Why the Szpiro conjecture over number fields doesn't depend on the discriminant of the number field?

Question

According to Hindry p.7 Conj 3.1 and Stein Szpiro's conjecture states that the Szpiro ratio is:

$$ \sigma_{E/K}=\frac{\log{|N_{K/Q}\Delta_{E/K}|}}{\log{|N_{K/Q} f_{E/K}}|}$$

Given $ \varepsilon >0$ there are only finitely many $ E/K$ with $ \sigma_{E/K}\geq 6+\varepsilon $. In particular, $ \sigma_{E/K}$ is bounded.

$\Delta_{E/K}$ is the minimal discriminant.

Szpiro's conjecture implies abc with a bit higher exponent over the integers.

Over number fields, the uniform abc conjecture depends on the discriminant of the number field. Maybe this is because every integer can be arbitrary large power in some number field. To make $d$ a $k$-th power, work with defining polynomial $x^k -d$. So start with EC over the rationals with discriminant $\Delta$ and compute $\sigma_{E/K}$ for $K$ with defining polynomial $x^k - \Delta$ for $k$ sufficiently large.

Unless the minimal discriminant takes care of this case, this will give unbounded Szpiro ratio over some number fields.

Why the Szpiro conjecture over number fields doesn't depend on the discriminant of the number field while the uniform abc conjecture depends?

Over $K$ with defining polynomial $x^{16} + 22384$ take $E/K : y^2 = x^3 + 7x -1$.

Not sure if the discriminant is minimal, but the norm is $2^{64} \cdot 1399^{16}$ while the norm of the conductor is $2^{10} \cdot 1399$ giving Szpiro ratio $11.305664847$

Some experiments in degree 14 and sage's global_minimal_model() suggest the global minimal model preserves the high powers in the discriminant:

a4,a6= -2 -1  global minimal model= Elliptic Curve defined by y^2 + (1/14*w^12-1/28*w^10+1/7*w^8+1/4*w^7-1/14*w^6+2/7*w^4-1/7*w^2-3/7)*y = x^3 + (243/56*w^13+17/16*w^12-297/28*w^11+51/2*w^10-561/14*w^9+111/2*w^8-1581/28*w^7+81/2*w^6+255/28*w^5-99*w^4+1530/7*w^3-374*w^2+3330/7*w-527)*x + (-531/112*w^13-9419/56*w^12+2945/7*w^11-40829/56*w^10+55889/56*w^9-31119/28*w^8+47939/56*w^7-1209/28*w^6-10470/7*w^5+53021/14*w^4-90715/14*w^3+125729/14*w^2-69170/7*w+107855/14) over Number Field in w with defining polynomial x^14 - 80 Delta= 2^32 * 5^14 f= 2^32 * 5 d_q= 2^4 * 5  ratio= 1.87946880927217
a4,a6= -2 1  global minimal model= Elliptic Curve defined by y^2 + (1/14*w^12-1/28*w^10+1/7*w^8+1/4*w^7-1/14*w^6+2/7*w^4-1/7*w^2-3/7)*y = x^3 + (243/56*w^13+17/16*w^12-297/28*w^11+51/2*w^10-561/14*w^9+111/2*w^8-1581/28*w^7+81/2*w^6+255/28*w^5-99*w^4+1530/7*w^3-374*w^2+3330/7*w-527)*x + (533/112*w^13+4709/28*w^12-5891/14*w^11+40819/56*w^10-55887/56*w^9+31125/28*w^8-47933/56*w^7+1185/28*w^6+10460/7*w^5-26504/7*w^4+90725/14*w^3-125767/14*w^2+69150/7*w-107857/14) over Number Field in w with defining polynomial x^14 - 80 Delta= 2^32 * 5^14 f= 2^18 * 5 d_q= 2^4 * 5  ratio= 3.17425556409935

The last two examples give the abc triples $c_4^3-c_6^2=w^{14}$

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Curves of Szpiro ratios $ > 66$ are in this question and they appear to contradict boundedness of the ratio.

Answer 1

Szpiro made his original conjecture in 1982. It was eventually published in Discriminant et conducteur des courbes elliptiques, Astérisque No. 183 (1990), pp7–18 (MR1065151). The conjecture was much weaker than what you've stated. It allowed for the possibility that even the $6$ could depend on $K$.

There is a whole tangled web of conjectures and implications related to Szpiro's conjecture. Some of the strongest conjectures are due to Lang and Vojta. They come from analogy with value distribution theory for meromorphic functions (aka Nevanlinna Theory). Even these seem to deal with one number field at a time.

Hindry has very nice article, Why is it difficult to compute the Mordell-Weil group?, about Mordell-Weil groups where he mentions Szpiro's conjecture. He also mentions the generalized Szpiro conjecture formulated by Frey and formulated in terms of Faltings height. Hindry used a theorem of Deligne about semiabelian schemes to generalize Faltings general abelian varieties. Conjecture 3.4 of Hindry's paper should read

Szpiro Conjecture for Abelian Varieties (Hindry, Conj. 3.4) $$h_{\text{Falt}}(A/K) \leq \left ( \frac{g}{2} + \epsilon \right ) \log N_{\Bbb Q}^K \mathcal F_{A/K} + C_{\epsilon,K}$$

where: $K$ is still a fixed number field, $A$ is a abelian variety over dimension $g$ over $K$, $h_{\text{Falt}}$ is the Faltings height of $A / K$. and $C_{\epsilon,K}$ is a constant that still depends on $K$ and $\epsilon$.

The good part is the remark just below! Assuming the conjecture is true for all abelian varieties over $\Bbb Q$ and playing with Weil restriction of scalars, Hindry obtains:

Uniform Szpiro Conjecture for Abelian Varieties (Hindry, Remark after Conj. 3.4) $$h_{\text{Falt}}(A/K) \leq \left (\frac{g}{2} + \epsilon\right) \log N_{\Bbb Q}^K \mathcal F_{A/K} + (g^2 + \epsilon) \log |\Delta_K| + C_{\epsilon,[K:\Bbb Q]}$$ where now $C_{\epsilon,[K : \Bbb Q]}$ depends on $[K:\Bbb Q]$ and $\epsilon$.

Your examples might shed some light on whether or not we really need the constant to depend on $[K : \Bbb Q]$.

Answer 2

A simple example: Choose arbitrarily $a,c\in \mathbb{Z}$ with $c\neq 0$. Let $K/\mathbb{Q}$ be a number field such that $b=\sqrt{a^3-c}\in \mathcal{O}_K$ and the elliptic curve $E$ defined by $y^2=x^3-3ax-2b$ has semistable reduction over $\mathcal{O}_K$. $E$ has the $j$-invariant $j(E)=a^3/c$. Then a computation as by Silverman/Pellarin gives for the stable Faltings height $$\log|a|\le \tfrac{1}{3}\log|c|+4h_F(E)+2\log\max\lbrace 1,h_F(E)\rbrace+7.$$ On the other hand side we have $$\tfrac{1}{[K:\mathbb{Q}]}\log|N_{K/\mathbb{Q}}\Delta_{E/K}|\le \log|2^4\cdot \mathrm{Disc}(x^3-3ax-2b)|=\log|c|+\log 3^3\cdot 2^6.$$ Since $K/\mathbb{Q}$ can be chosen with bounded degree, we obtain for $c=1$ and $a$ arbitrary large an Elliptic curve of arbitrary large Faltings height $h_F(E)$ and bounded discriminant $\log|N_{K/\mathbb{Q}}\Delta_{E/K}|$. That shows, that in Szpiro's conjecture $$h_F(E)\le (\tfrac{1}{2}+\epsilon)\log|N_{K/\mathbb{Q}}\mathcal{F}_{E/K}|+C_{\epsilon,K},$$ the constant $C_{\epsilon,K}$ should really depend on $K$ and not only on $[K:\mathbb{Q}]$.