Formal Jacobians of Universal Artin–Schreier Deformations

Definitions and mathematical statement

In the usual Artin--Schreier terminology, the ground field $k$ has characteristic $p>0$, and a degree-$p$ cover is generically described by an equation \[ y^p-y=f(x),\qquad f(x)\in k(x). \] The source considers a wildly ramified cover $\pi:C\to\mathbb P^1_k$ whose sole branch point is $\infty$. It does not specify $k$, $p$, the equation, or the ramification data of the cover.

The following notation makes one possible meaning of the requested completion precise, conditionally on deformation data that the source does not provide. Suppose a chosen deformation problem admits a universal smooth proper family of geometrically connected curves $\mathcal C\to S=\operatorname{Spec}R$, together with the required deformation of $\pi$. Let \[ J=\operatorname{Pic}^0_{\mathcal C/S},\qquad e:S\longrightarrow J \] be its relative Jacobian and identity section. The formal completion $\widehat J_e$ along the identity section is a commutative formal group over $R$. Its functor on nilpotent commutative $R$-algebras $N$, not necessarily with an identity, is \[ \widehat J_e(N)=\ker\!\left(J(R\oplus N)\longrightarrow J(R)\right), \] where $R\oplus N$ is the augmented $R$-algebra with multiplication $(r,n)(r',n')=(rr',rn'+r'n+nn')$.

Requested investigation, subject to unresolved specifications. Given the cover $\pi$, describe the formal completion of the Jacobian of its universal deformation in formal-group-theoretic terms, along the same lines as the formal-group-theoretic description of the deformation itself mentioned in the source. Under the conditional interpretation above, the object to describe would be $\widehat J_e$, including its dependence on the original cover and the deformation parameters.

The supplied record does not define the deformation functor, its coefficient category, its equivalences or markings, or the asserted universal object. It also does not identify the formal-group-theoretic description to which the comparison refers. Completion along the identity section, used conditionally above, is not explicitly specified in the source; completion along a special fibre or at a closed identity point would require a different specification. Consequently the record determines a research request, but not a unique fully specified mathematical target or a proposition with a prescribed conclusion.

Short English statement

For an Artin--Schreier cover with its only branch point at infinity, the source asks for a formal-group-theoretic description of the Jacobian in the universal deformation family. The deformation problem, the universal family, the locus of completion, and the intended comparison are unspecified. The identity-section completion above is therefore a conditional interpretation requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001111$, title Formal Jacobians of Universal Artin--Schreier Deformations, exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. No individual proposer is supplied. The proposed year is recorded as $2008$.
• Supplied report citation: Mini-Workshop: Group Actions on Curves: Reduction and Lifting (2009), Oberwolfach Reports, pp. 3029--3064, DOI 10.4171/owr/2008/54. Supplied source URL and literature-source link labelled "original OWR report": https://doi.org/10.4171/owr/2008/54. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No problem-specific page, speaker, or passage is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports that the cited report provides background but no resolving construction or classification, and that its literature search found no result settling the general question. This is supplied dated curation, not an independently verified assessment; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and explains that the complete statement was preserved while mathematical notation was placed in delimiters. This is supplied editorial metadata, not evidence that the missing deformation data or intended completion have been mathematically verified.