Major issue: make explicit the assumption k > h

[defeo]

p. 6 Prop 2.5: As stated this proposition places no constraints on k, but in what follows (e.g. p. 8 line 3) the authors are often assuming that k ≥ h. When k < h we have ℓ^h P = 0 and the Proposition is trivial (and this is also true if P has order less than ℓ^h). Not making this assumption explicit made the paper very confusing to read, as in several places they write expressions that may well be non-integral without this assumption in contexts where an integer is required (for example, ℤ_ℓ does not contain ℓ^(k-h) for k < h). In particular, as noted below, it is not obvious to me why, for example, line 9 of Algorithm 1 and line 7 of Algorithm 2 are well defined. On a related topic, I think there is an error in the proof of Proposition 2.5. The authors write the matrix of π | T_ℓ (E 0 ) ∈ GL₂ (ℤ_ℓ) as

  / λ    ℓ^(h-k) x(λ-μ) \
  \ 0           μ       /

where h = v_ℓ (λ - μ), and x is an integer. The entry in the upper right has ℓ-adic valuation v_ℓ (ℓ^(h-k) x(λ − μ)) = h - k + v_ℓ(x) + h, which will be negative if we choose k large and take P = R so that x = 1. I think they want to replace k - h with h - k in both the matrix and the condition on v_ℓ(x) ≥ k - h given later in the proof.

[plut]

• Alg. 1: fixed by commit 1c9eac7 ($$h$$ is mutable in the algorithm).
• Alg. 2: fixed by same commit (added missing hypothesis $$h \geqslant k+1$$).
• Proof of 2.5: fixed by commit 772ea7b (the hypotheses of the proof did not ensure that $$\varphi$$ was rational; note: the reviewer-suggested fix was wrong).
• Also fixed an error that went un-noticed in the statement of 2.5, in commit 8ef4429.

[defeo]

Why aren't we making the explicit assumption k>h in Algorithm 1 as requested by the reviewer? Is it because the height h is not known in advance?

If so, this would slightly contradict the intro to Sec. 3, which assumes we have already constructed the Kummer tower up to the right level. Maybe we should clarify this.

[defeo]

Also, the fix introduced this typo in the statement of Prop. 2.5:

ℓ^h P is an eigenvector of E → ℓ^h is an eigenvector of π

[plut]

@defeo Line 10 of Algorithm 1 computes h (or at least, h' = min (k, h), as stated in the algorithm output)

[defeo]

Isn't "previous discussion" better than "preceding discussion"? (frankly, I don't really know)

[defeo]

More typos:

We note here that the matrices → We note here that the matrix

[defeo]

And yet one more typo introduced in the proof of Prop. 2.7 (I think this is the last one):

we just note that since there exists a unique subgroup of order ℓ which either ascending or horizontal with a given direction

Maybe add "is" after "which"?