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A Lean formalization of Waldschmidt's subgroup theorem with period term for groups with elliptic factors

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theory

Contributed by Shawn Krisman

Version 3 / Nov 07, 2026 / CC BY 4.0

Abstract

We formalize in Lean 4 with Mathlib Waldschmidt's algebraic subgroup theorem with period term (Theorem 4.1 of his 1988 Durham paper) for G=Gad0×Gmd1×E1×⋯×EmG=\mathbb{G}_{\mathrm{a}}^{d_0}\times\mathbb{G}_{\mathrm{m}}^{d_1}\times E_1\times\dots\times E_m, with EjE_j elliptic curves with algebraic invariants, under the additional hypothesis 2N≥d1+2d0+κ2N\ge d_1+2d_0+\kappa. The proof uses only Lean's three standard axioms. Along the way we give what we believe are the first formal proofs of Schneider's theorems on elliptic functions and of the Schneider--Lang criterion, and the first Lean library for the Weierstrass σ\sigma and ζ\zeta functions. Restricted forms of the theorem cover some vector extensions and the Jacobian of y5=x(1−x)y^5=x(1-x). A companion certificate uses these to prove explicit cases of the four exponentials conjecture: for example, 2ϖ2^{\varpi} and 3ϖ3^{\varpi} are not both algebraic, where ϖ\varpi is the real period of y2=4x3−4xy^2=4x^3-4x, and likewise 2B(1/5,1/5)2^{B(1/5,1/5)} and 3B(1/5,1/5)3^{B(1/5,1/5)}.

Provenance statement

The research, paper and lean certificate were all done by Claude Opus 5.5

Tools used

Anthropic
Claude OpusVersion 5.5

References

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Version history

  1. v1 · 30 pagesSubmitted by Shawn KrismanInitial depositSupersededNov 06, 2026
  2. v2 · 31 pagesSubmitted by Shawn KrismanSubmitted as a new versionSupersededNov 07, 2026
  3. v3 · 31 pagesSubmitted by Shawn KrismanSubmitted as a new versionCurrentNov 10, 2026