Heights and isogenies of Drinfeld modules

Fabien Pazuki; Florian Breuer; Mahefason Heriniaina Razafinjatovo

REVIEW 2 major objections 4 minor 14 references

Heights and isogenies of Drinfeld modules

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves explicit bounds on how much an isogeny can change the height of a Drinfeld module, and applies them to finiteness and modular polynomials.

desk verdict Main explicit height bounds are new and plausible, but the proof of Corollary 3.3 contains a false inequality and Lemma 5.3 leans on unstated Gekeler estimates, so this deserves a careful revision rather than rejection. read the letter →

arxiv 1908.03485 v2 pith:OEA2KZFZ submitted 2019-08-09 math.NT

Florian Breuer , Fabien Pazuki , Mahefason Heriniaina Razafinjatovo This is my paper · ORCID

classification math.NT MSC 11G0911G5014G1714G40

keywords DrinfeldmodulesgradedheightisogeniesTaguchimodularpolynomialsfunctionfieldsBruhat-Titsbuildingfinitenessofisogenyclasses

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves a function-field analogue of the classical elliptic-curve statement that an isogeny changes the Faltings height by at most a small multiple of the logarithm of the degree. For Drinfeld $\mathbb{F}_q[t]$-modules of rank $r$ in generic characteristic, it shows that if $f:\varphi\to\varphi'$ is an isogeny with $\ker f \subset \varphi[N]$, then $|h_G(\varphi')-h_G(\varphi)|\leq \deg N+\left(\frac{q}{q-1}-\frac{q^r}{q^r-1}\right)$, where $h_G$ is the graded height computed from the coefficients of the module. The additive constant depends only on $q$ and $r$, so the height difference is controlled by the size of the kernel up to a rank-uniform constant. This comparison is then used to give an effective finiteness theorem for isogeny classes and, in rank two, an explicit upper bound on the coefficients of Drinfeld modular polynomials.

What carries the argument

The argument decomposes the height difference into a finite-place part governed by the isogeny lemma for the differential height and an infinite-place part that is handled by reducing each Drinfeld module to a reduced one and estimating the coefficient sizes of reduced modules. The load-bearing object is the fundamental domain for the moduli space and its associated Bruhat-Tits building, the simplicial complex encoding the relative sizes of the basis vectors of a lattice. The paper's Lemma 5.3 uses the building structure to show that $\log \max_i |g_i|^{1/(q^i-1)}$ lies between $q^r/(q^r-1)$ and $q/(q-1)$ for every reduced rank-$r$ module, and that two-sided bound converts the infinite-place term into the explicit additive constant in Theorem 3.1.

What would settle it

Find a reduced Drinfeld module of rank $r$ over $\mathbb{C}_\infty$ for which $\log \max_i |g_i|^{1/(q^i-1)}$ exceeds $\frac{q}{q-1}$ or falls below $\frac{q^r}{q^r-1}$; either example would refute Lemma 5.3 and with it the explicit constant in Theorem 3.1.

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Extended reading notes

Core claim

The paper's central claim is that the graded height $h_G$, the logarithmic size of the coefficient tuple viewed in weighted projective space, changes very little under isogeny. For an isogeny $f:\varphi\to\varphi'$ of rank $r$ modules with $\ker f \subset \varphi[N]$, Theorem 3.1 gives $$|h_G(\varphi')-h_G(\varphi)|\leq \deg N+\left(\frac{q}{q-1}-\frac{q^r}{q^r-1}\right),$$ where $N$ is the element of $\mathbb{F}_q[t]$ of minimal degree whose $N$-torsion contains the kernel. In rank two this yields the refined $j$-invariant inequality $$h(j')-h(j)\leq \frac{$q^{2}$-1}{2}\log\deg f+\frac{$q^{2}$-1}{2}\log\left(1+\frac{h(j')}{q}\right)+q,$$ with the same shape as the elliptic-curve comparison. From these bounds the paper derives an effective finiteness theorem for isogeny classes and an explicit upper bound on the coefficient height of Drinfeld modular polynomials.

Load-bearing premise

The proof assumes that a cited estimate pinning the coefficient sizes of reduced Drinfeld modules between two explicit bounds holds without extra hypotheses; if that estimate is wrong or incomplete, the additive constant in Theorem 3.1 does not follow.

Editorial extensions

If this is right

For a Drinfeld module defined over a finite extension of $\mathbb{F}_q(t)$, the height bound combines with the standard finiteness property of heights to give an effective version of the finiteness theorem: each isogeny class contains finitely many isomorphism classes, with a computable bound on their number.
For rank 2, the inequality for the $j$-invariant yields an explicit upper bound on the height $h(\Phi_m)$ of the coefficients of the modular polynomial $\Phi_m(X,Y)$; the paper records the asymptotic form $h(\Phi_m)<\left(\frac{q^2+4}{2}+\varepsilon\right)\psi(m)\deg(m)$, close to the known exact asymptotics for these polynomials.
The proof gives a template for controlling the difference between any two height functions that agree at finite places and differ only through infinite-place symmetries, whenever the moduli space has a fundamental domain with explicit bounds on the coordinate functions.
The theorem's rank-2 statement has the same logarithmic shape as the classical elliptic-curve comparison, indicating that the function-field analogue holds with a fully explicit constant rather than an asymptotic one.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Although the paper does not discuss optimality, if the coefficient bound in Lemma 5.3 is tight then the additive constant in Theorem 3.1 cannot be improved for generic kernels.
The same decomposition may apply to isogenies between Drinfeld modules of different ranks, or to heights of higher-rank modular polynomials, since the only rank-specific input is the two-sided coefficient bound for reduced modules.
A direct computation of reduced rank-3 modules with lattices at different vertices of the fundamental domain could test whether the extremes in Lemma 5.3 are actually attained, which would tell whether the constant in Theorem 3.1 is sharp.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

Here's my read on Breuer, Pazuki, and Razafinjatovo's paper. The real contribution is the pair of explicit inequalities in Theorem 3.1: the graded-height difference under an isogeny is bounded by deg N plus an explicit constant depending only on rank and field size, and the rank-2 variant in terms of j-invariants. These are genuinely new, quantitative analogues of Pazuki's elliptic curve result. The strategy is transparent: decompose the height difference into a Taguchi-height term, an analytic term, and a lattice term, and bound each piece with existing machinery (Taguchi's isogeny lemma, Gekeler's building estimates, and a new covolume lemma). The modular polynomial coefficient bound in Proposition 6.5 is also new and follows cleanly from the rank-2 inequality.

There is a real flaw in the proof of Corollary 3.3. The authors claim 'it is easy to see' that h(φ) ≤ (q^r−1)h_G(φ), where h(φ)=max_i h(g_i). This is false. For example, over F_2(t), take g1=t^{-N}, g2=t^{-3N} with r=2. Then h_G=0 (the local contributions cancel) while h(g2)=3N, so the inequality fails by an arbitrarily large factor. The proof of Corollary 3.3 collapses at that line. The corollary itself may be salvageable, since the finiteness result is already due to Taguchi, but the quantitative bound as stated needs a different argument.

The other soft spot is Lemma 5.3, the linchpin for the explicit constant in Theorem 3.1(1). The proof imports several claims from Gekeler's Corollaries 4.11 and 4.16: monotonicity of log|g_i| along the building, boundary values on strata, and equality at the origin. The paper doesn't quote the actual statements or address boundary cases. This may be fine, but a referee should ask the authors to state the precise Gekeler results or add a verification for small ranks. The constant depends entirely on this lemma.

The interpolation lemma in Section 6 has minor notational slips, but they don't affect the argument. The self-citation to [Paz19] is appropriate as a template rather than a premise.

Overall, the central theorems look sound and the quantitative bounds are of real interest to people working on Drinfeld modules. The Corollary 3.3 flaw is a genuine gap, but it's a secondary corollary and likely fixable. I'd send this to a serious referee, with instructions to ask for a corrected proof or a softened statement for Corollary 3.3, and to clarify the Gekeler input in Lemma 5.3. Not a desk reject.

Referee Report

2 major / 4 minor

Summary. The paper proves explicit bounds on the change of the graded height h_G of a Drinfeld module under an isogeny. The main result, Theorem 3.1(1), states that for an isogeny f: φ → φ' of rank r Drinfeld modules with ker f ⊂ φ[N], one has |h_G(φ') − h_G(φ)| ≤ deg N + (q/(q−1) − q^r/(q^r−1)). The proof combines Taguchi's isogeny lemma, the equality of the finite part of h_G with Taguchi's height, and analytic estimates on reduced lattices taken from Gekeler's work on Drinfeld modular forms and Bruhat–Tits buildings. In the rank 2 case the authors also prove a one-sided inequality involving the usual j-invariant and log deg f. These results are applied to obtain an effective finiteness statement for isogeny classes and, via an interpolation argument, an explicit upper bound on the height of Drinfeld modular polynomials Φ_m, complementing Hsia's asymptotic.

Significance. If the main theorem holds, it provides the first fully explicit function-field analogue of Pazuki's elliptic-curve height comparison, with the constant in Theorem 3.1(1) depending only on q and r. The paper is well structured: the heavy inputs (Taguchi's isogeny lemma, Gekeler's building estimates, David–Denis's effective isogeny theorem, Hsia's asymptotic) are clearly cited, and the authors are transparent about the analytic estimates that drive the explicit constant. The derivation of the modular polynomial bound from the height inequality is elegant and gives a concrete statement complementing Hsia's asymptotic. However, two points need attention: one inequality in the proof of Corollary 3.3 is false as stated, and Lemma 5.3 relies on building-theoretic statements that are not quoted precisely enough for the reader to verify the constant. These are load-bearing for the finiteness application and for the explicit constant, respectively.

major comments (2)

[§3, proof of Corollary 3.3] The proof uses the inequality h(φ) ≤ (q^r − 1)h_G(φ), where h(φ) = max_i h(g_i) is the maximum of the Weil heights of the coefficients. This inequality is false for arbitrary representatives. For example, take q = 2, r = 2, and φ_t(X) = tX + t^N X^2 + t^{3N} X^4 with N ≥ 1. Then h(φ) = 3N, while j_1 = g_1^3/g_2 = 1 and j_2 = 1, so h_G(φ) = 0. More generally, h_G is invariant under isomorphism, whereas max_i h(g_i) is not, so no such inequality can hold without fixing a normalized representative. Since the proof then uses [DD99, Thm. 1.3] to bound deg f in terms of h(φ) and converts the result into a bound in terms of h_G(φ), the derivation of Corollary 3.3 does not work as written. This also affects the new proof of Corollary 3.4. Please repair this step, for example by applying [DD99] to a height that is genuinely comparable to h_G, or by proving the needed comparison after choosing a suitable representative.
[§5.1, Lemma 5.3] The explicit constant in Theorem 3.1(1) rests entirely on Lemma 5.3, whose proof imports from [Gek17, Cor. 4.11 and 4.16] the assertions that each log|g_i(ω)| is non-increasing as λ(ω) moves away from the origin and that equations (21)–(22) hold on the stated strata. The paper does not quote the precise statements of those corollaries, nor does it address the boundary cases (for instance, fibres over boundary points of W(Q) or possible zeros of g_i). Since (19) enters both the bound on part (B) and the constant in Theorem 3.1, and since Theorem 3.1(2) and Proposition 6.5 inherit this constant, the proof would be substantially easier to certify if the exact statements from [Gek17] were reproduced or if a direct verification of the monotonicity and fibre-constancy claims were supplied. I am not asserting that the cited estimates are wrong, but as written Lemma 5.3 is a verification gap in the central argument.

minor comments (4)

[§6, Lemma 6.3] In the proof of Lemma 6.3, the sentence 'by Lemma 6.2 we have h(T_k) ≤ q^{nd}' should read h(T_k) ≤ nd (or h(T_k) = log max |a_j| ≤ nd). The subsequent '+2nd' in (24) only makes sense with the logarithmic bound.
[§5.1, equation (17)] The notation n_σ in (17) is used for the local degree of the embedding σ; it is not defined in the text. Please define n_σ explicitly when the sum over embeddings is introduced.
[§5.1, Lemma 5.3] Please clarify the normalization of the functions g_i(ω) in the building-coordinate argument: are these the coefficients of the Drinfeld module associated to the lattice generated by ω, and are they evaluated after fixing the basis (ω_1,...,ω_r) with ω_r = 1? The notation log|g_i(ω)| is otherwise ambiguous, since the coefficients depend on the lattice, not on the chosen basis.
[§6, Proposition 6.5] The displayed bound for h(Φ_m) is hard to parse because of the long bracketed expression. Adding a line break or a named auxiliary quantity (for example writing the max argument as B) would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 3.1 is proved from Taguchi's isogeny lemma, Gekeler's analytic building estimates, and an elementary lattice-covolume lemma, none of which encode the target inequality.

full rationale

The derivation of Theorem 3.1(1) is not circular. Equation (16) decomposes h_G(φ') − h_G(φ) into three parts: (A) h_Tag(φ') − h_Tag(φ), bounded by Taguchi's Isogeny Lemma (Lemma 4.4), quoted from [Tag93, Lemma 5.5] and not using Theorem 3.1; (B) the difference of infinite graded heights, bounded absolutely in Lemma 5.3 using Gekeler's building-theoretic estimates [Gek17, §4.6, Prop. 4.10, Cor. 4.11, Cor. 4.16]; and (C) the difference of Taguchi infinite-height contributions, bounded by Lemma 4.2, an elementary statement about covolumes of reduced lattices. The constant q/(q−1) − q^r/(q^r−1) is not fitted from the target quantity: it is obtained from the range of log|g_i|^{1/(q^i−1)} for reduced Drinfeld modules, as shown in Lemma 5.3. The proof does not use the desired height inequality as an assumption. The only self-citation is [Paz19], which supplies the elliptic-curve analogue and an organizational template; it is not invoked as a premise for the function-field result. The sceptical concern that some estimates imported from [Gek17] may require further verification is a potential correctness issue about an external cited result, not a circularity issue: the paper does not redefine its input to produce its output. Likewise, the possible issue with the inequality h(φ) ≤ (q^r−1)h_G(φ) in Corollary 3.3 is a substantive mathematical concern, not a circularity. Under the stated rules, an honest non-finding is appropriate, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The paper introduces no new free parameters or invented entities. The central bounds depend on a list of imported results from Gekeler, Taguchi, David-Denis, Potemine, and Hsia, all clearly cited and none of which are being proved in this paper.

assumptions (8)

domain assumption Gekeler's fundamental domain estimates ([Gek17, Cor. 4.11, 4.16; §4.6])
Lemma 5.3 depends on monotonicity and boundary values of log|g_i| on the building; the constant in Theorem 3.1 is imported from these results.
domain assumption Taguchi's Isogeny Lemma ([Tag93, Lemma 5.5])
Lemma 4.4 uses it to bound the difference of Taguchi heights under isogeny.
domain assumption Everywhere stable reduction after finite extension ([DD99, Lemme 2.10])
Section 5.1 invokes it to reduce to stable reduction before applying Taguchi's lemma.
domain assumption David-Denis effective isogeny bound ([DD99, Thm 1.3])
Corollary 3.3 uses it to bound the degree of an isogeny between two modules in the same isogeny class.
domain assumption Gekeler's estimate on |j(φ)| for reduced rank-2 lattices ([Gek97, Thm 2.17])
Lemma 5.4 uses it to bound D(Λ) in terms of h(j).
domain assumption Potemine's finiteness for J-invariants ([Pot98, Thm 2.2])
Proposition 2.2 uses it to pass from finitely many J-values to finitely many isomorphism classes.
domain assumption Hsia's asymptotic for modular polynomial height ([Hsi98])
Used as comparison for the new explicit bound in Section 6; not load-bearing.
standard math Product formula and Weil height Northcott theorem
Used throughout the definitions of heights and in Proposition 2.2.

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Cite this review

Pith. "Pith review of Heights and isogenies of Drinfeld modules." pith.science (2026). https://pith.science/paper/OEA2KZFZ

@misc{pith2026190803485,
  author       = {Pith},
  title        = {Pith review of: Heights and isogenies of Drinfeld modules},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OEA2KZFZ}},
  note         = {Machine review of arXiv:1908.03485}
}

read the original abstract

We provide explicit bounds on the difference of heights of isogenous Drinfeld modules. We derive a finiteness result in isogeny classes. In the rank 2 case, we also obtain an explicit upper bound on the size of the coefficients of modular polynomials attached to Drinfeld modules.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

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[2]

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and Denis, L

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[9]

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[13]

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[14]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Bae , On the modular equation for Drinfeld modules of rank 2

S. Bae , On the modular equation for Drinfeld modules of rank 2. J. Number Theory 42 (1992), 123--133

work page 1992

[2] [2]

Bae and S

S. Bae and S. Lee , On the Coefficients of the Drinfeld Modular Equation. J. Number Theory 66 (1997), 85--101

work page 1997

[3] [3]

and Denis, L

David, S. and Denis, L. , Isog\'enie minimale entre modules de Drinfel'd. Math. Ann. 315 (1999), 97--40

work page 1999

[4] [4]

atze f\"ur abelsche V ariet\

Faltings, G. , Endlichkeitss\"atze f\"ur abelsche V ariet\"aten \"uber Z ahlk\"orpern . Invent. Math. 73 (1983), 349--366

work page 1983

[5] [5]

, On the Drinfeld discriminant function

Gekeler, E.-U. , On the Drinfeld discriminant function. Compos. Math. 106 (1997), 181--202

work page 1997

[6] [6]

, On Drinfeld modular forms of higher rank

Gekeler, E.-U. , On Drinfeld modular forms of higher rank. J. Th\'eorie de Nombres Bordeaux 29 , No. 3 (2017), pp. 875--902

work page 2017

[7] [7]

, Towers of GL(r) -type of modular curves

Gekeler, E.-U. , Towers of GL(r) -type of modular curves. J. reine Angew. Math. 754 (2019), 87--141

work page 2019

[8] [8]

, On the coefficients of modular polynomials for Drinfeld modules

Hsia, L.-C. , On the coefficients of modular polynomials for Drinfeld modules. J. Number Theory 72 (1998), 236--256

work page 1998