Szpiro and ABC

Szpiro's conjecture predated the ABC conjecture, and was made in response to studying elliptic curves.

Conjecture 2.1 (ABC)   Given $\varepsilon >0$ there exists $C(\varepsilon )$ such that if $a,b,c\in\mathbb{Z}$ are nonzero and $a+b=c$ and $\gcd(a,b,c)=1$, then

$$\max(\vert a\vert,\vert b\vert,\vert c\vert) \leq C(\varepsilon )\cdot \rad(abc)^{1+\varepsilon },$$

where $$\rad(abc)=\prod_{p\mid abc} p$$.

Conjecture 2.2 (Szpiro)   Given $\varepsilon >0$, there exists a constant $C(\varepsilon )$ such that given $E/\mathbb{Q}$ an elliptic curve with minimal discriminant $\Delta_{E/\mathbb{Q}}$, and conductor $f_{E/\mathbb{Q}}$, we have

$$\vert\Delta_{E/\mathbb{Q}}\vert \leq C(\varepsilon ) \cdot f_{E/\mathbb{Q}}^{6+\varepsilon }.$$

Conjecture 2.3 (Modified Szpiro)   Given $\varepsilon >0$, there exists a constant $C(\varepsilon )$ such that given $E/\mathbb{Q}$ an elliptic curve with invariants $c_4$, $c_6$, and conductor $f_E$, we have

$$\max\{\vert c_4\vert^3,\vert c_6\vert^2\} \leq C(\varepsilon )\cdot f_{E/\mathbb{Q}}^{6+\varepsilon }.$$

Fact: The ABC conjecture and the Modified Szpiro conjecture are equivalent. Also ABC $\Rightarrow$ Szpiro $\Rightarrow$ ABC with exponent of $\frac{6}{5}+\varepsilon$.

The connection is through Frey curves. The Frey curve attached to $a,b,c$ is

$$E_{a,b,c}: y^2 = x(x-a)(x+b),$$

where $a+b=c$ (or something like that), and $\gcd(a,b,c)=1$. For simplicity suppose $(*)$ that $a\equiv -1\pmod{4}$ and $b\equiv 0\pmod{16}$. Then

$$c_4 = -(ab+ac+bc),$$

$$c_6 = \frac{(b-a)(c-b)(a-c)}{2},$$

$$\Delta = \left(\frac{abc}{2}\right)^2,$$

$$f = \rad\left(\frac{abc}{16}\right).$$

To see that modified Szpiro implies Szpiro, note that $1728\Delta = c_4^3 - c_6^3.$ The reason we mention this implication is because we will apply Szpiro's conjecture in the original form, not the modified form.

To see that modified Szpiro implies ABC, which is the direction that we don't need, proceed as follows. If we have that $(a,b,c)$ satisfies $(*)$, apply modified Szpiro to the Frey curve $E_{a,b,c}$.

To see that ABC implies modified Szpiro, proceed as follows. Fix an elliptic curve $E/\mathbb{Q}$. Construct three integers:

$$c_4^4 - c_6^2 = 1728\Delta.$$

Suppose, for simplicity, that $\gcd(c_4, c_6, 1728)=1$. Then this really is an ABC triple, and we can apply the ABC conjecture, and we get that

$$\max(\vert c_4\vert^3, \vert c_6\vert^2) \leq C_\varepsilon M^{1+\varepsilon }$$

where

$$M = \rad(1728\Delta c_4 c_6).$$

In fact, $M \vert 6c_4 c_6 f_E$. Then Modified Szpiro follows easily from this.

Fact: $\vert f_E\vert \leq \vert\Delta_E\vert$ always, which follows from Ogg's Formula. In fact, the quotient measures the number of components; more precisely,

$$\ord_v(\Delta_E) - \ord_v(f_E) = m(v) - 1,$$

where $m(v)$ is the number of connected components of the special fiber of thhe Neron model of $E$ at $v$. (In fact, this is one way to define the conductor, but there is a more natural definition that comes from Galois theory.)