1
$\begingroup$

Let $K$ be a number field and $E/K$ an elliptic curve with equation $Y^2Z = X^3 +AXZ^2+BZ^3$ in $\mathbf{P}^2_K$, where $A,B\in K$.

Let $S$ be non-empty finite set of finite places of $K$ and suppose that $E$ has bad reduction over $S$ and good reduction outside $S$. Moreover, let $L/K$ be a finite field extension such that $E_L$ has good reduction over $O_L$. (In particular, $E/K$ has potential good reduction.)

How similar is the reduction of the hyperelliptic curve $H$ of genus $g\geq 2$ given by $$Y^2 Z^{2g-1} = X^{2g+1} + AX Z^{2g} + B Z^{2g+1}$$ to the reduction of $E$?

Does $H_L$ have good reduction over $O_L$?

Does $H$ have good reduction outside $S$?

Does $H$ have bad reduction over $S$?

$\endgroup$

1 Answer 1

3
$\begingroup$

The first thing you can write already gives a negative answer to all your questions. Take $K=\mathbb{Q}, A=0, B=1, S=\lbrace 2,3 \rbrace$ and $L$ whatever it is the smallest field where $E$ has good reduction. Now take $g=2$. Then $H$ has bad reduction at $5$ and good reduction at $3$, so no to your last two questions. I haven't checked that $H_L$ has bad reduction at $5$ but I see no reason for it to be otherwise.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.