Formal Group Isomorphisms and Positive-Characteristic CIT

Definitions and mathematical statement

Retain the setting of the source's Theorem 2. Let $(K,D)$ be its Hasse--Schmidt field with absolute-constant field $C$, and let $A,B$ be commutative algebraic groups over $C$. In a one-parameter Hasse--Schmidt convention, $D=(D_i)_{i\geq0}$ has $D_0=\operatorname{id}$ and \[ D_r(xy)=\sum_{i+j=r}D_i(x)D_j(y),\qquad C=\bigcap_{i\geq1}\ker D_i. \] The source does not specify iterativity or other field/derivation hypotheses. Let $\phi:\widehat A\to\widehat B$ be the intended nowhere-algebraic formal isomorphism at the identities. The source writes $\phi:A\to B$, but a formal map must not be silently treated as an algebraic morphism on all points.

For $a\in A(K)$, $b\in B(K)$, retain the differential equation and dimension condition \[ U_\phi(l_D^A(a))=l_D^B(b),\qquad \operatorname{trdeg}_C(a,b)\leq n. \] The logarithmic-derivative maps $l_D^A,l_D^B$, induced map $U_\phi$, nowhere-algebraic condition and integer $n$ are not defined; in particular $n$ is not explicitly identified with either group dimension. Under the complete hypotheses of the missing Theorem 2, the supplied record reports a conclusion \[ b\in B_0(K)+c_B, \qquad B_0<B\text{ a proper algebraic subgroup over }C, \quad c_B\in B(C). \] This is attributed to that theorem's omitted setup, not proved from the abbreviated displayed hypotheses alone. The three requests are as follows.

1. If $\phi^{-1}$ is also nowhere algebraic, can the conclusion be strengthened on the $A$-side? The cleaned "likewise" reading asks for a proper algebraic subgroup $A_0<A$ over $C$ and a constant translate $c_A\in A(C)$ with $a\in A_0(K)+c_A$. The precise symmetry of the theorem's other hypotheses is not given, so application to the inverse is not asserted automatically.
2. Can the formal-map case from the multiplicative group $\mathbb G_m$ to an ordinary elliptic curve prove weak $\mathrm{CIT}_p$? The original statement expressly says "in characteristic zero," whereas the clean version drops that phrase and the title suggests positive characteristic. The meaning of ordinary, the relevant formal map and the exact weak $\mathrm{CIT}_p$ assertion are absent. The characteristic-zero qualifier is retained as original-source data and the tension is flagged, not silently changed to characteristic $p$.
3. What invariant should replace $\dim_{\mathbb{Q}}$ in a Schanuel-style reformulation? The source gives neither the candidate vector space nor the proposed inequality, so no dimension over a different field or endomorphism ring is chosen.

The literature triage checked 2026-08-21 reports no authoritative complete resolution found. Its open label and reported source verification remain dated metadata, not an independently verified theorem or a restoration of Theorem 2's missing assumptions.

Short English statement

The three tasks ask for a matching proper-subgroup coset on the first group, an application of the multiplicative-to-elliptic formal-map case to weak $\mathrm{CIT}_p$, and the correct dimension invariant for a Schanuel formulation. The theorem setup and several key definitions are missing. The original characteristic-zero qualifier is preserved despite its omission from the cleaned question.

Sources

• Local UnsolvedMath registry, permanent ID 30000670, source code catalogue entry, "Formal Group Isomorphisms and Positive-Characteristic CIT," including the three original Theorem 2 questions, the original characteristic-zero qualifier and expanded cleaned hypotheses.
• Model Theory and Groups (2007), Oberwolfach Reports, pp. 83--138, DOI 10.4171/owr/2007/02: https://doi.org/10.4171/owr/2007/02. This is also the sole supplied literature reference.
• Literature triage checked 2026-08-21; record updated 2026-09-16. No external source has been visited or independently checked here.