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

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 $$[1,6]$$?

• Well, you can start with a curve with large conductor $N$ and Szpiro ratio $>6$ and then quadratic twist by suitably chosen primes (coprime to $N$) $p$ so that $p^2$ shows up in the conductor and $p^6$ in the minimal discriminant. With care, this should give density in $[3,6)$. Starting with large conductor and Szpiro ratio $1$, then the same argument gives density in $(1,3)$. Feb 20, 2022 at 0:38
• @MikeBennett I think that does indeed answer my question, if you'd like please write it as an answer and I will accept. Thanks! Feb 23, 2022 at 17:56

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 $$[1,3]$$.