Szpiro ratios of elliptic curves over $\mathbb{Q}$

Question

For an elliptic curve $E/\mathbb{Q}$, let us denote by $\Delta_{\min}(E)$ the minimal discriminant of $E$ and $N(E)$ the conductor of $E$. Then it is well-known that $N(E) | \Delta_\min(E)$.

The Szpiro ratio of $E$ is defined as the ratio $$\displaystyle \beta_E = \frac{\log |\Delta_\min(E)|}{\log N(E)}.$$

This definition is motivated by Szpiro's conjecture, which asserts that for all $\epsilon > 0$ there exists only finitely many curves $E/\mathbb{Q}$ satisfying

$$\displaystyle \beta_E > 6 + \epsilon.$$

Of course, it is known that Szpiro's conjecture is equivalent to the $abc$-conjecture.

Note that the value of $6$ in the conjecture is sharp, because one can construct explicit families with Szpiro ratio converging to $6$. For example, if $E$ is given by a short Weierstrass model

$$\displaystyle E_{A,B} : y^2 = x^3 + Ax + B, A, B \in \mathbb{Z}$$

with the property that for all primes $p$ we have $p^4 | A \Rightarrow p^6 \nmid B$, then the discriminant is given by $\Delta(E) = 16(4A^3 - 27B^2)$. If $\Delta(E)$ is $12$-power-free, then $\Delta(E) = \Delta_\min(E)$. We can then find an integer $c_1$ such that the equation

$$\displaystyle 16(4x^3 - 27y^2) = c_1 z^6$$

defines an elliptic generalized Fermat equation with a solution, which then gives at least one family of parametrized solutions. In this family, the Szpiro ratio will approach $6$.

Similar constructions can be given for the following families: elliptic curves with a rational 2-torsion point, with 3 rational 2-torsion points, and those with an abelian 2-torsion field.

My question is: is the Szpiro ratio $\beta_E$, as $E$ varies over all $E/\mathbb{Q}$, dense in the interval $[1,6]$?

Answer 1

The answer is "yes". Use one of the various families with Szpiro ratio exceeding $6$ and large conductor $N$, and then twist by a prime $p$ of appropriate size, coprime to $N$ (which increases the conductor by $p^2$ and the minimal discriminant by $p^6$). With a certain amount of care, this gives density in the interval $[3,6]$. A similar argument, starting with a curve of Szpiro ratio arbitrarily close to $1$ (there are various ways to construct these) and large conductor then gives density in $[1,3]$.