Higher-Genus Multiple Zeta Values

• Higher-genus multiple zeta values are extensions of classical MZVs defined via iterated integrals on moduli spaces and compact Riemann surfaces, capturing new geometric and arithmetic phenomena.
• They employ techniques such as iterated Eisenstein integrals, modular graph forms, and Schottky uniformization to extend the mixed Tate structure of classical zeta values.
• The framework yields novel degeneration relations and zeta-like identities, bridging theories over number fields and function fields while offering concrete arithmetic applications.

Higher-genus multiple zeta values are not a single universally standardized object, but a family of extensions of classical multiple zeta values (MZVs) away from the genus-$0$ geometry of $\mathbb P^1\setminus\{0,1,\infty\}$. In the current literature, the phrase covers at least three distinct but overlapping directions: genus-$1$ modular and elliptic analogues in which MZVs are recovered from iterated Eisenstein integrals on $\mathcal M_{1,1}$ or from derivations acting on the once-punctured torus; positive-characteristic analogues attached to algebraic curves of genus $>0$ over finite fields; and, more recently, a direct higher-genus theory on compact Riemann surfaces in which one defines higher-genus multiple zeta values as $A$-cycle iterated integrals of Enriquez kernels (Saad, 2020, Dorigoni et al., 2024, Matsuzuki, 2023, Rodríguez et al., 2020, Baune et al., 29 Jul 2025). A separate literature uses “genus” in the cobordism-theoretic sense of complex genera rather than the genus of curves; there MZVs appear as coefficients of characteristic numbers, but that is a different notion (Li, 2021).

1. Classical MZVs and the genus-$1$ modular extension

Classical MZVs are periods of the motivic path torsor

$$\pi_1^{\mathrm{mot}}\bigl(\mathbb P^1\setminus\{0,1,\infty\},\vec1_0,-\vec1_1\bigr),$$

and are encoded by the motivic Drinfeld associator

$$\Phi_{01}^{\mathfrak m}=\sum_w \zeta^{\mathfrak m}(w)\,w.$$

This is the genus-$0$ model: iterated integrals of $\mathbb P^1\setminus\{0,1,\infty\}$0 and $\mathbb P^1\setminus\{0,1,\infty\}$1 on the thrice-punctured sphere (Saad, 2020).

The genus-$\mathbb P^1\setminus\{0,1,\infty\}$2 replacement is not merely a punctured elliptic curve, but the moduli stack $\mathbb P^1\setminus\{0,1,\infty\}$3 of elliptic curves together with the relative completion of

$\mathbb P^1\setminus\{0,1,\infty\}$4

Brown’s multiple modular values are periods of this relative completion, and their totally holomorphic part is built from iterated integrals of modular forms; the Eisenstein-only sector gives iterated Eisenstein integrals (Saad, 2020).

A precise bridge is established by the theorem that every motivic multiple zeta value of weight $\mathbb P^1\setminus\{0,1,\infty\}$5 and depth $\mathbb P^1\setminus\{0,1,\infty\}$6 is a $\mathbb P^1\setminus\{0,1,\infty\}$7-linear combination of motivic iterated Eisenstein integrals

$\mathbb P^1\setminus\{0,1,\infty\}$8

of length $\mathbb P^1\setminus\{0,1,\infty\}$9, total modular weight

$1$0

and multiplied by a power $1$1, where

$1$2

Passing to periods, every numerical MZV becomes a $1$3-linear combination of classical iterated Eisenstein integrals along the modular path corresponding to

$1$4

(Saad, 2020).

This modular absorption of genus-$1$5 periods is complemented by a Tannakian statement: the motivic Galois group acts faithfully on $1$6, equivalently $1$7 generates $1$8. In that sense, the genus-$1$9 modular side already generates the whole mixed Tate category over $\mathcal M_{1,1}$0 (Saad, 2020).

Concrete formulas illustrate the mechanism. The paper records, for example,

$\mathcal M_{1,1}$1

$\mathcal M_{1,1}$2

and

$\mathcal M_{1,1}$3

At the same time, the modular world is strictly larger than the MZV algebra, as shown by a concrete Eisenstein integral combination equal to the cusp-form value $\mathcal M_{1,1}$4 (Saad, 2020).

2. Canonical genus-$\mathcal M_{1,1}$5 zeta generators and non-holomorphic modular forms

A second genus-$\mathcal M_{1,1}$6 development is the canonicalization of zeta generators. In genus zero, odd zeta values are encoded by derivations of the free Lie algebra on two generators via Ihara derivations

$\mathcal M_{1,1}$7

A canonical choice of the corresponding Lie polynomials $\mathcal M_{1,1}$8 in each odd weight $\mathcal M_{1,1}$9 is characterized by pairing conditions against canonical subspaces of motivic MZVs, and these canonical genus-zero generators determine canonical genus-one derivations $>0$0 acting on the free Lie algebra $>0$1 of the once-punctured torus (Dorigoni et al., 2024).

The bridge from genus zero to genus one is induced by degeneration from the torus to the nodal sphere. At the Lie-algebra level one uses

$>0$2

with

$>0$3

For canonical genus-zero generators $>0$4, the canonical genus-one zeta generators are

$>0$5

and satisfy

$>0$6

(Dorigoni et al., 2024).

A basic structural theorem is that if $>0$7 is decomposed by total degree, then all contributions of degree different from the key degree $>0$8 lie in Tsunogai’s Lie algebra $>0$9 of geometric derivations dual to holomorphic Eisenstein series. The unique exceptional degree $A$0 contains the non-geometric arithmetic part $A$1, and

$A$2

The arithmetic part is fixed by a representation-theoretic condition: it is the one-dimensional irreducible $A$3-component of the key-degree part (Dorigoni et al., 2024).

This genus-$A$4 derivation formalism feeds directly into non-holomorphic modular objects built from iterated Eisenstein integrals. Equivariant iterated Eisenstein integrals are organized by generating series of the form

$A$5

and a special subclass gives an equivalent description of modular graph forms appearing in genus-one string amplitudes (Dorigoni et al., 2024).

The zeta-value content of these modular graph forms is controlled by single-valued MZVs. Products and higher-depth single-valued MZVs are generated from primitive odd zeta values through the group-like series $A$6, so that higher-depth content is not introduced independently but forced by the algebra of zeta generators (Dorigoni et al., 2024). Explicit formulas exhibit this structure: $A$7 and

$A$8

At modular depth three, indecomposable higher-depth single-valued MZVs such as $A$9 occur, while cusp-form completions introduce periods beyond classical MZVs and beyond critical or non-critical cusp-form $1$0-values (Dorigoni et al., 2024).

Taken together, these genus-$1$1 results show that elliptic and modular analogues are not merely reformulations of classical MZVs. They contain the MZV algebra, canonically organize it through zeta generators, and simultaneously enlarge it by modular and cuspidal period phenomena (Saad, 2020, Dorigoni et al., 2024, Dorigoni et al., 2024).

3. Higher-genus multiple zeta values on compact Riemann surfaces

A direct higher-genus theory is constructed for compact Riemann surfaces of genus $1$2. The central objects are higher-genus multiple zeta values defined from Enriquez’ kernels $1$3, obtained from a unique meromorphic flat connection $1$4 on the universal cover of the surface, valued in the free algebra on letters $1$5 (Baune et al., 29 Jul 2025).

Higher-genus multiple polylogarithms are iterated integrals

$1$6

with depth $1$7 and weight

$1$8

where $1$9 is the length of the multi-index minus the final target index. The corresponding higher-genus multiple zeta values are the $$\pi_1^{\mathrm{mot}}\bigl(\mathbb P^1\setminus\{0,1,\infty\},\vec1_0,-\vec1_1\bigr),$$0-cycle special values

$$\pi_1^{\mathrm{mot}}\bigl(\mathbb P^1\setminus\{0,1,\infty\},\vec1_0,-\vec1_1\bigr),$$1

These depend on the complex structure of the underlying surface, equivalently on its period matrix or Schottky data (Baune et al., 29 Jul 2025).

The formalism is made explicit through Schottky uniformization. If

$$\pi_1^{\mathrm{mot}}\bigl(\mathbb P^1\setminus\{0,1,\infty\},\vec1_0,-\vec1_1\bigr),$$2

with Schottky group $$\pi_1^{\mathrm{mot}}\bigl(\mathbb P^1\setminus\{0,1,\infty\},\vec1_0,-\vec1_1\bigr),$$3, then normalized holomorphic differentials and the Abel map are written as Poincaré series in Schottky coordinates, and Enriquez’ connection admits the Schottky representation

$$\pi_1^{\mathrm{mot}}\bigl(\mathbb P^1\setminus\{0,1,\infty\},\vec1_0,-\vec1_1\bigr),$$4

Expanding in the $$\pi_1^{\mathrm{mot}}\bigl(\mathbb P^1\setminus\{0,1,\infty\},\vec1_0,-\vec1_1\bigr),$$5 yields higher-genus kernels as Schottky sums of genus-one kernels: $$\pi_1^{\mathrm{mot}}\bigl(\mathbb P^1\setminus\{0,1,\infty\},\vec1_0,-\vec1_1\bigr),$$6 This gives a computationally explicit reduction of higher-genus kernels to weighted Schottky sums of genus-one kernels on Schottky subcovers (Baune et al., 29 Jul 2025).

A key issue is regularization. Only kernels $$\pi_1^{\mathrm{mot}}\bigl(\mathbb P^1\setminus\{0,1,\infty\},\vec1_0,-\vec1_1\bigr),$$7 have a pole at $$\pi_1^{\mathrm{mot}}\bigl(\mathbb P^1\setminus\{0,1,\infty\},\vec1_0,-\vec1_1\bigr),$$8, so endpoint divergences already arise at depth one, and for closed $$\pi_1^{\mathrm{mot}}\bigl(\mathbb P^1\setminus\{0,1,\infty\},\vec1_0,-\vec1_1\bigr),$$9-cycle integrals there are divergences at both ends. The regularization prescription uses the Schottky uniformization to reduce the singular part to genus-one regularization via the Abel map $$\Phi_{01}^{\mathfrak m}=\sum_w \zeta^{\mathfrak m}(w)\,w.$$0. For higher-genus multiple zeta values, this yields the striking depth-one formula

$$\Phi_{01}^{\mathfrak m}=\sum_w \zeta^{\mathfrak m}(w)\,w.$$1

More generally,

$$\Phi_{01}^{\mathfrak m}=\sum_w \zeta^{\mathfrak m}(w)\,w.$$2

Thus all-equal even-weight depth-one higher-genus values reduce to classical zeta values, while regularized odd-weight depth-one values become rational (Baune et al., 29 Jul 2025).

This theory is the first explicit construction in the supplied corpus where “higher-genus multiple zeta values” means iterated periods on compact Riemann surfaces of arbitrary genus rather than modular genus-$$\Phi_{01}^{\mathfrak m}=\sum_w \zeta^{\mathfrak m}(w)\,w.$$3 analogues or function-field variants. It retains shuffle structure and degeneration principles known from genus $$\Phi_{01}^{\mathfrak m}=\sum_w \zeta^{\mathfrak m}(w)\,w.$$4 and $$\Phi_{01}^{\mathfrak m}=\sum_w \zeta^{\mathfrak m}(w)\,w.$$5, but introduces genuinely new cycle combinatorics and regularization phenomena (Baune et al., 29 Jul 2025).

4. Degenerations and higher-genus relations

Degeneration is one of the main organizing principles of higher-genus multiple zeta values. Two cases are distinguished: non-separating degeneration, in which an $$\Phi_{01}^{\mathfrak m}=\sum_w \zeta^{\mathfrak m}(w)\,w.$$6-cycle is pinched and genus drops by one while two marked points remain, and separating degeneration, in which the surface splits into lower-genus components (Baune et al., 29 Jul 2025).

In a non-separating degeneration $$\Phi_{01}^{\mathfrak m}=\sum_w \zeta^{\mathfrak m}(w)\,w.$$7, if $$\Phi_{01}^{\mathfrak m}=\sum_w \zeta^{\mathfrak m}(w)\,w.$$8 then

$$\Phi_{01}^{\mathfrak m}=\sum_w \zeta^{\mathfrak m}(w)\,w.$$9

so kernels containing the pinched direction disappear in unaffected directions, and the others descend to genus $0$0. For example,

$0$1

while for $0$2,

$0$3

If all directions except $0$4 are pinched, one recovers the elliptic value on the $0$5-th genus-one subcover (Baune et al., 29 Jul 2025).

In a separating degeneration, the period matrix becomes block diagonal and genus $0$6 splits into genus $0$7 plus genus $0$8 in the formulation worked out in detail. In unaffected directions one again gets lower-genus higher-genus MZVs, while in the affected direction rescaled kernels reduce to genus-one elliptic kernels, and one finds

$0$9

Thus degeneration realizes higher-genus values as interpolating objects between hgMZVs of smaller genus and eMZVs (Baune et al., 29 Jul 2025).

Beyond degeneration-induced reductions, the higher-genus theory exhibits new identities. Since the kernels satisfy Fay-like relations generalizing the elliptic Fay identity, their $\mathbb P^1\setminus\{0,1,\infty\}$00-cycle values satisfy corresponding hgMZV identities. At genus two, an explicit example is

$\mathbb P^1\setminus\{0,1,\infty\}$01

(Baune et al., 29 Jul 2025).

A genuinely higher-genus phenomenon is the presence of cycle-exchange relations connecting integrals over different $\mathbb P^1\setminus\{0,1,\infty\}$02-cycles. For $\mathbb P^1\setminus\{0,1,\infty\}$03, $\mathbb P^1\setminus\{0,1,\infty\}$04, $\mathbb P^1\setminus\{0,1,\infty\}$05, $\mathbb P^1\setminus\{0,1,\infty\}$06,

$\mathbb P^1\setminus\{0,1,\infty\}$07

A simple instance is

$\mathbb P^1\setminus\{0,1,\infty\}$08

This indicates that the algebra of hgMZVs is not naturally decomposed cycle-by-cycle (Baune et al., 29 Jul 2025).

Hyperelliptic geometry yields further relations. If one chooses Schottky generators so that $\mathbb P^1\setminus\{0,1,\infty\}$09, then on a hyperelliptic surface one has

$\mathbb P^1\setminus\{0,1,\infty\}$10

for $\mathbb P^1\setminus\{0,1,\infty\}$11 and $\mathbb P^1\setminus\{0,1,\infty\}$12 even. The simplest case is

$\mathbb P^1\setminus\{0,1,\infty\}$13

The paper reports numerical checks for generic genus-two and genus-three hgMZVs of depth $\mathbb P^1\setminus\{0,1,\infty\}$14 and $\mathbb P^1\setminus\{0,1,\infty\}$15, total weight up to $\mathbb P^1\setminus\{0,1,\infty\}$16, using SchottkyTools (Baune et al., 29 Jul 2025).

5. Positive-characteristic higher-genus multiple zeta values

A different higher-genus direction arises over global function fields. Let $\mathbb P^1\setminus\{0,1,\infty\}$17 be a smooth projective curve of genus $\mathbb P^1\setminus\{0,1,\infty\}$18, $\mathbb P^1\setminus\{0,1,\infty\}$19 a rational point, and

$\mathbb P^1\setminus\{0,1,\infty\}$20

Then one defines higher-genus function-field multiple zeta values by summing over monic elements of $\mathbb P^1\setminus\{0,1,\infty\}$21, ordered by degree: $\mathbb P^1\setminus\{0,1,\infty\}$22 The role of degree is governed by the non-gap sequence $\mathbb P^1\setminus\{0,1,\infty\}$23 at $\mathbb P^1\setminus\{0,1,\infty\}$24, namely the Weierstrass semigroup (Matsuzuki, 2023).

A fundamental result is non-vanishing. If the non-gap sequence is either

$\mathbb P^1\setminus\{0,1,\infty\}$25

or

$\mathbb P^1\setminus\{0,1,\infty\}$26

then for every $\mathbb P^1\setminus\{0,1,\infty\}$27, the multiple zeta value $\mathbb P^1\setminus\{0,1,\infty\}$28 is nonzero. In particular, MZVs associated with elliptic curves are non-zero (Matsuzuki, 2023).

The proof proceeds through exact $\mathbb P^1\setminus\{0,1,\infty\}$29-adic valuation formulas for finite power sums

$\mathbb P^1\setminus\{0,1,\infty\}$30

where $\mathbb P^1\setminus\{0,1,\infty\}$31 runs over monic elements of degree $\mathbb P^1\setminus\{0,1,\infty\}$32. Under the semigroup hypotheses,

$\mathbb P^1\setminus\{0,1,\infty\}$33

with $\mathbb P^1\setminus\{0,1,\infty\}$34 a weighted sum determined by the non-gap sequence and $\mathbb P^1\setminus\{0,1,\infty\}$35 the no-carry multiples of $\mathbb P^1\setminus\{0,1,\infty\}$36 inherited from Sheats’s genus-zero analysis. Strict growth of these valuations yields a unique lowest-valuation term in the defining series, hence non-vanishing by ultrametricity (Matsuzuki, 2023).

A more delicate phenomenon is the existence of zeta-like identities for positive-genus function fields of class number one. In this setting a multizeta value is called zetalike if

$\mathbb P^1\setminus\{0,1,\infty\}$37

The paper on zeta-like multizeta values formulates a universal conjectural family: $\mathbb P^1\setminus\{0,1,\infty\}$38 for any class number one $\mathbb P^1\setminus\{0,1,\infty\}$39 with constant field $\mathbb P^1\setminus\{0,1,\infty\}$40 (Rodríguez et al., 2020).

It then proves the first nontrivial positive-genus zeta-like identities, including

$\mathbb P^1\setminus\{0,1,\infty\}$41

and

$\mathbb P^1\setminus\{0,1,\infty\}$42

for

$\mathbb P^1\setminus\{0,1,\infty\}$43

together with analogous formulas in three other class-number-one positive-genus cases (Rodríguez et al., 2020).

The paper emphasizes that the higher-genus mechanism differs from the rational function field case. In genus zero, zeta-like relations often already appear at the level of fixed-degree power sums $\mathbb P^1\setminus\{0,1,\infty\}$44. In the positive-genus examples proved, no such direct $\mathbb P^1\setminus\{0,1,\infty\}$45-level identity of the expected type exists; instead the result emerges only after introducing explicit rational interpolation functions on the curve and comparing leading terms in Frobenius-specialized identities. This suggests a different motivic mechanism from the Carlitz–Thakur setting (Rodríguez et al., 2020).

6. Related generalizations and terminological cautions

The phrase “higher-genus multiple zeta values” is used in adjacent literatures with different meanings, and the distinctions are mathematically significant.

One nearby direction concerns generalized multiple zeta values over number fields via plectic and graph-theoretic constructions. The basic object is a plectic Green function

$\mathbb P^1\setminus\{0,1,\infty\}$46

from which one forms graph-integrated higher plectic Green functions and then generalized multiple zeta values

$\mathbb P^1\setminus\{0,1,\infty\}$47

When $\mathbb P^1\setminus\{0,1,\infty\}$48 and $\mathbb P^1\setminus\{0,1,\infty\}$49 is a tree, these are finite $\mathbb P^1\setminus\{0,1,\infty\}$50-linear combinations of classical MZVs of depth $\mathbb P^1\setminus\{0,1,\infty\}$51 and weight $\mathbb P^1\setminus\{0,1,\infty\}$52 (Ai, 2018). This is structurally close to higher-genus correlator philosophy, but the geometry is plectic and toroidal rather than that of higher-genus curves.

Another distinct use of “genus” comes from complex genera in cobordism theory. There, for a complex genus $\mathbb P^1\setminus\{0,1,\infty\}$53, coefficients $\mathbb P^1\setminus\{0,1,\infty\}$54 in front of Chern numbers can be expressed in zeta-theoretic terms. For the $\mathbb P^1\setminus\{0,1,\infty\}$55-genus,

$\mathbb P^1\setminus\{0,1,\infty\}$56

so the coefficients are symmetrized MZVs; for the $\mathbb P^1\setminus\{0,1,\infty\}$57-genus,

$\mathbb P^1\setminus\{0,1,\infty\}$58

so the coefficients are symmetrized MZSVs (Li, 2021). Here “genus” means Hirzebruch genus, not the genus of a curve.

A third auxiliary strand is the study of extremal or maximal-height sectors of ordinary MZV theory. Explicit formulas for maximal-height MZVs and their finite analogues are given by alternating sums over refinements: $\mathbb P^1\setminus\{0,1,\infty\}$59 with parallel formulas for finite MZVs (Murahara et al., 2016). This work has no direct higher-genus content, but supplies combinatorial and derivation-based techniques that may be transferable.

Across these literatures, a common pattern emerges. Higher-genus or genus-extended zeta theories typically require a replacement for the genus-$\mathbb P^1\setminus\{0,1,\infty\}$60 path torsor by a moduli-theoretic or curve-theoretic structure, a regularization prescription for singular iterated integrals, and a mechanism—degeneration, relative completion, or graph expansion—linking new periods back to classical MZVs. In genus $\mathbb P^1\setminus\{0,1,\infty\}$61, those mechanisms are already strong enough to recover the mixed Tate MZV world inside modular geometry (Saad, 2020, Dorigoni et al., 2024, Dorigoni et al., 2024). In genus $\mathbb P^1\setminus\{0,1,\infty\}$62, the recent Riemann-surface theory shows that new cycle combinatorics, new regularization identities, and new degeneration patterns enter essentially, so higher-genus multiple zeta values are not merely elliptic multiple zeta values with more indices (Baune et al., 29 Jul 2025).