Weierstrass Curve tate A₄ eq eval Int
WeierstrassCurve.tateA₄_eq_evalInt
Plain-language statement
The Lambert series rearrangement ∑_{n≥1} n³qⁿ/(1-qⁿ) = ∑_{n≥1} σ₃(n)qⁿ for |q| < 1: the defining series of tateA₄ is the evaluation of the formal series a₄(q) = -5s₃(q) ∈ ℤ⟦q⟧.
Exact Lean statement
theorem WeierstrassCurve.tateA₄_eq_evalInt (q : k) (hq : valuation k q < 1) :
tateA₄ q = TateCurve.evalInt q TateCurve.a₄FormalFormal artifact
Lean source
theorem WeierstrassCurve.tateA₄_eq_evalInt (q : k) (hq : valuation k q < 1) : tateA₄ q = TateCurve.evalInt q TateCurve.a₄Formal := by have hF : ∀ n, PowerSeries.coeff n TateCurve.a₄Formal = ∑ d ∈ n.divisors, -5 * (d : ℤ) ^ 3 := by intro n rw [TateCurve.coeff_a₄Formal, ArithmeticFunction.sigma_apply] push_cast rw [Finset.mul_sum] rw [← TateCurve.tsum_lambert_eq_evalInt q hq _ hF] simp only [tateA₄] rw [← tsum_mul_left] exact tsum_congr fun m ↦ by push_cast; ring- Project
- Fermat's Last Theorem
- License
- Apache-2.0
- Commit
- 8dd808888295
- Source
- FLT/KnownIn1980s/EllipticCurves/TateCurve.lean:256-267
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