Tate defects and character splitting for critical groups of abelian graph covers

Definitions and mathematical statement

For a finite connected graph $X$, let $J(X)=\operatorname{Div}^0(X)/\operatorname{im}L_X$ be its critical group. The source asks, through Vic Reiner, how critical-group homomorphisms associated with Galois graph coverings relate to class field theory when the deck group is abelian. It supplies no particular reciprocity statement or proposed classification.

Let $p:Y\to X$ be a connected regular unramified graph cover with finite abelian deck group $A$, $|A|=n$. Pullback sums over vertex fibers and pushforward sends each vertex to its image. The induced maps satisfy \[ p_*p^*=n,\qquad p^*p_*=N_A:=\sum_{a\in A}a. \] Here pullback on critical groups is injective and pushforward surjective. Put $M=J(Y)$ and $I_AM=\langle am-m:a\in A,m\in M\rangle$. The attached aid's two integral defect identifications are \[ J(Y)^A/p^*J(X)\simeq\widehat H^0(A,M)=M^A/N_AM, \] \[ \ker(p_*)/I_AM\simeq\widehat H^{-1}(A,M)=\ker N_A/I_AM. \] Both groups are killed by $n$. This compares ambiguous classes and a Prym coinvariant defect with Tate norm formalism, without asserting an arithmetic reciprocity theorem.

For a prime $\ell\nmid n$, write $M_\ell$ for the $\ell$-primary component. The transfer idempotent $e=n^{-1}N_A$ gives the canonical splitting \[ J(Y)_\ell=p^*J(X)_\ell\oplus\ker(p_*)_\ell. \] After extension to a splitting unramified coefficient ring containing the required character values, the character idempotents $e_\chi=n^{-1}\sum_{a\in A}\chi(a)^{-1}a$ refine this to the full abelian-character decomposition. The integral defects can occur at primes dividing the covering degree, where the idempotent need not be integral.

For the cyclic cover $C_{mr}\to C_m$, $m\ge3$, $r\ge1$, the aid identifies $J(C_{mr})\simeq{\mathbb{Z}}/(mr)$, $J(C_m)\simeq{\mathbb{Z}}/m$, with pushforward reduction modulo $m$ and pullback multiplication by $r$. The deck action on $J(C_{mr})$ is trivial, so both Tate defects are ${\mathbb{Z}}/r$. The norm exact sequence \[ 0\longrightarrow{\mathbb{Z}}/r\longrightarrow{\mathbb{Z}}/(mr)\longrightarrow{\mathbb{Z}}/m\longrightarrow0 \] splits exactly when $\gcd(m,r)=1$. Since $C_m$ admits these connected cyclic covers for every $r$, its finite critical group alone cannot classify all abelian graph covers by its quotients. The source's broader class-field comparison remains a question beyond this precise bridge and obstruction.

Short English statement

Relate abelian graph-cover critical groups to class-field norm ideas. The supplied aid identifies two Tate-cohomological defects and prime-to-degree splittings, while cycle covers show why a fixed finite critical group cannot serve as a complete classifying reciprocity group.

Sources

• Local UnsolvedMath registry, permanent ID $20000993$, source code catalogue entry; canonical aim-combinatorics-notes.json, notes[117].
• American Institute of Mathematics, Problems from the AIM Chip-Firing Workshop, problem 28, attributed to Vic Reiner: https://aimath.org/pastworkshops/chipfiringproblems.pdf. The source cites covering-map results as [31], whose complete entry is absent; an appended page header is not part of the question. No PDF inspection is claimed here.
• Attached research attempt 1: partial result, run and registry status partially solved, rigor proved, novelty confidence low, classification PARTIAL-PROGRESS. Its Tate identifications and cycle calculation are machine generated and not peer reviewed; independently verify before citation. No authoritative full references are supplied.