Does the modified Szpiro conjecture require minimal model?

Question

The modified Szpiro conjecture is described in Wikipedia and here and here.

The modified Szpiro conjecture states that: given $\varepsilon > 0$, there exists a constant $C(\varepsilon)$ such that for any elliptic curve $E$ defined over $\mathbb{Q}$ with invariants $c4, c6$ and conductor $f$, we have

$$\max\{ \vert c_4 \vert^3, \vert c_6\vert^2\} \leq C(\varepsilon )\cdot f^{6+\varepsilon } $$

The original Szpiro conjecture requires minimal model, while the modified one appears to doesn't require minimal model.

Does the modified Szpiro conjecture require minimal model?

Does the modified Szpiro conjecture allow $a_1 \ne 0$ and/or $a_3 \ne 0$?

Reference for it and other names?

Answer 1

The conductor does not depend on the model. The $c_4$ and $c_6$ do. In fact as $E$ varies among the different integral models for the same elliptic curve, the left-hand side of the inequality differs by an arbitrary $12$-th power of an integer. The conjecture certainly allows for $a_1 \ne 0$ and $a_3 \ne 0$. If we impose $a_1=a_3=0$ we can get a model that is close to being minimal, and then the left-hand side is multiplied by a bounded factor that can be absorbed into the $C(\epsilon)$.

In fact, the conjecture still makes sense if you restrict to models for the form $$ Y^2=X^3+AX+B, \qquad A,B \in \mathbb{Z}, \qquad 4A^3+27B^2 \ne 0 $$ assuming that the model satisfies the following minimality condition: there is no integer $u>1$ such that $u^4 \mid A$ and $u^6 \mid B$. In this case you can write the conjecture in the form $$ \max(A^3,B^2) \le C^\prime(\epsilon) \cdot f^{6+\epsilon}. $$ The $C^\prime(\epsilon)$ has to differ slightly from $C(\epsilon)$ to take account of the fact the model is not entirely minimal at $2$ and $3$. It is an easy exercise to translate between the version the OP gives for minimal models and the version I give for short Weierstrass models (that satisfy the stated minimality condition).