# Thakur's automata and transcendence questions for function-field special values: status, one proof written out, and the open residue-class case

Date: 2026-10-23, America/New_York. Written by an AI agent (claude-opus-5-5,
Claude Code fork) for Taylor. Saved to disk by the parent agent because a hook
blocked the fork's own write. The report separates literature results, proofs
written here, computations and conjectural claims. Not human-reviewed; no
independent second-agent audit or formal verification.

**Scope exclusion (human instruction):** Diophantine approximation exponents and
continued fractions are off limits because someone else is working on them. This
report does not touch them. Christol's theorem is used only as an algebraicity
criterion.

**Status:** not fully resolved. Theorem A is proved in full but is not new
(Wen–Yao Thm 8). The general case is still open; the two missing steps are
stated as R1 and R2 in §4.

## 0. Negative-result register (read first)

- **N1 — Not every special value is transcendental.** It is natural to expect
  every v-adic special value to be transcendental, and hence non-automatic. In
  fact ζ_v(n) = 0 whenever (q−1) | n (Goss 1979, as reported in [CM19] §1.2.1);
  this is the analogue of Kubota–Leopoldt ζ_p vanishing at even integers.
  Thakur's Gross–Koblitz analogue also makes Π_v(n) algebraic whenever the
  q-adic digits of n are eventually d-periodic ([Th-CRM] §4.1; [WY02] §10).
  Those values are automatic. The real question is which values are
  non-automatic.
- **N2 — "In t" is ill-posed for ∞-adic values.** ∞-adic values (ζ(n), π̃, Γ,
  MZVs) lie in F_q((1/t)), and Christol's theorem applies to the 1/t-expansion.
  "Expansion in t" makes sense only at v = t or after a change of variable. In
  the two-variable Anderson–Thakur/Pellarin setting the coefficients lie in
  C_∞, which is not a finite field, so Christol does not apply. §1.4 gives the
  substitute statement.
- **N3 — The partial result here is not new.** Theorem A is Wen–Yao 2002,
  Thm 8 [WY02 §10], whose proof they left "as a good exercise to the reader".
  §3 supplies the proof. Only that write-up and the computations are
  contributions of this dump; no novelty is claimed.
- **N4 — The open conjecture is Wen–Yao's.** Thakur asked whether his
  hypothesis (H2) is best possible ([Th-CRM] §4.1). The precise conjecture for
  all n ∈ Z_p is in [WY02 §10]; Thakur's notes credit it to Yao. PAMQ 2010
  (p. 16) says the general case "is still open".
- **N5 — Period methods reach only rational arguments.** [CWY24] proves
  results only at rational arguments (Thm 1.4.1, Cor 1.4.2). Thakur–Wen–Yao–Zhao
  say the period method does not apply to non-fractional arguments (search
  snippet).
- **N6 — Per-residue reduction is circular.** The shift identity
  D_{j+1,v} = D_{j,v}^q·[j+1]/[j+1−d] gives F(b) = G^{q^r}·W. Since
  d log(G^{q^r}) = 0, the log-derivative just returns the residue-r series
  (§4).
- **N7 — The naive log-derivative series diverges.** The series
  −Σ (n_j − n_{j+d})/[j] does not converge v-adically, because 1/[j] does not
  tend to 0 at v. Only the telescoped grouping of §3 converges.

Inherited refutations: zero. The parent's informal list item 9 is refined by
N1, N2 and N4.

## 1. Question-to-answer map

**Notation.** A = F_q[t], K = F_q(t), v a monic prime of degree d,
[j] = t^{q^j} − t, D_j = ∏_{m≤j} [m]^{q^{j−m}}. For j ≥ d,
D_{j,v} = D_j/(v^{q^{j−d}} D_{j−d}); for j < d, D_{j,v} = D_j. For
n = Σ n_j q^j ∈ Z_p, Π_v(n) = ∏ (−D_{j,v})^{n_j}. Christol: f ∈ F_Q[[x]] is
algebraic over F_Q(x) ⟺ its coefficients are p-automatic ⟺ its Q-kernel is
finite.

| Value family | Question | Status | Source |
|---|---|---|---|
| Carlitz period π̃ | transcendental? | Settled: yes (Wade 1941; automata proof Allouche 1990) | [Th-CRM], [WY02] ref. 2 |
| Carlitz ζ(n), ∞-adic | transcendental? | Settled. ζ(n) is transcendental for all n. ζ(n)/π̃^n ∈ K iff (q−1) \| n, and it is transcendental otherwise (Yu 1991). Automata proofs only for 1 ≤ n ≤ q−2 (Berthé). | [CM19] §1.2.1, [Ber93] |
| ζ_v(n), v-adic | transcendental? | Settled: 0 if (q−1) \| n (Goss); transcendental otherwise (Yu) | [CM19] §1.2.1 |
| Arithmetic Γ at ∞, fractions | which monomials are algebraic? | Settled by automata (Thakur, Allouche 1996). Full algebraic relations with ζ: CPTY 2010. | [Th-CRM] Thm 4.1, [CPTY10] |
| Arithmetic Γ at ∞, n ∈ Z_p \ N | transcendental? | Settled: yes (Mendès France–Yao 1997). A period-method proof is still wanted (methodological). | [Th-CRM] Thm 4.4 |
| Geometric Γ at ∞, fractions | all relations? | Settled (Anderson–Brownawell–Papanikolas 2004) | [ABP04] |
| **Π_v(n), v-adic, n ∈ Z_p** | **algebraic iff digits eventually d-periodic?** | **Open in general.** Known: d = 1; residue-0 case (WY Thm 8, §3 here); rational n (via [CWY24] Cor 1.4.2, §2). | §§2–4 |
| v-adic geometric Γ | algebraic iff …? | Open ([CWY24] Rem. 1.4.3, reported) | [CWY24] |
| Thakur MZVs at ∞ | transcendental? | Settled: each is transcendental (Chang 2014 plus Thakur's nonvanishing; deduction in §1.3) | [C14] |
| v-adic MZVs | vanishing / transcendence? | Partly open. Furusho analogue [CM21]; deg v = 1 criterion [Chen25]. Transcendence for deg v ≥ 2 in depth ≥ 2 is open, to our knowledge. | §1.3 |
| Tate period / Voloch | automata proof? | Settled (Thakur 1996; Allouche–Thakur 1999) | [Th-PAMQ] |
| Γ(0) for the shtuka gamma, general A | nature of the value? | Open ([Th-CRM] §4.3); outside automata | [Th-CRM] |

### 1.3 MZVs

Chang [C14] proves that MZVs of different weights are linearly independent over
K̄, and Thakur proved ζ(s) ≠ 0. If ζ(s) = α ∈ K̄, then α·1 − ζ(s) = 0 would be
a K̄-linear relation between weight 0 and weight w, which is forbidden. So ζ(s)
is transcendental and, by Christol, not automatic. This is the fork's own
deduction; whether [C14]'s statement explicitly includes weight 0 was not
checked.

For v-adic MZVs, Eulerian ones vanish ([CM21], reported). For deg v = 1,
transcendence of nonzero values is plausibly a corollary of [Chen25]
(unverified). For deg v ≥ 2 it is open. [CM19] p. 4 calls the higher-depth
v-adic criterion an open question (2017).

### 1.4 "In t", done correctly

The Anderson–Thakur function
ω(t) = (−θ)^{1/(q−1)} ∏_{i≥0} (1 − t/θ^{q^i})^{−1} has infinitely many poles. A
meromorphic function algebraic over C_∞(t) has its poles among the zeros of the
leading coefficient of its minimal polynomial, so only finitely many. Hence ω
is transcendental over C_∞(t). This is elementary and surely known.

## 2. The open conjecture and what is known

**Conjecture (Wen–Yao 2002).** Let v be a monic prime of degree d and
n = Σ n_j q^j ∈ Z_p. Then Π_v(n) is algebraic over K iff n_{j+d} = n_j for all
large j.

- **"If":** Thakur's Gross–Koblitz analogue; see also [WY02] §10.
- **"Only if", d = 1:** [WY02] Thm 2; [Th-CRM] Thm 4.5.
- **"Only if", n ∈ Q:** write n = a/b with p ∤ b. The digits of n have eventual
  period ord_b(q). So they are eventually d-periodic ⟺ ord_b(q) | d
  ⟺ b | q^d − 1. Since Π_v(n) = Γ_v(n+1), [CWY24] Cor 1.4.2 (algebraic iff
  b | q^{deg v} − 1) settles this case. Adam 2023 reportedly proves the same
  independently (not verified).
- **Residue-0 case:** Theorem A below.
- **Open:** irrational n whose digits are not eventually d-periodic on some
  residue class r ≠ 0.

## 3. Theorem A ([WY02] Thm 8), with a complete proof

**Theorem A.** If for every r ≠ 0 the digit subsequence (n_{kd+r})_k is
eventually constant, then Π_v(n) is algebraic iff (n_{kd})_k is eventually
constant.

**Proof.**

**Step 0.** Let α be a root of v in F_Q, Q = q^d, and s = t − α. Then K_v
embeds in F_Q((s)), and algebraic over K is the same as algebraic over F_Q(s).
Christol applies over F_Q.

**Step 1.** For a bounded integer sequence c, set F(c) = ∏ (−D_{j,v})^{c_j}.
This converges because −D_{j,v} → 1 at v. F is a homomorphism,
F(p^k c) = F(c)^{p^k}, and finitely supported c give F(c) ∈ K^×.

**Step 2.** The residue classes r ≠ 0 are eventually d-periodic, so their
factors are algebraic by Gross–Koblitz. This reduces to c supported on dN.

**Step 3.** Let δ_k = c_{kd} − c_{(k+1)d}. If δ is eventually 0, we are in the
algebraic case. Otherwise let κ = liminf v_p(δ_k) over nonzero δ_k; then
κ < log_p q. Subtract the periodic sequence γ·1_{dN}, where γ ≡ c_{kd} mod p^κ
for large k; its F is algebraic. Changing finitely many entries costs a factor
in K^×. This leaves p^κ·b, which is valid because x^{p^κ} is algebraic ⟺ x is
algebraic. Now ε_k := b_{kd} − b_{(k+1)d} is not eventually ≡ 0 mod p.

**Step 4.** Use ′ = d/dt. It is continuous, and it preserves algebraic elements
because K_v/K is separable. Since [m]′ = −1, d log D_j = −1/[j]. So
d log D_{j,v} = 1/[j−d] − 1/[j] for j > d, and
d log D_{d,v} = −1/[d] − v′/v. For d | m, [m] = s^{q^m} − s, so
1/[kd] = −1/s − G_k with G_k = s^{Q^k−2}/(1 − s^{Q^k−1}). Then
d log D_{kd,v} = G_k − G_{k−1}. Abel summation, using G_K → 0, gives
L := F(b)′/F(b) ≡ Σ_{k≥2} ε_k G_k (mod F_Q(s)).

**Step 5.** s·Σ ε_k G_k = Σ_m c(m) s^m with c(m) = Σ_{(Q^k−1) | m} ε_k. This
is Thakur's Theorem 4.3 pattern with q replaced by Q.

*Lemma.* For Q ≥ 2 and a, b, c ≥ 1, (Q^c − 1) | Q^a(Q^b − 2) + 1 iff
c | gcd(a, b).

*Proof.* The quantity reduces to X = Q^A + 1 − 2Q^B with A = (a+b) mod c and
B = a mod c. Since |X| < Q^c − 1, divisibility means X = 0, which forces
A = B = 0.

For a > b with ε_a, ε_b ≠ 0, let h be the least divisor of a that does not
divide b and has ε_h ≠ 0. Then
c(Q^a(Q^h−2)+1) − c(Q^b(Q^h−2)+1) = ε_h ≠ 0. So infinitely many kernel
sequences are distinct, the Q-kernel is infinite, and the series is
transcendental. Therefore Π_v(n) is transcendental. ∎

## 4. Where the general case stops

For r ≠ 0, set β_r = α^{q^r} − α ≠ 0. For j ≡ r mod d,
[j] = β_r − s + s^{q^j} is a unit at v. Telescoping gives
d log F(b^{(r)}) ≡ Σ_k ε_k u_{kd+r}, where

u_j = Σ_{m≥1} (−1)^m s^{m q^j} (β_r − s)^{−m−1},

with s^N-coefficient
Σ_m (−1)^m C(N − m(q^j−1), m)·β_r^{−(N−m(q^j−1))−1}. Two steps are needed:

- **(R1)** each residue-r series is transcendental whenever ε is not
  eventually 0;
- **(R2)** different residue classes cannot cancel.

Pure q-power sums such as Σ s^{Q^k} are algebraic in characteristic p, so
transcendence must come from the binomial twist, presumably through Lucas's
theorem. R1 and R2 are open here. A Hasse-derivative refinement of order q^r is
a possible route that was not carried out (see N6).

## 5. Computations (evidence, not proof)

Files: `vadic_gamma_kernel.sage`, `run_checks.sage`, `dlog_res_r.sage`,
`check_telescoping.sage`, and `out_*.txt`; SageMath 10.8. Π_v(n) is computed
exactly mod s^P. The table counts distinct p-kernel prefixes of length L for
k = 0…K.

| (q, v, P, L) | d-periodic (algebraic) | residue-0 non-periodic (Thm A) | residue-1 non-periodic (open) | period d+1, rational | random |
|---|---|---|---|---|---|
| (2, t²+t+1, 8192, 64) | stays at 2 | …, 64, 122–127 | …, 64, 128 | 128 | 128 |
| (3, t²+1, 6561, 27) | ≤ 5 | 1, 3, 8, 22, 81, 242 | 1, 3, 9, 27, 75–81, 243 | 243 | 243 |
| (2, t³+t+1, 8192, 64) | ≤ 15, non-monotone | 128 | 128 | 128 | 128 |

Additional checks:

- The log-derivative formula holds to O(s^398).
- The telescoped identity holds exactly to O(s^{P−2}) for
  (q, d) = (2, 2), (2, 3), (3, 2).
- The residue-1 log-derivative series grow maximally, which supports R1.
- The divisibility lemma had no violations for
  Q ∈ {2, 3, 4, 5, 7, 8, 9, 16, 27}, a, b ≤ 8, c ≤ 12.
- Calibration: the algebraic series Σ s^{Q^k} gives bounded counts (3).

Limits: finite windows cannot prove transcendence, and only small q and d were
tested.

## 6. Carry-forward

The parent's item 9 is carried in refined form (1 carried, 0 excluded). The
off-limits Diophantine topic is excluded by instruction.

## 7. Unresolved questions

1. R1 and R2, which would give the full Wen–Yao conjecture.
2. A period-method proof of Mendès France–Yao.
3. Transcendence of nonzero v-adic MZVs for deg v ≥ 2.
4. The v-adic geometric gamma conjecture ([CWY24] Rem. 1.4.3).
5. Thakur's Γ(0) for the shtuka gamma.

## 8. Sources (inspection level)

**Read:**

- [Th-CRM] Thakur, *Arithmetic of gamma, zeta and multizeta values for
  function fields*,
  https://www.sas.rochester.edu/mth/sites/dinesh-thakur/crmDec12.pdf
  (§§1.2, 4.1–4.3).
- [Th-PAMQ] Thakur, PAMQ 6(1) (2010),
  https://www.sas.rochester.edu/mth/sites/dinesh-thakur/PAMQtate.pdf (p. 16).
- [WY02] Wen–Yao, Acta Arith. 101 (2002) 39–51, doi:10.4064/aa101-1-3 (§1,
  Thm 2 proof, §10).
- [CM19] Chang–Mishiba, IMRN 2019, arXiv:1511.03490 (§1).

**Read via an HTML summary:** [CWY24] arXiv:2407.15024.

**Abstract or search summary only:**

- [CM21] Chang–Mishiba, *On a conjecture of Furusho over function fields*,
  Invent. Math. (2021).
- [Chen25] Y.-T. Chen, MRL 32(4) (2025), arXiv:2308.05825.
- [C14] Chang, Compositio 150 (2014), arXiv:1207.2326.
- [CPTY10] Chang–Papanikolas–Thakur–Yu, Adv. Math. 223 (2010),
  arXiv:0909.0096.
- [ABP04] Anderson–Brownawell–Papanikolas, Ann. of Math. 160 (2004).
- [Ber93] Berthé, JTNB 5 (1993); Astérisque 209 (1992).
- [TWC25] T.-W. Chang, arXiv:2502.01109.
- [TWYZ] Thakur–Wen–Yao–Zhao,
  https://www.sas.rochester.edu/mth/sites/dinesh-thakur/hyperf.pdf.

**Not located:** [Ada23] Adam 2023. Thakur's "Automata and transcendence"
(Contemp. Math.) and *Function Field Arithmetic* ch. 11 could not be
retrieved.

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