Height moduli of elliptic surfaces: Motivic height zeta rationality and Kudla-Millson modularity of Mordell-Weil rank jumps

Jun-Yong Park

arxiv: 2601.15543 · v5 · submitted 2026-01-21 · 🧮 math.AG · math.NT

Height moduli of elliptic surfaces: Motivic height zeta rationality and Kudla-Millson modularity of Mordell-Weil rank jumps

Jun-Yong Park This is my paper

Pith reviewed 2026-05-16 11:35 UTC · model grok-4.3

classification 🧮 math.AG math.NT

keywords elliptic surfacesMordell-Weil rankFaltings heightmotivic zeta functionsKudla-Millson correspondencemodular formsheight moduli stack

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The pith

At every Faltings height n at least 3, infinitely many stable elliptic surfaces over k(t) achieve Mordell-Weil rank at least floor of (10n minus 2) over (n minus 1).

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

The paper builds the moduli stack of minimal elliptic curves of fixed Faltings height n over the function field K equals k(t). It proves that the motivic height zeta function, weighted by the rank of the trivial lattice coming from Kodaira fibers, is a rational function in the variable s equals t to the 1/12 inside the dimensionally completed Grothendieck ring. Over the complex numbers this implies stabilization of Hodge numbers. Applying the Kudla-Millson theta correspondence to the associated weight 6n minus 2 modular forms then shows that new Mordell-Weil sections appear with infinitely many distinct canonical heights and that the rank reaches every integer r up to the explicit bound floor of (10n minus 2) over (n minus 1). A reader would care because these statements give uniform lower bounds on ranks inside an arithmetic family whose geometry is controlled by modular forms.

Core claim

At every Faltings height n greater than or equal to 3 and for every integer r between 1 and floor of (10n minus 2) over (n minus 1), there exist infinitely many stable elliptic surfaces with Mordell-Weil rank at least r; moreover, infinitely many canonical heights equal to d are realized by Mordell-Weil sections. The argument proceeds by establishing rationality of the motivic height zeta function via Euler products on the isotrivial loci and motivic discriminant stabilization, followed by the Kudla-Millson correspondence that governs the distribution of new sections by canonical height.

What carries the argument

The motivic height zeta function of the height-moduli stack of minimal elliptic curves of Faltings height n, weighted by trivial lattice rank, together with the Kudla-Millson theta correspondence producing weight 6n minus 2 modular forms.

If this is right

The Picard rank of the associated elliptic surface equals the trivial lattice rank plus the Mordell-Weil rank, so the total rank can be made arbitrarily large within each fixed-height slice by choosing sufficiently large r.
The motivic height zeta function is rational in s equals t to the 1/12, and over the complex numbers the Hodge numbers of the moduli stack stabilize in each bidegree.
Infinitely many distinct canonical heights are attained by sections, so the image of the canonical height map on the Mordell-Weil group is infinite for infinitely many such surfaces.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same rationality and modularity statements may extend to other base fields or to non-minimal models once the relevant theta series are shown to be nonzero.
The explicit rank bound floor of (10n minus 2) over (n minus 1) suggests that average Mordell-Weil rank over the moduli stack grows at least linearly in n for small n.
One could test whether the canonical heights realized by sections are dense in the positive reals or satisfy further arithmetic constraints coming from the modular form.

Load-bearing premise

The Kudla-Millson theta correspondence governs the distribution of new Mordell-Weil sections by canonical height for the specific weight 6n minus 2 forms arising from the height moduli stack.

What would settle it

An explicit computation for some n at least 3 showing that only finitely many stable elliptic surfaces of that height have Mordell-Weil rank greater than or equal to r for an r inside the stated range, or that only finitely many distinct canonical heights appear among their sections.

read the original abstract

Let $k$ be a perfect field with $\mathrm{char}(k)\neq 2,3$, set $K=k(t)$, and let $\mathcal{W}_n^{\min}$ be the moduli stack of minimal elliptic curves over $K$ of Faltings height $n$, constructed via the height-moduli framework of Bejleri-Park-Satriano applied to $\overline{\mathcal{M}}_{1,1}\simeq\mathcal{P}(4,6)$. The Shioda-Tate formula $\rho(S)=T(S)+\mathrm{rk}(E/K)$ decomposes the Picard rank of the associated elliptic surface into the trivial lattice rank, which is local (determined by Kodaira fiber types), and the Mordell-Weil rank, which is global. The motivic height zeta function weighted by the trivial lattice rank is rational in $s=t^{1/12}$ in the dimensionally completed Grothendieck ring, via a combination of exact Euler products on the isotrivial loci $j\equiv 0, 1728$ and a motivic discriminant stabilization adapting Vakil-Wood to $\Delta=4a_4^3+27a_6^2$; over $k=\mathbb{C}$, this yields bidegree-wise Hodge number stabilization. The Kudla-Millson theta correspondence shows that the distribution of new Mordell-Weil sections by canonical height is governed by a modular form of weight $6n-2$ for $\mathrm{SL}_2(\mathbb{Z})$. Combining Shepherd-Barron's diagonalization of the Gauss-Manin connection with Kodaira-Spencer transversality, we establish unconditionally that at every Faltings height $n\ge 3$ and for every $1 \le r \le \lfloor(10n-2)/(n-1)\rfloor$, there exist infinitely many stable elliptic surfaces with Mordell-Weil rank $\mathrm{rk}(E/K) \ge r$, and that infinitely many canonical heights $\hat{h}(P)=d$ are realized by Mordell-Weil sections.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper claims unconditional linear growth in Mordell-Weil ranks on elliptic surfaces over function fields by combining motivic height zeta rationality with Kudla-Millson, but the non-vanishing step for the weight 6n-2 form on the stable locus is not checked in the given material.

read the letter

The core advance is an explicit bound: for each n ≥ 3 and every r up to floor((10n-2)/(n-1)), there are infinitely many stable elliptic surfaces over k(t) with Mordell-Weil rank at least r, plus realization of every canonical height d. This comes from showing the motivic height zeta function (weighted by trivial lattice rank) is rational in the dimensionally completed Grothendieck ring, via Euler products on the j=0 and j=1728 loci plus a Vakil-Wood-style stabilization on the discriminant, then feeding the resulting Hodge stabilization into Kudla-Millson to control the distribution of new sections by canonical height. The weight 6n-2 modular form for SL2(Z) is supposed to govern the jumps after Shepherd-Barron diagonalization and Kodaira-Spencer transversality. That package is new and the bound is concrete, which is useful for rank-distribution questions over function fields. The framework builds directly on Bejleri-Park-Satriano, so the overlap is expected rather than hidden. The soft spot is the one flagged in the stress test. The existence statements rest on the Fourier coefficients of the Kudla-Millson lift being positive on the stable (non-isotrivial) locus after motivic stabilization. The abstract invokes the correspondence but gives no argument that the theta series attached to W_n^min is non-zero or that it remains supported away from the isotrivial components. If that form vanishes or concentrates on the isotrivial part, the coefficient counts drop and the linear rank growth fails. Without the full derivations or an explicit non-vanishing check, it is impossible to tell whether this is a minor gap or a load-bearing assumption. This is for specialists in arithmetic surfaces, moduli stacks, and automorphic forms who already know Kudla-Millson and motivic zeta functions. A reader who wants test cases for rank growth over function fields will find the statements worth checking. It deserves a serious referee because the claims are sharp and the tools are advanced; the non-vanishing issue is exactly the sort of thing a referee can press on with the full manuscript in hand.

Referee Report

1 major / 1 minor

Summary. The manuscript constructs the moduli stack W_n^min of minimal elliptic curves of Faltings height n over K=k(t) via the height-moduli framework, establishes rationality of the motivic height zeta function (weighted by trivial lattice rank) in the dimensionally completed Grothendieck ring via Euler products on isotrivial loci and motivic discriminant stabilization, and invokes the Kudla-Millson theta correspondence to conclude that Mordell-Weil section heights are governed by a weight 6n-2 form for SL_2(Z). Combining this with Shepherd-Barron diagonalization and Kodaira-Spencer transversality, it claims that for every n≥3 and 1≤r≤⌊(10n−2)/(n−1)⌋ there exist infinitely many stable elliptic surfaces with rk(E/K)≥r, and that every canonical height d is realized by some Mordell-Weil section.

Significance. If the claims hold, the work supplies explicit, unconditional bounds on Mordell-Weil ranks for elliptic surfaces of fixed Faltings height together with a modular-form description of realized canonical heights. The motivic rationality result and the link between height moduli and Kudla-Millson lifts would constitute a substantive advance connecting motivic methods with arithmetic geometry of elliptic surfaces.

major comments (1)

[Abstract / Kudla-Millson section] Abstract and Kudla-Millson application: the statement that the theta correspondence governs the distribution of new Mordell-Weil sections by canonical height (and thereby produces the stated infinitude of ranks and heights) requires non-vanishing of the weight 6n-2 theta series on the stable (non-isotrivial) locus after motivic discriminant stabilization. The manuscript provides no verification or reference establishing this non-vanishing, which is load-bearing for the coefficient counts underlying the existence claims.

minor comments (1)

[Abstract] The explicit bound ⌊(10n−2)/(n−1)⌋ is stated without a short derivation or reference to its origin in the weight or dimension of the relevant space of forms; adding one sentence would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful and constructive report. The major comment identifies a point that requires clarification in the Kudla-Millson application. We address it below and will incorporate the necessary additions in the revised manuscript.

read point-by-point responses

Referee: [Abstract / Kudla-Millson section] Abstract and Kudla-Millson application: the statement that the theta correspondence governs the distribution of new Mordell-Weil sections by canonical height (and thereby produces the stated infinitude of ranks and heights) requires non-vanishing of the weight 6n-2 theta series on the stable (non-isotrivial) locus after motivic discriminant stabilization. The manuscript provides no verification or reference establishing this non-vanishing, which is load-bearing for the coefficient counts underlying the existence claims.

Authors: We agree that an explicit reference or short argument for non-vanishing strengthens the exposition. The Kudla-Millson theta correspondence, applied to the orthogonal group associated to the height moduli stack after motivic discriminant stabilization, produces a holomorphic modular form of weight 6n-2 whose Fourier coefficients count the relevant section heights. This form is a non-zero multiple of the Eisenstein series E_{6n-2} (up to the contribution of the isotrivial loci, which is already factored out by the Euler-product decomposition). The Eisenstein series E_k for k ≥ 4 has strictly positive Fourier coefficients in all degrees, a standard fact recorded in Kudla-Millson (Intersection numbers of cycles on locally symmetric spaces and Fourier coefficients of holomorphic modular forms in several variables, §3) and in the classical theory of theta series for SL_2(Z). Consequently the coefficients governing the infinitude statements for 1 ≤ r ≤ ⌊(10n-2)/(n-1)⌋ are positive, and the existence claims follow. In the revised version we will add a short paragraph (new subsection 4.3) recalling this positivity and citing the relevant references, thereby supplying the missing verification without altering any statements. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external theorems after setup citation

full rationale

The paper defines the central moduli stack W_n^min by citing the height-moduli framework of Bejleri-Park-Satriano (author overlap) and then derives rationality of the motivic height zeta via Euler products plus Vakil-Wood adaptation, and the rank/height existence claims via Kudla-Millson theta correspondence plus Shepherd-Barron and Kodaira-Spencer. These steps invoke independent external results rather than reducing any prediction or first-principles claim to a fitted input, self-definition, or unverified self-citation chain. The self-citation supports only the initial construction and is not load-bearing for the modularity or existence statements, which remain falsifiable via the cited theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard assumptions of algebraic geometry over perfect fields of characteristic not 2 or 3, the Shioda-Tate formula, and the applicability of Kudla-Millson and Vakil-Wood results; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption k perfect with char(k) ≠ 2,3
Required for the construction of the moduli stack of minimal elliptic curves and the height-moduli framework.
standard math Shioda-Tate formula decomposes Picard rank into trivial lattice plus Mordell-Weil rank
Invoked to separate local fiber contributions from global sections.

pith-pipeline@v0.9.0 · 5695 in / 1548 out tokens · 35539 ms · 2026-05-16T11:35:54.939410+00:00 · methodology

Height moduli of elliptic surfaces: Motivic height zeta rationality and Kudla-Millson modularity of Mordell-Weil rank jumps

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)