A Constructive Proof of Masser's Theorem

TL;DR

This paper provides a constructive proof demonstrating the infinite existence of certain elliptic curves, called good Frey curves, related to the Modified Szpiro Conjecture and the abc Conjecture.

Contribution

It offers a constructive proof that there are infinitely many good Frey elliptic curves, advancing understanding of the Modified Szpiro Conjecture.

Findings

Constructive proof of infinite good Frey curves

Supports the Modified Szpiro Conjecture

Links to the abc Conjecture

Abstract

The Modified Szpiro Conjecture, equivalent to the $abc$ Conjecture, states that for each $\epsilon>0$, there are finitely many rational elliptic curves satisfying $N_{E}^{6+\epsilon}<\max\!\left\{ \left\vert c_{4}^{3}\right\vert,c_{6}^{2}\right\} $ where $c_{4}$ and $c_{6}$ are the invariants associated to a minimal model of $E$ and $N_{E}$ is the conductor of $E$. We say $E$ is a good elliptic curve if $N_{E}^{6}<\max\!\left\{ \left\vert c_{4}^{3}\right\vert,c_{6}^{2}\right\} $. Masser showed that there are infinitely many good Frey curves. Here we give a constructive proof of this assertion.

Tables

Table 1. Table 1. Universal Elliptic Curve 𝒳 t ​ ( T ) subscript 𝒳 𝑡 𝑇 \mathcal{X}_{t}\!\left(T\right)

${𝒳}_t\left(T\right):y^2+\left(1-g\right)xy-fy=x^3-f{x}^2$
$f$	$g$	$T$
$\frac{2t^4-7t^3+12t^2-7t+2}{2{\left(-1+t\right)}^4}$	$1$	$C_2\times C_4$
$\frac{-2t^3+14t^2-22t+10}{{\left(t+3\right)}^2{\left(t-3\right)}^2}$	$\frac{-2t+10}{\left(t+3\right)\left(t-3\right)}$	$C_2\times C_6$
$\frac{16t^3+16t^2+6t+1}{{\left(8t^2-1\right)}^2}$	$\frac{16t^3+16t^2+6t+1}{2t\left(4t+1\right)\left(8t^2-1\right)}$	$C_2\times C_8$

Table 2. Table 2. Table for Example 5.2

$T$	$a_1$	$b_1$	$q\left(P_1^T\right)$	$q\left(P_2^T\right)$	$q\left(P_3^T\right)$
$C_2\times C_2$	$2^5{11}^{2}14657$	$3^8{13}^4$	$1.0755$	$1.0324$	$1.015$
$C_2\times C_4$	$2^{12}{7}^4$	$3^8{17}^2$	$1.2425$	$1.0531$	$1.0130$
$C_2\times C_6$	$2^{16}{3}^9{17}^{3}61$	$5^9{7}^{12}11\cdot 27127$	$1.1211$	$1.0278$	$-$
$C_2\times C_8$	$2^{12}{11}^8$	$7\cdot 31\cdot 503\cdot 1951\cdot {14657}^2$	$1.0331$	$1.0040$	$-$

Table 3. Table 3. The Invariant c 4 subscript 𝑐 4 c_{4} of F T subscript 𝐹 𝑇 F_{T}

$c_{4,T}$	$T$
$a^8+60a^6{b}^2+134a^4{b}^4+60a^2{b}^6+b^8$	$C_2\times C_2$
$a^8+14a^4{b}^4+b^8$	$C_2\times C_4$
$9a^8+228a^6{b}^2+30a^4{b}^4-12a^2{b}^6+b^8$	$C_2\times C_6$
$a^{16}-8a^{14}{b}^2+12a^{12}{b}^4+8a^{10}{b}^6+230a^8{b}^8+8a^6{b}^{10}+12a^4{b}^{12}-8a^2{b}^{14}+b^{16}$	$C_2\times C_8$

Table 4. Table 4. Example of Good Frey Curves

$T$	$C_2\times C_2$	$C_2\times C_4$	$C_2\times C_6$	$C_2\times C_8$
${\sigma }_m\left(F_T(a_1,b_1)\right)$	$6.4204$	$7.4219$	$6.7269$	$6.1985$
${\sigma }_m\left(F_T(a_2,b_2)\right)$	$6.1912$	$6.3124$	$6.1666$	$6.0241$
${\sigma }_m\left(F_T(a_3,b_3)\right)$	$6.0901$	$6.0769$

Table 5. Table 5. Admissible Change of Variables for Lemma 4.3

$T$	$u_T$	$r_T$	$s_T$	$w_T$	$t_T$
$C_2\times C_2$	$a^2$	$0$	$0$	$0$	$\frac{b}{a}$
$C_2\times C_4$	$2{\left(a-b\right)}^2$	$-2ab{\left(a-b\right)}^2$	${\left(a-b\right)}^2$	$-2ab{\left(a-b\right)}^2\left(a^2+b^2\right)$	$\frac{b}{a}$
$C_2\times C_6$	$9a^2-b^2$	$-4a^2\left(a+b\right)\left(-3a+b\right)$	$5a^2-b^2$	$36a^6-40a^4{b}^2+4a^2{b}^4$	$\frac{9a+b}{a+b}$
$C_2\times C_8$	$\frac{1}{2a\left(a+b\right)\left(b^2-2ab-a^2\right)}$	$\frac{ab\left(a^2+b^2\right)}{{\left(a+b\right)}^2\left(b^2-2ab-a^2\right)}$	$\frac{a^4+4a^{3}b-b^4}{2a\left(a+b\right)\left(b^2-2ab-a^2\right)}$	$\frac{a{b}^2{\left(a^2+b^2\right)}^2}{{\left(a+b\right)}^3{\left(b^2-2ab-a^2\right)}^2}$	$\frac{a}{2\left(b-a\right)}$

Table 6. Table 6. Polynomials and Rational Functions

$T$	$C_2\times C_2$	$C_2\times C_4$	$C_2\times C_6$	$C_2\times C_8$
${𝔄}_T$	$8ab\left(a^2+b^2\right)$	${\left(2ab\right)}^2$	$16a^{3}b$	${\left(2ab\right)}^4$
${𝔅}_T$	${\left(a-b\right)}^4$	${\left(a^2-b^2\right)}^2$	${\left(a+b\right)}^3\left(b-3a\right)$	$(a^4-6a^2{b}^2+b^4){\left(a^2+b^2\right)}^2$
${ℭ}_T$	${\left(a+b\right)}^4$	${\left(a^2+b^2\right)}^2$	$(3a+b){\left(b-a\right)}^3$	${\left(a^2-b^2\right)}^4$
${𝔇}_T$	$b^4-a^4$	$b^4-a^4$	$(b^2-a^2)\left(b^2-9a^2\right)$	$(a^4-6a^2{b}^2+b^4)\left(b^4-a^4\right)$
${𝔄}_T^r$	$ab\left(a^2+b^2\right)$	$ab$	$ab$	$ab$
${𝔅}_T^r$	$\left(a-b\right)$	$a^2-b^2$	$(a+b)\left(b-3a\right)$	$(a^4-6a^2{b}^2+b^4)\left(a^2+b^2\right)$
${ℭ}_T^r$	$a+b$	$a^2+b^2$	$(3a+b)\left(b-a\right)$	$a^2-b^2$
$f_T$	$\frac{{\left(1-t\right)}^4}{8t\left(1+t^2\right)}-t$	$\frac{{\left(1-t^2\right)}^2}{{\left(2t\right)}^2}-t$	$\frac{{\left(1+t\right)}^3\left(t-3\right)}{16t}-t$	$\frac{(1-6t^2+t^4){\left(1+t^2\right)}^2}{{\left(2t\right)}^4}-t$
$g_T$	$4t^3+6t^2+4t+2$	$2t^2+2$	$4t^2+8t-12$	$2t^6+6t^4-10t^2+2$
${\theta }_T$	$1$	$1$	$4.87517$	$3.17374$
$U_T$	$\begin{array}{c}5a^{3}r+20a^{2}br+29a{b}^{2}r+\\ 16b^{3}r+16a^{3}s+29a^{2}bs+\\ 20a{b}^{2}s+5b^{3}s\end{array}$	$\begin{array}{c}{a}^{2}r+2b^{2}r+\\ 2a^{2}s+b^{2}s\end{array}$	$\begin{array}{c}-54a^{3}r+144a^{2}br-117a{b}^{2}r+\\ 24b^{3}r-8a^{3}s+\\ 6a^{2}bs-b^{3}s\end{array}$	$\begin{array}{c}4a^{6}r-15a^4{b}^{2}r+20a^2{b}^{4}r-\\ 10b^{6}r-10a^{6}s+\\ 20a^4{b}^{2}s-15a^2{b}^{4}s+4b^{6}s\end{array}$
$V_T$	$\begin{array}{c}-5a^{3}r+20a^{2}br-29a{b}^{2}r+\\ 16b^{3}r+16a^{3}s-\\ 29a^{2}bs+20a{b}^{2}s-5b^{3}s\end{array}$	$\begin{array}{c}-a^{2}r+2b^{2}r+\\ 2a^{2}s-b^{2}s\end{array}$	$\begin{array}{c}54a^{3}r+144a^{2}br+\\ 117a{b}^{2}r+24b^{3}r-\\ 8a^{3}s-6a^{2}bs+b^{3}s\end{array}$	$\begin{array}{c}-4a^{6}r+15a^4{b}^{2}r+44a^2{b}^{4}r+\\ 26b^{6}r+26a^{6}s+\\ 44a^4{b}^{2}s+15a^2{b}^{4}s-4b^{6}s\end{array}$
$W_T$	$32\left(r{a}^7+s{b}^7\right)$	$4\left(r{a}^6+s{b}^6\right)$	$48\left(r{a}^7+s{b}^7\right)$	$16\left(r{a}^{14}+s{b}^{14}\right)$

Equations

$$F_{P_j^T}:y^2=x\left(x-a_j\right)\left(x+b_j\right)$$

$$a^{m_T}{\mathfrak{A}}_T\left(1,\frac{b}{a}\right)={\mathfrak{A}}_T\left(a,b\right)$$

$$f_T\left(\frac{b}{a}\right)=\frac{{\mathfrak{B}}_T\left(1,\frac{b}{a}\right)}{{\mathfrak{A}}_T\left(1,\frac{b}{a}\right)}-\frac{b}{a}.$$

$$\mathrm{rad}\left({\mathfrak{A}}_T\right)=\mathrm{rad}\left({\mathfrak{A}}_T^r\right),\mathrm{rad}\left({\mathfrak{B}}_T\right)=\mathrm{rad}\left({\mathfrak{B}}_T^r\right),\mathrm{rad}\left({\mathfrak{C}}_T\right)=\mathrm{rad}\left({\mathfrak{C}}_T^r\right).$$

$$\mathrm{rad}\left({\mathfrak{A}}_T{\mathfrak{B}}_T{\mathfrak{C}}_T\right)=\mathrm{rad}\left({\mathfrak{A}}_T^r{\mathfrak{B}}_T^r{\mathfrak{C}}_T^r\right)<\left|{\mathfrak{D}}_T\right|.$$

$${\mathfrak{C}}_T-\mathrm{rad}\left({\mathfrak{A}}_T{\mathfrak{B}}_T{\mathfrak{C}}_T\right)>{\mathfrak{C}}_T-{\mathfrak{D}}_T=a^{m_T}\left({\mathfrak{C}}_T\left(1,\frac{b}{a}\right)-{\mathfrak{D}}_T\left(1,\frac{b}{a}\right)\right)>0$$

$$P_j^T=\left(a_j,b_j,c_j\right)=\left({\mathfrak{A}}_T\left(a_{j-1},b_{j-1}\right),{\mathfrak{B}}_T\left(a_{j-1},b_{j-1}\right),{\mathfrak{C}}_T\left(a_{j-1},b_{j-1}\right)\right)\text{for }j\ge 1.$$

$${\mathcal{X}}_t\left(T\right):y^2=x^3+\left(t^4-12t^3+6t^2-12t+1\right)x^2-8t{\left(t-1\right)}^4\left(t^2+1\right)x.$$

$$t^{\prime }=\frac{t}{2{\left(t-1\right)}^2}$$

$$y^2=x\left(x-8ab\left(a^2+b^2\right)\right)\left(x+{\left(a-b\right)}^4\right)$$

$$F_P:y^2=x\left(x-a\right)\left(x+b\right).$$

$$\left|{\mathcal{X}}_t\left(T\right){\left(\mathbb{Q}\right)}_{\text{tors}}\right|\left|{\mathcal{X}}_t\left(T_4\right)\left(\mathbb{Q}\right)\left[\varphi \right]\right|=\left|{\mathcal{X}}_t\left(T_4\right){\left(\mathbb{Q}\right)}_{\text{tors}}\right|\left[{\mathcal{X}}_t\left(T_4\right){\left(\mathbb{Q}\right)}_{\text{tors}}:\varphi \left({\mathcal{X}}_t\left(T\right){\left(\mathbb{Q}\right)}_{\text{tors}}\right)\right].$$

$$y^2=x\left(x-8ab\left(a^2+b^2\right)\right)\left(x+{\left(a-b\right)}^4\right).$$

$$8ab\left(a^2+b^2\right)+{\left(a-b\right)}^4={\left(a+b\right)}^4.$$

$$q\left(P\right)=\frac{\log \left(c\right)}{\log \left(\mathrm{rad}\left(abc\right)\right)}.$$

$$N_E^{6+ϵ}<\max \left\{\left|c_4^3\right|,c_6^2\right\}$$

$${\sigma }_m\left(E\right)=\frac{\log \max \left\{\left|c_4^3\right|,c_6^2\right\}}{\log N_E}\text{and}\sigma \left(E\right)=\frac{\log \left|{\Delta }_E^{\text{min}}\right|}{\log N_E}$$

$$F_T:y^2=x\left(x-{\mathfrak{A}}_T\right)\left(x+{\mathfrak{B}}_T\right)$$

$$N_T=\mathrm{rad}\left({\mathfrak{A}}_T{\mathfrak{B}}_T{\mathfrak{C}}_T\right)<{\mathfrak{D}}_T$$

$$\frac{c_{4,T}^3-N_T^6}{{\mathfrak{D}}_T{\left(1,t\right)}^6}>\frac{c_{4,T}{\left(1,t\right)}^3-{\mathfrak{D}}_T{\left(1,t\right)}^6}{{\mathfrak{D}}_T{\left(1,t\right)}^6}\text{ for }t=\frac{b}{a}$$

$$P_j^T=\left(a_j,b_j,c_j\right)=\left({\mathfrak{A}}_T\left(a_{j-1},b_{j-1}\right),{\mathfrak{B}}_T\left(a_{j-1},b_{j-1}\right),{\mathfrak{C}}_T\left(a_{j-1},b_{j-1}\right)\right).$$

Full text

1. Introduction

By an $ABC$ triple, we mean a triple of positive integers $\left(a,b,c\right)$ such that $a,b,$ and $c$ are relatively prime positive integers with $a+b=c$. The $ABC$ Conjecture [CR01, 5.1] states that for any $\epsilon>0$, there are only finitely many $ABC$ triples that satisfy $\operatorname{rad}\!\left(abc\right)^{1+\epsilon}<c$ where $\operatorname{rad}\!\left(n\right)$ denotes the product of the distinct primes dividing $n$. We say that an $ABC$ triple is good if $\operatorname{rad}\!\left(abc\right)<c$. For instance, the triple $\left(1,8,9\right)$ is a good $ABC$ triple and more generally the triple $\left(1,9^{k}-1,9^{k}\right)$ is a good $ABC$ triple for each positive integer $k$ [CR01]. In $1988,$ Oesterlé [Oes88] proved that the $ABC$ Conjecture is equivalent to the modified Szpiro conjecture which states that for $\epsilon>0$, there are only finitely many elliptic curves $E$ such that $N_{E}^{6+\epsilon}<\max\left\{\left|c_{4}^{3}\right|,c_{6}^{2}\right\}$ where $N_{E}$ denotes the conductor of the elliptic curve and $c_{4}$ and $c_{6}$ are the invariants associated to a minimal model of $E$. As with $ABC$ triples, we define a good elliptic curve to be an elliptic curve $E$ that satisfies the inequality $N_{E}^{6}<\max\left\{\left|c_{4}^{3}\right|,c_{6}^{2}\right\}$. In the special case of Frey curves, that is, a rational elliptic curve that has a Weierstrass model of the form $y^{2}=x\left(x-a\right)\left(x+b\right)$ where $a$ and $b$ are relatively prime integers, Masser [Mas90] showed that there are infinitely many good Frey curves. In this article, we provide a constructive proof of Masser’s Theorem. Moreover, the torsion subgroup of a Frey curve can only take on four possibilities due to Mazur’s Torsion Theorem [Maz77], namely $E\!\left(\mathbb{Q}\right)_{\text{tors}}\cong C_{2}\times C_{2N}$ where $C_{m}$ denotes the cyclic group of order $m$ and $N=1,2,3,$ or $4$. With this we state our main theorem:

• Theorem 1

For each of the four possible torsion subgroups $T=C_{2}\times C_{2N}$ where $N=1,2,3,$ or $4$, there are infinitely many good elliptic curves such that $E\!\left(\mathbb{Q}\right)_{\text{tors}}\cong T$.

This is equivalent to Theorem 6.3, where the main theorem is given in its constructive form. As a consequence we get examples akin to the infinitely many good $ABC$ triples $\left(1,9^{k}-1,9^{k}\right)$ for each positive integer $k$. For each of the four possible $T$, we use rational maps of modular curves to construct a recursive sequence of $ABC$ triples $P_{j}^{T}=\left(a_{j},b_{j},c_{j}\right)$ such that if $P_{j}^{T}$ is a good $ABC$ triple satisfying certain congruences, then $P_{j}^{T}$ is a good $ABC$ triple for each nonnegative integer $j$. Once this is proven, we prove our main Theorem by showing that the associated Frey curve

[TABLE]

is a good elliptic curve for each positive integer $j$ with $F_{P_{j}^{T}}\!\left(\mathbb{Q}\right)_{\text{tors}}\cong T$.

2. Certain Polynomials

In this section we establish a series of technical results which will ease the proofs in the sections that are to follow. Let $T=C_{2}\times C_{2N}$ where $N=1,2,3,4$. For each $T$ let $\mathfrak{A}_{T}=\mathfrak{A}_{T}\!\left(a,b\right),$ $\mathfrak{B}_{T}=\mathfrak{B}_{T}\!\left(a,b\right),$ $\mathfrak{C}_{T}=\mathfrak{C}_{T}\!\left(a,b\right),$ $\mathfrak{D}_{T}=\mathfrak{D}_{T}\!\left(a,b\right),$ $\mathfrak{A}_{T}^{r}=\mathfrak{A}_{T}^{r}\!\left(a,b\right),\ \mathfrak{B}_{T}^{r}=\mathfrak{B}_{T}^{r}\!\left(a,b\right),$ $\mathfrak{C}_{T}^{r}=\mathfrak{C}_{T}^{r}\!\left(a,b\right),$ $U_{T}=U_{T}\!\left(a,b,r,s\right),$ $V_{T}=V_{T}\!\left(a,b,r,s\right),$ and $W_{T}=W_{T}\left(a,b,r,s\right)$ be the polynomials in $R=\mathbb{Z}\!\left[a,b,r,s\right]$ defined in Table 6.

For a fixed $T$, the polynomials $\mathfrak{A}_{T},$ $\mathfrak{B}_{T},$ $\mathfrak{C}_{T},$ and $\mathfrak{D}_{T}$ are homogenous polynomials in $a$ and $b$ of the same degree $m_{T}$. In particular, we have the equalities

[TABLE]

The first result can be verified via a computer algebra system and we note that we are considering $\mathfrak{A}_{T}\!\left(1,t\right),\ \mathfrak{B}_{T}\!\left(1,t\right),\ \mathfrak{C}_{T}\!\left(1,t\right),\ \mathfrak{D}_{T}\!\left(1,t\right)$ as functions from $\mathbb{R}$ to $\mathbb{R}$.

Lemma 2.1**.**

For $T=C_{2}\times C_{2N}$ with $N=1,2,3,4$, let $f_{T},g_{T}:\mathbb{R}\rightarrow\mathbb{R}$ be the function in the variable $t$ defined in Table 6. Let $\theta_{T}$ be the greatest real root of $f_{T}\!\left(t\right)$. The (approximate) value of $\theta_{T}$ is found in Table 6. Then for each $T$,

1. (1)

$\mathfrak{A}_{T}\in 4R$; 2. (2)

$\mathfrak{A}_{T}+\mathfrak{B}_{T}=\mathfrak{C}_{T}$; 3. (3)

$U_{T}\mathfrak{B}_{T}+V_{T}\mathfrak{C}_{T}=W_{T}$; 4. (4)

$f_{T}\!\left(\frac{b}{a}\right)=\frac{\mathfrak{B}_{T}\!\left(a,b\right)}{\mathfrak{A}_{T}\left(a,b\right)}-\frac{b}{a}$; 5. (5)

$g_{T}\!\left(t\right)=\mathfrak{C}_{T}\!\left(1,t\right)-\mathfrak{D}_{T}\!\left(1,t\right)$; 6. (6)

$f_{T}\!\left(t\right),g_{T}\!\left(t\right),\ \mathfrak{A}_{T}\!\left(1,t\right),\ \mathfrak{B}_{T}\!\left(1,t\right),\ \mathfrak{C}_{T}\!\left(1,t\right),\ \mathfrak{D}_{T}\!\left(1,t\right)>0$* for $t>\theta_{T}$;* 7. (7)

For $T=C_{2}\times C_{2N}$ for $N=1,2$, $f_{T}\!\left(t\right),g_{T}\!\left(t\right),\ \mathfrak{A}_{T}\!\left(1,t\right),\ \mathfrak{B}_{T}\!\left(1,t\right),\ \mathfrak{C}_{T}\!\left(1,t\right)$, $\mathfrak{D}_{T}\!\left(1,t\right)>0$ for $t$ in $\left(0,1\right)$.

3. Good $ABC$ Triples

Definition 3.1**.**

By an $ABC$ triple, we mean a triple $P=\left(a,b,c\right)$ such that $a,b,$ and $c$ are relatively prime positive integers with $a+b=c$. We say $P=\left(a,b,c\right)$ is good if $\operatorname{rad}\!\left(abc\right)<c$.

Lemma 3.2**.**

For each $T=C_{2}\times C_{2N}$, let $P=\left(a,b,a+b\right)$ be an $ABC$ triple with $a$ even and $\frac{b}{a}>\theta_{T}$ where $\theta_{T}$ is as defined in Lemma 2.1. Suppose further that $a\equiv 0\ \operatorname{mod}3$ if $N=3$. Then $\left(\mathfrak{A}_{T},\mathfrak{B}_{T},\mathfrak{C}_{T}\right)$ is an $ABC$ triple with $\mathfrak{A}_{T}\equiv 0\ \operatorname{mod}16,\ \mathfrak{B}_{T}\equiv 1\ \operatorname{mod}4,$ and$\ \frac{\mathfrak{B}_{T}}{\mathfrak{A}_{T}}>\theta_{T}$. Moreover, if $N=3$, then $\mathfrak{A}_{T}\equiv 0\ \operatorname{mod}3$.

Proof.

Since $a$ and $b$ are relatively prime, there exist integers $r$ and $s$ such that $ra^{n}+sb^{n}=1,$ for any positive integer $n$. Therefore, by Lemma 2.1, $\gcd\!\left(\mathfrak{B}_{T},\mathfrak{C}_{T}\right)$ divides $32$ if $N\neq 3$ and $\gcd\!\left(\mathfrak{B}_{T},\mathfrak{C}_{T}\right)$ divides $48$ if $N=3$. Since $a$ is even and $a\equiv 0\ \operatorname{mod}3$ when $N=3$, we conclude that $\gcd\!\left(\mathfrak{B}_{T},\mathfrak{C}_{T}\right)=1$. Next, observe that

[TABLE]

Since $\frac{b}{a}>\theta_{T}$, we have by Lemma 2.1 that $f_{T}\!\left(\frac{b}{a}\right)$ is positive and therefore $\frac{\mathfrak{B}_{T}}{\mathfrak{A}_{T}}>\frac{b}{a}>\theta_{T}$. By Lemma 2.1 we also have that $\mathfrak{A}_{T}+\mathfrak{B}_{T}=\mathfrak{C}_{T}$ for each $T$ and therefore $\left(\mathfrak{A}_{T},\mathfrak{B}_{T},\mathfrak{C}_{T}\right)$ is an $ABC$ triple. Since $a$ is even it is easily verified that $\mathfrak{A}_{T}\equiv 0\ \operatorname{mod}16$. Similarly, when $N=3$, $\mathfrak{A}_{T}\equiv 0\ \operatorname{mod}3$ since $a\equiv 0\ \operatorname{mod}3$. It easily checked that for each $T$, $\mathfrak{B}_{T}\equiv b^{2k}\ \operatorname{mod}4$ for some integer $k$. Since $b$ is odd, it follows that $\mathfrak{B}_{T}\equiv 1\ \operatorname{mod}4$. ∎

Lemma 3.3**.**

Let $P=\left(a,b,a+b\right)$ be a good $ABC$ triple and assume the statement of Lemma 3.2. Then $\left(\mathfrak{A}_{T},\mathfrak{B}_{T},\mathfrak{C}_{T}\right)$ is a good $ABC$ triple.

Proof.

Since $a$ is assumed to be even, we have that $\operatorname{rad}\!\left(2^{n}ax\right)=\operatorname{rad}\!\left(ax\right)$ for some integer $x$. Therefore

[TABLE]

Since $\left(a,b,a+b\right)$ is a good $ABC$ triple, we have that $\operatorname{rad}\!\left(ab\!\left(a+b\right)\right)<a+b$. From this and the fact that $\operatorname{rad}\!\left(xy^{k}\right)=\operatorname{rad}\!\left(xy\right)\leq xy$ for positive integers $k,x,y$, we have that for each $T$, we attain

[TABLE]

Since $\frac{b}{a}>\theta_{T}$, $\mathfrak{D}_{T}\!\left(1,\frac{b}{a}\right)$ is positive by Lemma 2.1. In particular, $\mathfrak{D}_{T}$ is positive since $a^{m_{T}}\mathfrak{D}_{T}\!\left(1,\frac{b}{a}\right)=\mathfrak{D}_{T}$ where $m_{T}$ is the homogenous degree of $\mathfrak{D}_{T}$. Now observe that

[TABLE]

where the positivity follows from Lemma 2.1. Hence $\left(\mathfrak{A}_{T},\mathfrak{B}_{T},\mathfrak{C}_{T}\right)$ is a good $ABC$ triple since $\operatorname{rad}\!\left(\mathfrak{A}_{T}\mathfrak{B}_{T}\mathfrak{C}_{T}\right)<\mathfrak{C}_{T}$. ∎

Proposition 3.4**.**

Let $\left(a_{0},b_{0},c_{0}\right)$ be a good $ABC$ triple with $a_{0}$ even. For each $T$ define the triple $P_{j}^{T}$ recursively by

[TABLE]

Assume further that $\frac{b_{0}}{a_{0}}>\theta_{T}$ and that $b_{0}\equiv 0\ \operatorname{mod}3$ if $T=C_{2}\times C_{6}$. Then for each $j\geq 1$, $P_{j}^{T}$ is a good $ABC$ triple with $a_{j}\equiv 0\ \operatorname{mod}16,\ b_{j}\equiv 1\ \operatorname{mod}4,$ and $\frac{b_{j}}{a_{j}}>\theta_{T}$. Additionally, if $T=C_{2}\times C_{6}$, then $a_{j}\equiv 0\ \operatorname{mod}3$.

Proof.

This follows automatically from Lemmas 3.2 and 3.3. ∎

4. Frey Curves

As before, we suppose $T=C_{2}\times C_{2N}$ and define for $t\in\mathbb{P}^{1}$, the mapping $\mathcal{X}_{t}$ as the mapping which takes $T$ to the elliptic curve $\mathcal{X}_{t}\!\left(T\right)$ where the Weierstrass model of $\mathcal{X}_{t}\!\left(T\right)$ is given in Table 1. Our parameterizations for $T=C_{2}\times C_{2N}$ where $N=3,4$ are those found in [HLP00, Table 3] which expands the implicit expressions for the parameters $b$ and $c$ in [Kub76, Table 3] to express the universal elliptic curves for the modular curves $X_{1}\!\left(2,2N\right)$ in terms of a single parameter $t$. Similarly, our model for $T=C_{2}\times C_{4}$ differs by a linear change of variables from the model given for $W_{4}$ in [Sil97, $\S 4$] which parameterizes elliptic curves $E$ with $C_{4}\times C_{4}\hookrightarrow E\!\left(\mathbb{Q}\!\left(i\right)\right)_{\text{tors}}$. In particular, $\mathcal{X}_{t}\!\left(T\right)$ is a one-parameter family of elliptic curves with the property that if $t\in K$ for some field $K$, then $\mathcal{X}_{t}\!\left(T\right)$ is an elliptic curve over $K$ and $T\hookrightarrow\mathcal{X}_{t}\!\left(T\right)\!\left(K\right)_{\text{tors}}$.

For $T=C_{2}\times C_{2}$, define

[TABLE]

Lemma 4.1**.**

If $t\in\mathbb{Q}$ such that $\mathcal{X}_{t}\!\left(T\right)$ is an elliptic curve, then $T\hookrightarrow\mathcal{X}_{t}\!\left(T\right)\!\left(\mathbb{Q}\right)_{\text{tors}}$.

Proof.

Recall that the modular curve $X_{1}\!\left(2,2N\right)$ (with cusps removed) for $N=2,3,4$ parameterizes isomorphism classes of pairs $\left(E,P,Q\right)$ where $E$ is an elliptic curve having full $2$-torsion, $P$ and $Q$ are torsion points of order $2$ and $2N$, respectively, and $\left\langle P,N\cdot Q\right\rangle=E\!\left[2\right]$.

For $T=C_{2}\times C_{2N}$ where $N=3,4$, we note that our parameterizations are those of the universal elliptic curve for the modular curve $X_{1}\!\left(2,2N\right)$ [HLP00, Table 3]. Thus $T\hookrightarrow\mathcal{X}_{t}\!\left(T\right)\left(\mathbb{Q}\right)_{\text{tors}}$.

For $T=C_{2}\times C_{4}$, let

[TABLE]

so that $\mathcal{X}_{t}\!\left(T\right)$ is equal to the Weierstrass model given for the universal elliptic curve over $X_{1}\!\left(2,4\right)$ given in [HLP00, Table 3] with parameter $t^{\prime}$. Hence $T\hookrightarrow\mathcal{X}_{t}\!\left(T\right)\!\left(\mathbb{Q}\right)_{\text{tors}}$.

For $T=C_{2}\times C_{2}$, let $t=\frac{b}{a}$ and consider the admissible change of variables $x\longmapsto\frac{1}{a^{4}}x$ and $y\longmapsto\frac{1}{a^{6}}y$. This gives a $\mathbb{Q}$-isomorphism between $\mathcal{X}_{t}\!\left(T\right)$ and the elliptic curve

[TABLE]

which has $\left\langle\left(8ab\left(a^{2}+b^{2}\right),0\right),\left(0,0\right)\right\rangle\cong C_{2}\times C_{2}$. Thus $T\hookrightarrow\mathcal{X}_{t}\!\left(T\right)\!\left(\mathbb{Q}\right)_{\text{tors}}$. ∎

Definition 4.2**.**

For an $ABC$ triple $P=\left(a,b,c\right)$, let $F_{P}=F_{P}\!\left(a,b\right)$ be the Frey curve given by the Weierstrass model

[TABLE]

Lemma 4.3**.**

Let $\left(a,b,c\right)$ be an $ABC$ triple which satisfies the assumptions of Lemma 3.2. Then for each $T$, the Frey curve $F_{P}$ with $P=\left(\mathfrak{A}_{T},\mathfrak{B}_{T},\mathfrak{C}_{T}\right)$ has torsion subgroup $F_{P}\!\left(\mathbb{Q}\right)_{\text{tors}}\cong T$.

Proof.

Let $\mathcal{X}_{t}\!\left(T\right)$ be as defined in Table 1 for $T=C_{2}\times C_{2N}$ for $N=2,3,4$ and as defined in (4.1) for $N=1$. In addition, let $u_{T},r_{T},s_{T},w_{T},$ and $t_{T}$ be as defined in Table 5. We now proceed by cases.

Case I. Suppose $T=C_{2}\times C_{2N}$ for $N=2,3,4$. Then the admissible change of variables $x\longmapsto u_{T}^{2}x+r_{T}$ and $y\longmapsto u_{T}^{3}y+u_{T}^{2}s_{T}x+w_{T}$ gives a $\mathbb{Q}$-isomorphism from $F_{P}$ onto $\mathcal{X}_{t_{T}}\!\left(T\right)$. In particular, $T\hookrightarrow F_{P}\!\left(\mathbb{Q}\right)_{\text{tors}}$ by Lemma 4.1. By Mazur’s Torsion Theorem [Maz77] we conclude that $F_{P}\!\left(\mathbb{Q}\right)_{\text{tors}}\cong C_{2}\times C_{2N}$ for $N=3,4$ and that $F_{P}\!\left(\mathbb{Q}\right)_{\text{tors}}$ is isomorphic to either $C_{2}\times C_{4}$ or $C_{2}\times C_{8}$ if $T=C_{2}\times C_{4}$. For the latter, we observe that our model for $\mathcal{X}_{t}\!\left(T\right)$ parametrizes elliptic curves $E$ over $\mathbb{Q}\!\left(i\right)$ with $C_{4}\times C_{4}\hookrightarrow E\!\left(\mathbb{Q}\!\left(i\right)\right)_{\text{tors}}$ [Sil97, $\S 4$]. By Kamienny’s Torsion Theorem [Kam92] we conclude that $E\!\left(\mathbb{Q}\!\left(i\right)\right)_{\text{tors}}\cong C_{4}\times C_{4}$. Thus $\mathcal{X}_{t}\!\left(T\right)\!\left(\mathbb{Q}\!\left(i\right)\right)_{\text{tors}}\cong C_{4}\times C_{4}$ and therefore $C_{2}\times C_{8}\not\hookrightarrow\mathcal{X}_{t}\!\left(T\right)\!\left(\mathbb{Q}\!\left(i\right)\right)_{\text{tors}}$. Hence $\mathcal{X}_{t}\!\left(T\right)\!\left(\mathbb{Q}\right)_{\text{tors}}\cong C_{2}\times C_{4}$.

Case II. Suppose $T=C_{2}\times C_{2}$ and $T_{4}=C_{2}\times C_{4}$. Then there is a $2$-isogeny $\phi:\mathcal{X}_{t}\!\left(T_{4}\right)\rightarrow\mathcal{X}_{t}\!\left(T\right)$ obtained by applying Vélu’s formulas [V7́1] to the elliptic curve $\mathcal{X}_{t}\!\left(T_{4}\right)$ and its torsion point $2P$ where $P=\left(0,0\right)$ is the torsion point of order $4$ of $\mathcal{X}_{t}\!\left(T_{4}\right)$.

Next, observe that via the First Isomorphism Theorem:

[TABLE]

By Case I above we have that $\left|\mathcal{X}_{t}\!\left(T_{4}\right)\!\left(\mathbb{Q}\right)_{\text{tors}}\right|=8$ which implies that the only prime dividing $\left|\mathcal{X}_{t}\!\left(T\right)\!\left(\mathbb{Q}\right)_{\text{tors}}\right|$ is $2$ since $\phi$ is a $2$-isogeny.

Next, we consider the admissible change of variables $x\longmapsto u_{T}^{2}x+r_{T}$ and $y\longmapsto u_{T}^{3}y+u_{T}^{2}s_{T}x+w_{T}$ which gives a $\mathbb{Q}$-isomorphism from $F_{P}$ onto $\mathcal{X}_{t_{T}}\!\left(T\right)$. In particular, $C_{2}\times C_{2}\hookrightarrow F_{P}\!\left(\mathbb{Q}\right)_{\text{tors}}$ by Lemma 4.1. By the proof of Lemma 4.1, $\mathcal{X}_{t}\!\left(T\right)$ is $\mathbb{Q}$-isomorphic to the elliptic curve given by the Weierstrass model

[TABLE]

This model satisfies the assumptions of [Ono96, Main Theorem 1] and therefore we have that $\mathcal{X}_{t}\!\left(T\right)\!\left(\mathbb{Q}\right)_{\text{tors}}\cong C_{2}\times C_{2}$ if $8ab\left(a^{2}+b^{2}\right)$ is not a square. If it were a square we would have a nontrivial integer solution to the Diophantine equation $x^{4}-y^{4}=z^{2}$ since

[TABLE]

This contradicts Fermat’s Theorem and therefore $\mathcal{Y}_{t}\!\left(T\right)\!\left(\mathbb{Q}\right)_{\text{tors}}\cong C_{2}\times C_{2}$. ∎

Theorem 4.4**.**

Let $T=C_{2}\times C_{2N}$ for $N=1,2,3,4$ and consider the sequence of good $ABC$ triples $P_{j}^{T}$ defined in Proposition 3.4. Then for each $j\geq 1$, the Frey curve $F_{P_{j}^{T}}$ determined by $P_{j}^{T}$ has torsion subgroup $F_{P_{j}^{T}}\!\left(\mathbb{Q}\right)_{\text{tors}}\cong C_{2}\times C_{2N}$.

Proof.

In Proposition 3.4, we saw that each $P_{j}^{T}$ satisfies the assumptions of Lemma 3.2. Consequently, the Theorem follows from Lemma 4.3. ∎

The case of $N=2,4$ in Theorem 4.4 was proven by the author alongside Watts and Tillman [BTW10] as part of the Mathematical Sciences Research Institute Undergraduate Program.

5. Examples of Good ABC Triples

Definition 5.1**.**

For an $ABC$** **triple $P=\left(a,b,c\right)$, define the quality $q\!\left(P\right)$ of $P$ to be

[TABLE]

In particular, $P$ is a good $ABC$ triple is equivalent to $q\!\left(P\right)>1$.

Example 5.2**.**

For $T=C_{2}\times C_{2N}$ where $N=1,2$ let $P_{0}=\left(2^{5},7^{2},3^{4}\right)$. Then $P_{0}$ is a good $ABC$ triple since $q\!\left(P\right)\approx 1.1757$. By Proposition 3.4, this good $ABC$ triple results in two distinct infinite sequences of good $ABC$ triples $P_{j}^{T}$.

For $T=C_{2}\times C_{6}$, let $P_{0}=\left(2^{4}3^{3},17^{3}61,5^{3}7^{4}\right)$. Then $P_{0}$ is a good $ABC$ triple since $q\!\left(P\right)\approx 1.0261$. Moreover, $\frac{17^{3}61}{2^{4}3^{3}}>\theta_{T}$. By Proposition 3.4, this good $ABC$ triple results in an infinite sequence of good $ABC$ triples $P_{j}^{T}$.

For $T=C_{2}\times C_{8}$, let $P_{0}=\left(2^{2},11^{2},5^{3}\right)$. Then $P_{0}$ is a good $ABC$ triple since $q\!\left(P\right)\approx 1.0272$. Moreover, $\frac{121}{4}>\theta_{T}$. By Proposition 3.4, this good $ABC$ triple results in an infinite sequence of good $ABC$ triples $P_{j}^{T}$.

Table 2 gives $a_{1}$ and $b_{1}$ of $P_{j}^{T}=\left(a_{j},b_{j},c_{j}\right)$ as well as the quality $q\!\left(P_{j}^{T}\right)$ for $j=1,2,3$. We note that the values of $a_{j}$ and $b_{j}$ are not given for $j\geq 2$ due to the size of these quantities. For $T=C_{2}\times C_{2N}$ for $N=3,4$, we only compute $q\!\left(P_{j}^{T}\right)$ for $j=1,2$ due to computational limitations.

6. Infinitely Many Good Frey Curves

Recall that the $ABC$ Conjecture is equivalent to the modified Szpiro conjecture which states that for every $\epsilon>0$ there are finitely many rational elliptic curves $E$ satisfying

[TABLE]

where $N_{E}$ is the conductor of $E$ and $c_{4}$ and $c_{6}$ are the invariants associated to a minimal model of $E$. The following definition gives the analog of good $ABC$ triples and the quality of an $ABC$ triple in the context of elliptic curves.

Definition 6.1**.**

Let $E$ be a rational elliptic curve with minimal discriminant $\Delta_{E}^{\text{min}}$ and associated invariants $c_{4}$ and $c_{6}$. Define the modified Szpiro ratio $\sigma_{m}\!\left(E\right)$ and Szpiro ratio $\sigma\!\left(E\right)$ of $E$ to be the quantities

[TABLE]

where $N_{E}$ is the conductor of $E$. We say that $E$ is good if $\sigma_{m}\!\left(E\right)>6$.

Let $P=\left(a,b,c\right)$ be an $ABC$ triple with $a$ even and $b\equiv 1$ $\operatorname{mod}4$. For $T=C_{2}\times C_{2N}$ where $N=1,2,3,4$, let $\mathfrak{A}_{T}=\mathfrak{A}_{T}\!\left(a,b\right),$ $\mathfrak{B}_{T}=\mathfrak{B}_{T}\!\left(a,b\right),\ \mathfrak{C}_{T}=C_{T}\!\left(a,b\right),$ and $\mathfrak{D}_{T}=\mathfrak{D}_{T}\!\left(a,b\right)$ be as defined in Table 6. Assume further that $a\equiv 0\ \operatorname{mod}3$ if $T=C_{2}\times C_{6}$. Then the elliptic curve $F_{T}=F_{T}\!\left(a,b\right)$ given by the Weierstrass model

[TABLE]

satisfies $F_{T}\!\left(\mathbb{Q}\right)_{\text{tors}}\cong T$ by Lemma 4.3. Moreover, the congruences on $\mathfrak{A}_{T}$ and $\mathfrak{B}_{T}$ imply that the Frey curve $F_{T}$ is semistable with minimal discriminant $\Delta_{T}=\left(16^{-1}\mathfrak{A}_{T}\mathfrak{B}_{T}\mathfrak{C}_{T}\right)^{2}$ [Sil09, Exercise 8.23]. Consequently, the conductor $N_{T}\mathfrak{\ }$of $F_{T}$ satisfies $N_{T}=\operatorname{rad}\!\left(\Delta_{T}\right)<\left|\mathfrak{D}_{T}\right|$ and the invariant $c_{4,T}=c_{4,T}\!\left(a,b\right)$ associated with a global minimal model of $F_{T}$ is as given in Table LABEL:ta:c4THT.

Lemma 6.2**.**

Let $P=\left(a,b,c\right)$ be a good $ABC$ triple satisfying $a\equiv 0\ \operatorname{mod}2$, $b\equiv 1\ \operatorname{mod}4,$ and $\frac{b}{a}>\theta_{T}$ where $\theta_{T}$ is as given in Lemma 2.1. Assume further that $a\equiv 0\ \operatorname{mod}3$ if $T=C_{2}\times C_{6}$. Then the Frey curve $F_{T}=F_{T}\!\left(\mathfrak{A}_{T},\mathfrak{B}_{T}\right)$ is good and $F_{T}\!\left(\mathbb{Q}\right)_{\text{tors}}\cong T$.

Proof.

By Lemma 4.3, $F_{T}\!\left(\mathbb{Q}\right)_{\text{tors}}\cong T$. Since $F_{T}$ is a Frey curve we have that the invariants $c_{4}$ and $c_{6}$ associated to a global minimal model of $F_{T}$ satisfy $\max\!\left\{\left|c_{4}^{3}\right|,c_{6}^{2}\right\}=c_{4}^{3}$ since $c_{4}$ and $\Delta_{F_{T}}^{\text{min}}$ are always positive [Sil09, Lemma VIII.11.3]. The congruences on $a$ and $b$ imply that $c_{4}=c_{4,T}$. It, therefore, suffices to show that $c_{4,T}^{3}-N_{T}^{6}>0$ where $N_{T}$ is the conductor of $F_{T}$. Since $F_{T}$ is semistable,

[TABLE]

by Lemma 3.3. Note that $\mathfrak{D}_{T}$ is positive since $\frac{b}{a}>\theta_{T}$. Thus

[TABLE]

Lastly, for each $T$, the polynomial $c_{4,T}\!\left(1,t\right)^{3}-\mathfrak{D}_{T}\!\left(1,t\right)^{6}$ is positive on the open interval $\left(\theta_{T},\infty\right)$ from which we conclude that $F_{T}$ is a good elliptic curve. ∎

Theorem 6.3**.**

For each $T$, let $P_{0}^{T}=\left(a_{0},b_{0},c_{0}\right)$ be a good $ABC$ triple satisfying $a_{0}\equiv 0\ \operatorname{mod}2$, $b_{0}\equiv 1\ \operatorname{mod}4,$ and $\frac{b_{0}}{a_{0}}>\theta_{T}$ where $\theta_{T}$ is as given in Lemma 2.1. Assume further that $a_{0}\equiv 0\ \operatorname{mod}3$ if $T=C_{2}\times C_{6}$. For $j\geq 1$, define $P_{j}^{T}$ recursively by

[TABLE]

Then for each $j$, the Frey curve $F_{T}\!\left(a_{j},b_{j}\right)$ is good and $F_{T}\!\left(a_{j},b_{j}\right)\left(\mathbb{Q}\right)_{\text{tors}}\cong T$.

Proof.

By Proposition 3.4, $P_{j}^{T}=\left(a_{j},b_{j},c_{j}\right)$ satisfies $a_{j}\equiv 0\ \operatorname{mod}2$, $b_{j}\equiv 1\ \operatorname{mod}4,$ and $\frac{b_{j}}{a_{j}}>\theta_{T}$ for each $j$. For $T=C_{2}\times C_{6}$, if $a_{0}\equiv 0\ \operatorname{mod}3$, then $a_{j}\equiv 0\ \operatorname{mod}3$ for each $j$. Hence $P_{j}^{T}$ is a good $ABC$ triple for each $j$ by Proposition 3.4. Therefore the result follows by Lemma 6.2. ∎

In Example 5.2 we began with a good $ABC$ triple $P_{0}=\left(a_{0},b_{0},c_{0}\right)$. For each $T$, we constructed an infinite sequence of good $ABC$ triples $P_{j}^{T}=\left(a_{j},b_{j},c_{j}\right)$. By Theorem 6.3, each Frey curve $F_{T}\!\left(a_{j},b_{j}\right)\left(\mathbb{Q}\right)_{\text{tors}}$ is a good elliptic curve with torsion subgroup isomorphic to $T$. Table 4 lists the modified Szpiro ratios of the Frey curves corresponding to $P_{j}^{T}$. Due to computational limitations, we could only compute these ratios up to $j=3$.

7. Table of Polynomials