Motivic Height Zeta Functions

• Motivic height zeta functions are generating series in Grothendieck rings that encode moduli spaces of sections of prescribed degree over smooth projective curves.
• They capture geometric analogues of classical height zeta functions via explicit formulas, functional equations, and rationality properties in settings like projective bundles and toric varieties.
• Analytic methods, including Poisson summation and weight convergence, reveal pole structures and asymptotic behaviors that extend the Manin–Peyre principle in arithmetic geometry.

Searching arXiv for recent and foundational papers on motivic height zeta functions. Motivic height zeta functions are generating series in Grothendieck rings that encode spaces of sections or maps of prescribed degree over the function field of a curve. In the simplest form, one fixes a smooth projective curve $C$ over a field $k$, a flat projective morphism $\pi:X\to C$, and a line bundle or log-anticanonical class that plays the role of a height; one then forms a series whose $d$-th coefficient is the class of the moduli space of sections of degree $d$. This construction is a geometric analogue of classical height zeta functions over number fields and function fields, but its coefficients lie in motivic coefficient rings such as $K_0(\mathrm{Var}_k)$, localizations by the Lefschetz motive, or their completions. The subject has developed along several parallel lines: explicit formulas and functional equations for projective bundles, rationality and pole analysis for equivariant compactifications of vector groups, Poisson-summation methods for split toric varieties, and weighted refinements for elliptic surfaces (Maruyama, 2015, Chambert-Loir et al., 2013, Bilu et al., 3 Apr 2026).

1. Basic setup and definitions

A foundational formulation starts with a smooth projective curve $C/k$, a function field $F=k(C)$, and a family $X\to C$ whose generic fibre $X_\eta$ carries a natural height datum. For each integer $k$0, one considers a moduli space of sections of degree $k$1, typically quasi-projective by a Hilbert-scheme argument, and defines a generating series in a Grothendieck ring. In the additive-group setting, if $k$2 is the log-anticanonical bundle and $k$3 denotes sections $k$4 with $k$5 and $k$6, then

$k$7

In the projective-bundle setting, if $k$8 is a vector bundle on $k$9 and $\pi:X\to C$0 is the associated projective bundle, then

$\pi:X\to C$1

where $\pi:X\to C$2 is the quasi-projective scheme of sections $\pi:X\to C$3 with $\pi:X\to C$4 (Chambert-Loir et al., 2013, Maruyama, 2015).

The coefficient rings vary with the analytic regime under study. For projective bundles one passes from $\pi:X\to C$5 to $\pi:X\to C$6, the quotient identifying radicial surjections, sets $\pi:X\to C$7, and localizes to $\pi:X\to C$8. In toric and convergence-theoretic work one instead uses

$\pi:X\to C$9

or its completion by the virtual-dimension filtration. Over finite fields, the count-points morphism or zeta measure sends the motivic series to more classical counting series, so the motivic formalism refines the arithmetic one rather than replacing it (Maruyama, 2015, Bilu et al., 3 Apr 2026).

A second basic feature is that “height” is not unique. In projective-bundle and additive-group settings it is usually a scalar degree. For split toric varieties one often has a multidegree height with values in $d$0, and the single-variable anticanonical series is obtained by specializing along the anticanonical ray. This multivariable viewpoint is essential for the explicit Euler-product and residue formulas in the toric case (Bilu et al., 3 Apr 2026).

2. Projective bundles, Kapranov zeta, and functional equations

For a vector bundle $d$1 of rank $d$2 on $d$3, the projective bundle $d$4 provides one of the cleanest motivic height-zeta theories. The key auxiliary series is Kapranov’s motivic zeta function of the base curve,

$d$5

with $d$6. A Schwarzenberger-type construction produces varieties $d$7, where $d$8 is the $d$9 of the coherent sheaf

$d$0

on $d$1, and yields the identity

$d$2

in $d$3 (Maruyama, 2015).

The main motivic theorem states that in $d$4, the product $d$5 is rational and satisfies a motivic functional equation. More precisely,

$d$6

is a polynomial of degree $d$7, and

$d$8

Thus the denominator has exactly two critical factors, $d$9 and $K_0(\mathrm{Var}_k)$0, and the involution $K_0(\mathrm{Var}_k)$1 exchanges the two sides (Maruyama, 2015).

The structural input is a motivic Riemann–Roch relation for the family $K_0(\mathrm{Var}_k)$2: $K_0(\mathrm{Var}_k)$3 where $K_0(\mathrm{Var}_k)$4. The local system

$K_0(\mathrm{Var}_k)$5

has rank difference $K_0(\mathrm{Var}_k)$6, so the corresponding projective bundles differ motivically by a projective-space class. Summing in $K_0(\mathrm{Var}_k)$7 forces both the two-term denominator and the involution in the functional equation. The paper presents this as the exact analogue of the Arakelov–Tate Riemann–Roch method used earlier in the analytic setting (Maruyama, 2015).

The ruled-surface case $K_0(\mathrm{Var}_k)$8, $K_0(\mathrm{Var}_k)$9 with $C/k$0, gives the Hirzebruch surface $C/k$1. In that case

$C/k$2

and one again obtains a functional equation together with the denominator $C/k$3. The geometry of the ruled surface stratifies the amplitudes of poles of $C/k$4, and the two critical factors $C/k$5 and $C/k$6 allow one to read off the decomposition of the effective cone of $C/k$7 (Maruyama, 2015).

3. Additive-group compactifications and motivic Poisson summation

A broader theory treats equivariant compactifications of the vector group $C/k$8. Here $C/k$9 is an algebraically closed field of characteristic zero, $F=k(C)$0 is a smooth projective connected curve, $F=k(C)$1 is a dense open subset, $F=k(C)$2, and $F=k(C)$3 is a smooth projective $F=k(C)$4-equivariant compactification of $F=k(C)$5. The boundary

$F=k(C)$6

is assumed to have strict normal crossings, with each $F=k(C)$7 smooth and geometrically irreducible. One chooses a flat model $F=k(C)$8 of $F=k(C)$9 over $X\to C$0, extends to a good model $X\to C$1, and denotes the vertical irreducible components over $X\to C$2 by $X\to C$3. The relevant height is defined by the log-anticanonical divisor

$X\to C$4

and $X\to C$5 is the constructible moduli space of sections $X\to C$6 of height $X\to C$7 whose generic value lies in $X\to C$8 and whose restriction to $X\to C$9 lands in $X_\eta$0 (Chambert-Loir et al., 2013).

The main rationality theorem asserts that $X_\eta$1 lies in the subring $X_\eta$2 of rational power series generated by inverses of $X_\eta$3. More precisely, there exist integers $X_\eta$4, $X_\eta$5, and a polynomial $X_\eta$6 in $X_\eta$7 such that $X_\eta$8 is an effective nonzero element of $X_\eta$9 and

$k$00

The same work identifies the largest pole: it is at $k$01, and its order is

$k$02

where $k$03 is the analytic Clemens complex of the special fibre at $k$04, with vertices given by the horizontal components $k$05 and the vertical components $k$06, and faces given by their nonempty intersections (Chambert-Loir et al., 2013).

The proof is adelic and Fourier-theoretic. One rewrites the section-counting problem as a sum over $k$07 with local intersection data, decomposes

$k$08

and studies each $k$09 as a product of local Fourier transforms $k$10. The motivic Poisson summation formula, in the Hrushovski–Kazhdan framework, replaces classical harmonic analysis on $k$11. Local analysis shows that the $k$12 term has the maximal possible pole, contributed by the full Clemens complex, while for $k$13 the pole order drops by at least $k$14. This isolates the geometric origin of the leading pole (Chambert-Loir et al., 2013).

This setting is explicitly described as the geometric analogue of a number-field result of Chambert-Loir and Tschinkel. Counting sections $k$15 of bounded log-anticanonical degree parallels counting integral points of bounded height, and motivic integration on arc spaces replaces the classical local Igusa integrals appearing in arithmetic Poisson summation (Chambert-Loir et al., 2013).

4. Split toric varieties and the motivic Manin–Peyre picture

For a split projective toric variety $k$16 over a field $k$17 of characteristic zero, with open torus $k$18, fan $k$19, character lattice $k$20, and cocharacter lattice $k$21, the height can be made multigraded by recording local intersection indices with the $k$22-invariant divisors $k$23, $k$24. The motivic height zeta function is then

$k$25

or, in the single-variable anticanonical case,

$k$26

where $k$27 denotes the moduli of maps of multidegree $k$28 (Bilu et al., 3 Apr 2026).

The central analytic tool is a motivic Poisson formula for split algebraic tori. If $k$29, a Schwartz–Bruhat motivic function on $k$30 is an element of $k$31 supported on finitely many cocharacters, and its Fourier transform is

$k$32

Applied fibrewise on symmetric products of $k$33, this gives a global Poisson identity that rewrites the height zeta function as a motivic integral of the Fourier transform over characters trivial on principal divisors. The local Fourier transform has the explicit form

$k$34

with $k$35 determined by the fan $k$36 (Bilu et al., 3 Apr 2026).

The resulting analytic statements parallel the classical Batyrev–Tschinkel theory. In multivariable form,

$k$37

for some $k$38. After specializing to the anticanonical ray $k$39, the series converges for $k$40 and has a unique principal pole at $k$41. Its order is $k$42, and the residue is expressed as

$k$43

where $k$44 is the volume of the dual effective cone and $k$45 is the motivic Tamagawa factor involving $k$46, $k$47, and the product over places $k$48 (Bilu et al., 3 Apr 2026).

A direct consequence is a motivic Manin–Peyre statement: as $k$49 tends to infinity in the effective cone, the classes

$k$50

stabilize in the virtual-dimension completion to the constant $k$51. For $k$52, the toric theory recovers

$k$53

with principal pole at $k$54 of order $k$55 and residue $k$56 (Bilu et al., 3 Apr 2026).

5. Topologies, convergence, and asymptotic extraction

Motivic height zeta functions are studied not only as formal rational expressions but also as analytic objects in completed coefficient rings. One important framework uses the weight filtration coming from mixed Hodge theory. Over $k$57, the Hodge realization

$k$58

pulls back the standard weight filtration to a filtration on $k$59, and the completion $k$60 is defined so that a series $k$61 converges when the weights of $k$62 tend to $k$63 fast enough. In the setting of partial equivariant compactifications of $k$64, there exist a nonzero $k$65 and $k$66 such that

$k$67

converges for $k$68 and takes the value $k$69 at $k$70. In particular, $k$71 has a pole of order $k$72 at $k$73 and no other poles in that disc; the integer $k$74 is expressed in terms of $k$75 and the simplicial complexes $k$76 attached to bad fibres (Bilu, 2018).

This convergence theorem has asymptotic consequences for Hodge–Deligne polynomials of the section spaces. If $k$77 has such a controlled pole, then for each congruence class modulo a fixed integer either $k$78, or else

$k$79

for some integer $k$80, and the number of components of top dimension grows like $k$81. Moreover, a positive proportion of congruence classes achieve the maximal dimension growth. In this way, the motivic pole structure is converted into geometric asymptotics for the moduli spaces of sections (Bilu, 2018).

A different completion arises from the Witt-ring and Hadamard-function viewpoint. Let $k$82 be the rational Witt ring of rational functions $k$83 with $k$84, equipped with the Hadamard topology defined by the divisor norm

$k$85

Its completion is identified with a space $k$86 of meromorphic functions called Hadamard functions. Over a finite field, the zeta measure sends motivic classes to rational Witt vectors, and therefore maps completed motivic rings to $k$87. For the toric motivic height-zeta series, the renormalized series

$k$88

converges in the dimensional topology to a nonzero limit, and after applying the zeta measure one obtains convergence of

$k$89

in the point-counting topology and, for $k$90 above an explicit combinatorial bound, also in the Hadamard topology. This is presented as evidence for a meta-conjecture that natural sequences whose zeta-images converge in both the weight and point-counting topologies should converge in the stronger Hadamard topology as well (Bilu et al., 2020).

These two analytic regimes show that “pole” and “convergence” are not uniform notions in the subject. Rationality in a localized Grothendieck ring, convergence in the weight topology, and convergence in the Hadamard topology are distinct statements, each tailored to a different part of the motivic-height-zeta formalism (Bilu, 2018, Bilu et al., 2020).

6. Weighted refinements, elliptic surfaces, and limits of rationality

A recent refinement replaces plain section-counting by weighted counting in a moduli stack of elliptic curves over $k$91. Let $k$92 be the Deligne–Mumford stack parametrizing $k$93-rational points of $k$94-height $k$95 on $k$96, equivalently minimal elliptic curves $k$97 of discriminant degree $k$98. If $k$99 is the associated relatively minimal elliptic surface with section, one defines

$\pi:X\to C$00

where $\pi:X\to C$01 is the rank of the trivial lattice in $\pi:X\to C$02 and $\pi:X\to C$03 is the Mordell–Weil rank. The Shioda–Tate formula reads

$\pi:X\to C$04

Thus the series refines the height distribution simultaneously by trivial-lattice rank and Mordell–Weil rank (Park, 21 Jan 2026).

The rationality result applies to the trivial-lattice specialization

$\pi:X\to C$05

Because $\pi:X\to C$06 is determined by fibre types, the moduli stack decomposes into finitely many Kodaira strata on which $\pi:X\to C$07 is constant. After introducing a multivariate series indexed by elementary local patterns and using a power-structure Euler product, one obtains a finite Euler product

$\pi:X\to C$08

in $\pi:X\to C$09. In particular, $\pi:X\to C$10 is rational in $\pi:X\to C$11 after inverting $\pi:X\to C$12 (Park, 21 Jan 2026).

By contrast, the Mordell–Weil and Néron–Severi specializations are conjectured to be irrational: $\pi:X\to C$13 The stated reason is geometric: on a fixed Kodaira stratum the trivial lattice is constant, whereas $\pi:X\to C$14 and $\pi:X\to C$15 vary along Noether–Lefschetz or Hodge jump loci, which are in general countable unions of closed algebraic subsets rather than a finite stratification. This creates a sharp distinction between locally determined weights, which admit finite Euler products, and globally varying lattice invariants, which are expected not to (Park, 21 Jan 2026).

$\pi:X\to C$16

Taken together, these developments delineate the current scope of the subject. Projective bundles exhibit explicit functional equations and low-degree denominators; equivariant compactifications of $\pi:X\to C$17 yield rationality and pole orders governed by Clemens complexes; split toric varieties admit a motivic Poisson theory compatible with a motivic Manin–Peyre principle; convergence can be studied in weight and Hadamard topologies; and weighted variants for elliptic surfaces show that rationality can depend sensitively on whether the chosen invariant is locally stratifiable or controlled by global jump loci (Maruyama, 2015, Chambert-Loir et al., 2013, Bilu et al., 3 Apr 2026, Bilu, 2018, Bilu et al., 2020, Park, 21 Jan 2026).