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Related to this question.

According to Hindry p.7 Conj 3.1 and Stein's notes Szpiro's conjecture over number fields 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 and $f_{E/K}$ is the conductor.

To my knowledge over the rationals the largest known Szpiro ratio is about $9.01$.

Consider the elliptic curve with cremona label 37b1 and ainvariants $[0, 1, 1, -23, -50]$. Over the rationals the discriminant is $37^3$. Over the number field with defining polynomial $x^{22} - 37^3$ the norm of the minimal discriminant is $37^{66}$ and the norm of the conductor is $37$ while the global minimal model is the same as over the rationals, giving Szpiro ratio $66$. If this pattern continue for number fields of the form $x^n - 37^3$, the Szpiro ratio probably will be unbounded.

Computation with sage, possibly assuming GRH since the number fields are not certified suggest the following large Szpiro ratios {33,39,42,48,51,66}:

a_i= [0, 1, 1, -23, -50]  ratio= 33.00000 Delta= 37^33 f= 37 d_E/Q= 37^3  global minimal model= Elliptic Curve defined by y^2 + y = x^3 + x^2 + (-23)*x + (-50) over Number Field in w with defining polynomial x^11 - 50653 
a_i= [0, 1, 1, -23, -50]  ratio= 39.00000 Delta= 37^39 f= 37 d_E/Q= 37^3  global minimal model= Elliptic Curve defined by y^2 + y = x^3 + x^2 + (-23)*x + (-50) over Number Field in w with defining polynomial x^13 - 50653
a_i= [0, 1, 1, -23, -50]  ratio= 42.00000 Delta= 37^42 f= 37 d_E/Q= 37^3  global minimal model= Elliptic Curve defined by y^2 + y = x^3 + x^2 + (-23)*x + (-50) over Number Field in w with defining polynomial x^14 - 50653
a_i= [0, 1, 1, -23, -50]  ratio= 48.00000 Delta= 37^48 f= 37 d_E/Q= 37^3  global minimal model= Elliptic Curve defined by y^2 + y = x^3 + x^2 + (-23)*x + (-50) over Number Field in w with defining polynomial x^16 - 50653
a_i= [0, 1, 1, -23, -50]  ratio= 51.00000 Delta= 37^51 f= 37 d_E/Q= 37^3  global minimal model= Elliptic Curve defined by y^2 + y = x^3 + x^2 + (-23)*x + (-50) over Number Field in w with defining polynomial x^17 - 50653
a_i= [0, 1, 1, -23, -50]  ratio= 66.00000 Delta= 37^66 f= 37 d_E/Q= 37^3  global minimal model= Elliptic Curve defined by y^2 + y = x^3 + x^2 + (-23)*x + (-50) over Number Field in w with defining polynomial x^22 - 50653

Q1 Are the computations correct?

Q2 Does increasing the degree of the number field continue to increase the ratio?

Q3 (New) Is this a counterexample this formulation of Szpiro's conjecture? My concern is bounded, not infinitely many?

Working will large degree fields is not fast for me.


Small search found 46 ratios greater than $24$.

The largest is $134.6199$ from ainvariants $[1, 1, 1, -110, -880]$

 a_i= (1, 1, 1, -110, -880)  ratio= 134.6199 Delta= -1 * 3^304 * 5^19 f= 3 * 5 d_E/Q= -1 * 3^16 * 5  global minimal model= Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 + (-110)*x + (-880) over Number Field in w with defining polynomial x^19 + 215233605

Added Addressing Jamie Weigandt answer.

With his code and possibly assuming GRH, I get the same ratio.

To assume GRH, replace K.<a> = NumberField(x^11 - 50653) with K.<a> = NumberField(x^11 - 50653,check=False). This avoids certification of the NF with pari's bnfcertify.

Here is the output of

Nf.<w>=NumberField(x**D-d)
Nf.<w>=NumberField(x**D-d,check=False) #assume GRH
print 'a_i=',ai,' ratio=',szpiro_ratio(E2),Nf

a_i= [0, 1, 1, -23, -50]  ratio= 30.0 Number Field in w with defining polynomial x^10 - 50653
a_i= [0, 1, 1, -23, -50]  ratio= 33.0 Number Field in w with defining polynomial x^11 - 50653
a_i= [0, 1, 1, -23, -50]  ratio= 39.0 Number Field in w with defining polynomial x^13 - 50653
a_i= [0, 1, 1, -23, -50]  ratio= 42.0 Number Field in w with defining polynomial x^14 - 50653
a_i= [0, 1, 1, -23, -50]  ratio= 48.0 Number Field in w with defining polynomial x^16 - 50653
a_i= [0, 1, 1, -23, -50]  ratio= 51.0 Number Field in w with defining polynomial x^17 - 50653
a_i= [0, 1, 1, -23, -50]  ratio= 57.0 Number Field in w with defining polynomial x^19 - 50653
a_i= [0, 1, 1, -23, -50]  ratio= 60.0 Number Field in w with defining polynomial x^20 - 50653
a_i= [0, 1, 1, -23, -50]  ratio= 66.0 Number Field in w with defining polynomial x^22 - 50653
a_i= [0, 1, 1, -23, -50]  ratio= 69.0 Number Field in w with defining polynomial x^23 - 50653
a_i= [0, 1, 1, -23, -50]  ratio= 75.0 Number Field in w with defining polynomial x^25 - 50653
a_i= [0, 1, 1, -23, -50]  ratio= 78.0 Number Field in w with defining polynomial x^26 - 50653
a_i= [0, 1, 1, -23, -50]  ratio= 84.0 Number Field in w with defining polynomial x^28 - 50653

According to pari's documentation, the comment: #Don't do this. It will take forever probably will mean GRH fails for this NF, since in this case pari's bnfcertify doesn't return.

Regarding his comment about Q2 and 32a1: 32a1 has $a_6=0$, so the ratio degrades for trivial reasons. I didn't claim high ratio for it.

The claim that the ratio is bounded over number fields is from here and William Stein' notes

Similar definition of the Szpiro's conjecture is in Silverman's The Arithmetic of Elliptic Curves, p. 275.

With the above notation, there is constant $k$, depending on $\epsilon$ and $K$ s.t.:

$$ N_{K/Q} \Delta_{E/K} \le k(N_{K/Q} f_{E/K})^{6+ \epsilon}\qquad (2) $$

Silverman asks "It is very interesting to ask how the constants $k$ appearing in these conjectures depend on the field $K$."

In analogy with the uniform abc conjecture, probably Silverman should take $k$ the discriminant of $K$.

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The correct discriminant ideal to compute is a locally defined ideal $$\mathcal D_{E/K}^\text{min} = \prod_{\frak {p}} {\frak{p}}^{v_\frak{p}}$$ where $v_{\frak p}$ is the valuation of the discriminant of a local minimal model of $E$ at $p$. For general number fields, global minimal models do not exist. They exist if and only if the class number is 1.

This may or may not bring down your Szpiro ratios. If it doesn't, I still don't think this is inconsistent with Szpiro's conjecture since the finite set of counterexamples depends both on $\epsilon$ and the number field $K$. For every number field $K$ there is a finite (and sometimes non-empty) set of elliptic curves over $K$ with everywhere good reduction. These curves have conductor ideal $(1)$ so the Szpiro ratio can arguably be taken to be $\infty$.

I can't say whether your discriminant calculations are correct because I'm not sure how exactly you computed them, and working in these number fields is a bit too hard for me to do the calculations myself today. Here's code to compute the honest Szpiro ratio:

def szpiro_ratio(E):
    #Only works over number fields.
    N = E.conductor()
    nN = N.norm()
    S = N.prime_factors() 
    local_norms =[p.norm()^E.local_minimal_model(p).discriminant().ord(p) for p in S]
    nD = prod(local_norms)
    return log(RDF(abs(nD)))/log(RDF(abs(nN)))

To do these calculations, I would try:

sage: E = EllipticCurve('37b1')
sage: K.<a> = NumberField(x^11 - 50653)
sage: EK = E.base_extend(K)
sage: szpiro_ratio(EK) #Don't do this. It will take forever.

Then I would give up after 10 minutes because it would be clear that this is not a feasible approach for such high degree number fields. There might be a way around this difficulty with high degree number fields. I'd look at the code for local_minimal_model?? and try doing it by hand for primes in $K$ above $37$ to understand what's going it.

Here's an example where the code does work, and gives a negative answer to Q2.

sage: E = EllipticCurve('32a1').quadratic_twist(97)
sage: K.<a> = QuadraticField(97)
sage: L.<b> = NumberField(x^4 - 97)
sage: szpiro_ratio(E.base_extend(K))
2.4
sage: szpiro_ratio(E.base_extend(L))
2.25

EDIT: (Fixed some inaccuracies, and added a more reasonable explanation for the $2-2g$ term.

Look at the Shioda-Tate formula: $$ N = \frac{D}{6} + 2 - 2g + r + (h^{1,1} - \rho).$$ This is the elliptic surface analog of Szpiro's conjecture. Here $N$ is the conductor degree, $D$ is the discriminant degree, $r$ is the Mordell-Weil rank, $(h^{1,1} - \rho)$ is a non-negative number we don't need to worry about. We do have to worry about $2-2g$ where $g$ is the genus of the base curve.

There are two things this could be reflecting in the number field case.

(1) The negative $2g$ could be accounting for the difference between the minimal discriminant ideal and the discriminant ideal attached to a global model, since it measures the size of the Picard group of the base.

(2) (2-2g) could be correspond to something like $(1 + \epsilon)\log|\text{Disc}_{K/\Bbb Q}|$. Maybe both are related to the degree of some canonical divisor. I suppose this would be an Arakelov divisor in the number field case.

I think the second interpretation seems more reasonable, and agrees with what you're trying to say.

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    $\begingroup$ I fixed the code to deal with 37b1, not 37a1. My apologies to 37b, the most under-appreciated isogeny class of elliptic curves. $\endgroup$ – Jamie Weigandt Sep 20 '14 at 22:07
  • $\begingroup$ I get the same ratios with your code and possibly assuming GRH. Edited the question. Your code is fast if one doesn't certify the NF (assume GRH). $\endgroup$ – joro Sep 21 '14 at 7:17
  • $\begingroup$ Do you think this leads to counterexample of Stein's formulation of Szpiro conjecture? $\endgroup$ – joro Sep 21 '14 at 10:34
  • $\begingroup$ About unit conductor: won't in this case the discriminant be unit too leading to $\frac00$ which might not be $\infty$? $\endgroup$ – joro Sep 21 '14 at 10:57
  • $\begingroup$ Regarding 0/0: That's why I said "arguably". Regarding the reference you linked to: These are notes William Stein live-TeXed at a talk given by Matt Baker. It's unfair to William to call this his formulation of Szpiro's conjecture. Even then, Conjecture 3.1 is for a specific field $K$, while Theorem 3.2 is for all number fields. Everything seems to be written for a specific field $K$ here too: webusers.imj-prg.fr/~marc.hindry/MarcJoe.pdf $\endgroup$ – Jamie Weigandt Sep 21 '14 at 19:29

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