11.6. Supporting machinery developed in the project🔗

This section records the principal named theorems developed in the project — including pieces that were not used on the critical path to the FLT37 statement above. They are part of the project's deliverable formalisation of cyclotomic-field arithmetic and are referenced from multiple chapters of the wider blueprint.

11.6.1. Sinnott's index formula🔗

Theorem11.6.1.1

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Under the Kummer–Dirichlet regulator identity (the analytic input, proved via the deleted-Fourier formula), Sinnott's index formula holds: [\mathcal{O}_{\mathbb{Q}(\zeta_p)^+}^\times : C^+] = 2^{(p-3)/2}\, h^+.

Lean code for Theorem11.6.1.1●1 theorem

theoremdefined in BernoulliRegular/FLT37/LehmerVandiver/PlusCoprime/Sinnott/IndexFormula.lean

complete

theorem BernoulliRegular.FLT37.Sinnott.sinnottIndexFormula_of_regulatorIdentity.{u_1}
  (p : ℕ) [hp : Fact (Nat.Prime p)] (K : Type u_1) [Field K]
  [NumberField K] [IsCyclotomicExtension {p} ℚ K]
  [NumberField.IsCMField K] (hp_odd : p ≠ 2) (hp_three : 3 ≤ p)
  (h :
    BernoulliRegular.FLT37.Sinnott.SinnottRegulatorIdentity p K hp_odd
      hp_three) :
  BernoulliRegular.FLT37.Sinnott.SinnottIndexFormula p K hp_odd hp_three

theorem BernoulliRegular.FLT37.Sinnott.sinnottIndexFormula_of_regulatorIdentity.{u_1}
  (p : ℕ) [hp : Fact (Nat.Prime p)]
  (K : Type u_1) [Field K] [NumberField K]
  [IsCyclotomicExtension {p} ℚ K]
  [NumberField.IsCMField K]
  (hp_odd : p ≠ 2) (hp_three : 3 ≤ p)
  (h :
    BernoulliRegular.FLT37.Sinnott.SinnottRegulatorIdentity
      p K hp_odd hp_three) :
  BernoulliRegular.FLT37.Sinnott.SinnottIndexFormula
    p K hp_odd hp_three

**Equivalence of Sinnott formula formulations**: the index version
follows from the regulator version (both encode the same content via
`regOfFamily_div_regulator`).

Theorem11.6.1.2

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Sinnott's formula transports the divisibility p \mid h^+ to the cyclotomic-unit index p \mid [E^+ : C^+] (and conversely), giving the unit-side characterisation used to bypass Vandiver's conjecture at p = 37 via a direct Bernoulli-table check.

Theorem 11.6.1.1

Lean code for Theorem11.6.1.2●1 theorem

theoremdefined in BernoulliRegular/CyclotomicUnits/IndexFormula.lean

complete

theorem BernoulliRegular.cyclotomicUnitIndex_primeConductor_pPrimary_of_sinnottIndexFormula
  {p : ℕ} [Fact (Nat.Prime p)] {K : Type} [Field K] [NumberField K]
  [IsCyclotomicExtension {p} ℚ K] [NumberField.IsCMField K]
  (hp_odd : p ≠ 2) (hp_three : 3 ≤ p)
  (hSinnott :
    BernoulliRegular.FLT37.Sinnott.SinnottIndexFormula p K hp_odd
      hp_three) :
  p ∣
      (BernoulliRegular.cyclotomicUnitIndexSubgroup hp_odd
          hp_three).index ↔
    p ∣ BernoulliRegular.hPlus K

theorem BernoulliRegular.cyclotomicUnitIndex_primeConductor_pPrimary_of_sinnottIndexFormula
  {p : ℕ} [Fact (Nat.Prime p)] {K : Type}
  [Field K] [NumberField K]
  [IsCyclotomicExtension {p} ℚ K]
  [NumberField.IsCMField K]
  (hp_odd : p ≠ 2) (hp_three : 3 ≤ p)
  (hSinnott :
    BernoulliRegular.FLT37.Sinnott.SinnottIndexFormula
      p K hp_odd hp_three) :
  p ∣
      (BernoulliRegular.cyclotomicUnitIndexSubgroup
          hp_odd hp_three).index ↔
    p ∣ BernoulliRegular.hPlus K

P-primary form of the real cyclotomic-unit index formula, conditional on
the existing explicit Sinnott index formula.

11.6.2. Thaine's theorem and circular units🔗

Theorem11.6.2.1

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Sinnott's and Washington's two conventions for the group of circular units of \mathbb{Q}(\zeta_p)^+ coincide. The identification underwrites the use of either convention in downstream Thaine-style annihilation arguments.

Lean code for Theorem11.6.2.1●1 theorem

theoremdefined in BernoulliRegular/Thaine/CircularUnits.lean

complete

theorem BernoulliRegular.Thaine.circularSubgroupKplus_eq_sinnott_eq_washington.{u_1}
  (p : ℕ) [Fact (Nat.Prime p)] (K : Type u_1) [Field K] [NumberField K]
  [IsCyclotomicExtension {p} ℚ K] [NumberField.IsCMField K]
  (hp_three : 3 ≤ p) :
  BernoulliRegular.Thaine.circularSubgroupKplus p K hp_three =
    BernoulliRegular.Thaine.circularSubgroupKplus p K hp_three

theorem BernoulliRegular.Thaine.circularSubgroupKplus_eq_sinnott_eq_washington.{u_1}
  (p : ℕ) [Fact (Nat.Prime p)]
  (K : Type u_1) [Field K] [NumberField K]
  [IsCyclotomicExtension {p} ℚ K]
  [NumberField.IsCMField K]
  (hp_three : 3 ≤ p) :
  BernoulliRegular.Thaine.circularSubgroupKplus
      p K hp_three =
    BernoulliRegular.Thaine.circularSubgroupKplus
      p K hp_three

**Marker theorem: for prime conductor, `circularSubgroupKplus` serves
as both Sinnott's `C_S(K⁺)` and Washington's `C_cl(K⁺)`**.

The two are definitionally the same group when the conductor is prime
([Wash97 §8.2] / [Sinnott 1980]). The naming `circularSubgroupKplus`
avoids the distinction.

Theorem11.6.2.2

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For every prime \ell in the Chebotarev-positive density set \{\ell : \ell \equiv 1 \pmod p, \ell \text{ totally split in some auxiliary field}\}, the Thaine auxiliary construction produces an annihilator of the class group \mathrm{Cl}(\mathbb{Q}(\zeta_p)^+) drawn from a circular unit at \ell. Combined with a finite-set exclusion, this gives existence of Thaine auxiliaries outside any fixed finite set of bad primes.

Lean code for Theorem11.6.2.2●1 theorem

theoremdefined in BernoulliRegular/Thaine/AuxiliaryPrimes.lean

complete

theorem BernoulliRegular.Thaine.thaineAuxiliaryExistence_of_prime (p : ℕ)
  (hp : Nat.Prime p) (n : ℕ) (hn : 1 ≤ n) :
  BernoulliRegular.Thaine.ThaineAuxiliaryExistence p n

theorem BernoulliRegular.Thaine.thaineAuxiliaryExistence_of_prime
  (p : ℕ) (hp : Nat.Prime p) (n : ℕ)
  (hn : 1 ≤ n) :
  BernoulliRegular.Thaine.ThaineAuxiliaryExistence
    p n

**Generic Thaine-auxiliary existence via Dirichlet density**
(unconditional). For any prime `p` and any `n ≥ 1`, there exists a
prime ℓ ≠ p with `p^n ∣ ℓ - 1`. Proof: Dirichlet's theorem (mathlib's
`Nat.setOf_prime_and_eq_mod_infinite`) says the set of primes ℓ with
`(ℓ : ZMod (p^n)) = 1` is infinite, hence in particular non-empty;
moreover, `ℓ ≠ p` is automatic since `(p : ZMod (p^n)) = 0 ≠ 1` for any
prime `p` (since `p ∣ p^n` already, so `p ≡ 0` not `1` modulo `p^n`).

Theorem11.6.2.3

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The set of Thaine auxiliary primes is infinite.

Theorem 11.6.2.2

Lean code for Theorem11.6.2.3●1 theorem

theoremdefined in BernoulliRegular/Thaine/AuxiliaryPrimes.lean

complete

theorem BernoulliRegular.Thaine.infinite_setOf_thaineAuxiliary (p : ℕ)
  (hp : Nat.Prime p) (n : ℕ) (hn : 1 ≤ n) :
  {ℓ |
      Nat.Prime ℓ ∧
        BernoulliRegular.Thaine.IsThaineAuxiliary p n ℓ}.Infinite

theorem BernoulliRegular.Thaine.infinite_setOf_thaineAuxiliary
  (p : ℕ) (hp : Nat.Prime p) (n : ℕ)
  (hn : 1 ≤ n) :
  {ℓ |
      Nat.Prime ℓ ∧
        BernoulliRegular.Thaine.IsThaineAuxiliary
          p n ℓ}.Infinite

**Infinitude of Thaine-auxiliary primes** (unconditional). For any
prime `p` and any `n ≥ 1`, the set of primes ℓ with `IsThaineAuxiliary
p n ℓ` is infinite. Proof: the set of primes ℓ with `(ℓ : ZMod (p^n)) = 1`
is infinite by Dirichlet density, and removing the (at most one) prime
equal to `p` (which doesn't satisfy the congruence anyway, by the
argument in `thaineAuxiliaryExistence_of_prime`) leaves an infinite set.

11.6.3. Reflection / Spiegelungssatz🔗

Theorem11.6.3.1

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The weak reflection principle on the unit side: divisibility of the plus class number p \mid h^+ implies a non-trivial \epsilon_i-eigencomponent in the minus class group for some reflection-component index i. This is the project's bridge from unit-side data to class-side data, used as one input to the Lehmer–Vandiver descent.

Lean code for Theorem11.6.3.1●1 theorem

theoremdefined in BernoulliRegular/CyclotomicUnits/UnitsReflection.lean

complete

theorem BernoulliRegular.weakReflection_dvd_hMinus_of_dvd_hPlus_units {p : ℕ}
  [Fact (Nat.Prime p)] {K : Type} [Field K] [NumberField K]
  [IsCyclotomicExtension {p} ℚ K] [NumberField.IsCMField K]
  (hp_odd : p ≠ 2) (hplus : p ∣ BernoulliRegular.hPlus K) :
  p ∣ BernoulliRegular.hMinus K

theorem BernoulliRegular.weakReflection_dvd_hMinus_of_dvd_hPlus_units
  {p : ℕ} [Fact (Nat.Prime p)] {K : Type}
  [Field K] [NumberField K]
  [IsCyclotomicExtension {p} ℚ K]
  [NumberField.IsCMField K]
  (hp_odd : p ≠ 2)
  (hplus : p ∣ BernoulliRegular.hPlus K) :
  p ∣ BernoulliRegular.hMinus K

CU-18: weak reflection by contrapositive from the cyclotomic-unit route.

Theorem11.6.3.2

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Conditional on the weak reflection principle, p \mid h is equivalent to p dividing a Bernoulli numerator in the Kummer range — an alternative form of Kummer's criterion routed through the reflection / Spiegelungssatz machinery rather than the Sinnott / circular-unit route.

Theorem 11.6.3.1

Lean code for Theorem11.6.3.2●1 theorem

theoremdefined in BernoulliRegular/Reflection/ClassNumberCriterion.lean

complete

theorem BernoulliRegular.dvd_h_iff_exists_dvd_bernoulli_of_weakReflection.{u_1}
  {p : ℕ} [hp : Fact (Nat.Prime p)] (hp_odd : p ≠ 2) {K : Type u_1}
  [Field K] [NumberField K] [IsCyclotomicExtension {p} ℚ K]
  [NumberField.IsCMField K] :
  p ∣ BernoulliRegular.h K ↔
    ∃ k, 1 ≤ k ∧ 2 * k ≤ p - 3 ∧ ↑p ∣ (bernoulli (2 * k)).num

theorem BernoulliRegular.dvd_h_iff_exists_dvd_bernoulli_of_weakReflection.{u_1}
  {p : ℕ} [hp : Fact (Nat.Prime p)]
  (hp_odd : p ≠ 2) {K : Type u_1}
  [Field K] [NumberField K]
  [IsCyclotomicExtension {p} ℚ K]
  [NumberField.IsCMField K] :
  p ∣ BernoulliRegular.h K ↔
    ∃ k,
      1 ≤ k ∧
        2 * k ≤ p - 3 ∧
          ↑p ∣ (bernoulli (2 * k)).num

**T045a/T045b via weak reflection.**

This is the class-number/Bernoulli bridge using the explicit weak-reflection
component theorem rather than a prebuilt `ReflectionMinusNontrivialityBridge`.
The only reflection input is `weakReflection_componentNontrivial`.

11.6.4. The CM splitting h-plus divides h🔗

Theorem11.6.4.1

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In the CM field \mathbb{Q}(\zeta_p)/\mathbb{Q}(\zeta_p)^+, the plus class number divides the full class number, h(\mathbb{Q}(\zeta_p)^+) \mid h(\mathbb{Q}(\zeta_p)). This is proved unconditionally via faithful flatness of \mathcal{O}_{\mathbb{Q}(\zeta_p)}/\mathcal{O}_{\mathbb{Q}(\zeta_p)^+} and the injectivity of the induced map of class groups.

Lean code for Theorem11.6.4.1●1 theorem

theoremdefined in BernoulliRegular/TotallyRealSubfield/ClassGroup.lean

complete

theorem BernoulliRegular.hPlus_dvd_h.{u_1} (p : ℕ) [hp : Fact (Nat.Prime p)]
  (hp_odd : p ≠ 2) (K : Type u_1) [Field K] [NumberField K]
  [IsCyclotomicExtension {p} ℚ K] [NumberField.IsCMField K] :
  BernoulliRegular.hPlus K ∣ BernoulliRegular.h K

theorem BernoulliRegular.hPlus_dvd_h.{u_1} (p : ℕ)
  [hp : Fact (Nat.Prime p)]
  (hp_odd : p ≠ 2) (K : Type u_1)
  [Field K] [NumberField K]
  [IsCyclotomicExtension {p} ℚ K]
  [NumberField.IsCMField K] :
  BernoulliRegular.hPlus K ∣
    BernoulliRegular.h K

The class number of `K⁺` divides the class number of `K`.