M in text {SL} _2 ( mathbb {Z}) $Can you show that $ S (F, G_M) $ is uniformly limited in terms of $ M, g $?