Removing the not-a-$p$-th-power hypothesis on the trace formula and the structure-theorem dependency in Serre's characteristic-$p$ proof of the modulus theorem (Algebraic Groups and Class Fields, Chapter III)

In Chapter III of Algebraic Groups and Class Fields, Serre proves the trace formula $\mathrm{Tr}_g(\alpha_i) = f'^*(\omega_i)$ (Lemma 4, no. 6) only for rational functions $g$ that are not $p$-th powers in characteristic $p$, remarking on p. 39 that the lemma is likely valid for arbitrary $g$ (cf. Barsotti, thm. 4.2), and proves the modulus theorem (Theorem 1: every regular map from a curve minus a finite set of points to a commutative algebraic group admits a modulus) in characteristic $p$ only through a case reduction via Chevalley's structure theorem for algebraic groups and Kolchin's decomposition of commutative linear groups, closing the chapter's bibliographic note with the wish to avoid these structure theorems as one can in characteristic 0. The problem is to settle and repair both gaps: establish (or refute) the trace formula for arbitrary $g$ in characteristic $p$, and give a proof of the modulus theorem in characteristic $p$ that does not rest on the structure theory of commutative algebraic groups.

Background

Let $k$ be an algebraically closed field and $X$ a smooth, projective, connected curve over $k$ with function field $F = k(X)$. For a finite separable extension $E \subset F$, where $E = k(Y)$ is the function field of a curve $Y$ (equivalently, a non-constant morphism $X \to Y$ makes $X$ a ramified covering of $Y$), Serre defines a trace on differential forms as follows (Algebraic Groups and Class Fields, Chapter II, no. 12): since $F/E$ is separable, the natural map $\Omega^1_E \otimes_E F \to \Omega^1_F$ is an isomorphism, and applying the field trace $\mathrm{Tr}_{F/E}$ to the second factor transports a differential of $X$ to one of $Y$; explicitly, for $\omega = f\,\mathrm{d}\varphi$ with $\varphi$ a separating element of $E$, one has $\mathrm{Tr}_{F/E}(\omega) = \mathrm{Tr}_{F/E}(f)\,\mathrm{d}\varphi$. This trace satisfies the residue formula $\sum_{Q \mapsto P}\mathrm{Res}_Q(\omega) = \mathrm{Res}_P(\mathrm{Tr}(\omega))$ (Chapter II, no. 12, Lemma 4). In characteristic $p > 0$ the construction breaks down for inseparable coverings: a rational function $g = h^p$ has $\mathrm{d}g = 0$, the extension $F/k(g)$ is not separable, and the map on differentials is no longer an isomorphism; the standard auxiliary device in this setting is the Cartier operator, which for $\omega = f\,\mathrm{d}t$ with $f = \sum_{j=0}^{p-1} f_j^{\,p} t^j$ is defined by $C(\omega) = f_{p-1}\,\mathrm{d}t$ and inverts the $p$-power operation on differentials, but it plays no role in Serre's book. The book instead develops geometric class field theory for curves over $k$: to each effective divisor $\mathfrak{m}$ on $X$ (a modulus) it attaches the generalized Jacobian $J_\mathfrak{m}$, an algebraic group fitting into an exact sequence $0 \to L_\mathfrak{m} \to J_\mathfrak{m} \to J \to 0$ with $L_\mathfrak{m}$ a commutative linear algebraic group determined by $\mathfrak{m}$, whose points are degree-zero divisors prime to the support of $\mathfrak{m}$ modulo the principal divisors of functions congruent to 1 modulo $\mathfrak{m}$ (French original Groupes algébriques et corps de classes, Hermann 1975; English translation, GTM 117, 1988). Chapter III proves the foundational theorem of the subject, due to Rosenlicht (Theorem 1): every regular map $f : X - S \to G$ from the complement of a finite set $S$ of points to a commutative algebraic group $G$ admits a modulus supported on $S$ — equivalently (cf. Chapter I, Theorem 1), $f$ factors through some generalized Jacobian $J_\mathfrak{m}$; this is what makes the generalized Jacobians the receptors of the reciprocity law of geometric class field theory (Chapter VI). The characteristic-0 proof (no. 6, Proposition 10) runs through differential forms: pulling back a basis $\omega_1, \dots, \omega_r$ of translation-invariant differentials on $G$ along $f$ and putting $\alpha_i = f^*(\omega_i)$, it invokes the trace formula of Lemma 4, $\mathrm{Tr}_g(\alpha_i) = f'^*(\omega_i)$, where $g$ is a non-constant rational function on $X$ vanishing modulo $\mathfrak{m}$, viewed as a finite covering $g : X \to \mathbb{A}$ of the projective line $\mathbb{A} = \mathbb{P}^1$, and $f' = \mathrm{Tr}_g f$ is the trace of the map $f$; the residue formula then forces $f'^*(\omega_i) = 0$, hence $f'$ constant, hence $f$ has a modulus. The proof of Lemma 4 is valid in any characteristic provided $g$ is not a $p$-th power, and Serre remarks (p. 39): 'Indeed, it is likely that lemma 4 is valid without hypothesis on g, cf. Barsotti [6], thm. 4.2' — Barsotti's Repartitions on abelian varieties (1958) announced exactly such a device, a trace of a differential from a field to its $p$-th power. In characteristic $p$ the book abandons the differential route: no. 7 reduces Theorem 1, via Chevalley's structure theorem (a connected algebraic group contains a connected linear normal subgroup with abelian-variety quotient, Proposition 11) and Kolchin's decomposition of connected commutative linear groups into a torus and a unipotent group (Proposition 12), to two special cases, treated in nos. 8–10 by regularity arguments on symmetric products that apply to arbitrary coverings. The bibliographic note at the end of the chapter records that Rosenlicht first proved Theorem 1, that Chevalley himself never published his proof, and closes: 'It would be interesting to avoid these structure theorems, as one can do in characteristic 0.' The two questions — is Lemma 4 valid for arbitrary $g$ in characteristic $p$, and can the modulus theorem be proved in characteristic $p$ without the structure theorems — are thus recorded by Serre as, respectively, likely-true-but-unproved and desirable-to-avoid.

Problem Statement

Two linked questions from Chapter III of Serre's Algebraic Groups and Class Fields. Throughout, $k$ is an algebraically closed field, $X$ a smooth projective connected curve over $k$, $S \subset X$ a finite set of points, and $f : X - S \to G$ a regular map to a commutative algebraic group $G$; let $\omega_1, \dots, \omega_r$ be a basis of the translation-invariant differential forms on $G$ and $\alpha_i = f^*(\omega_i)$ their pullbacks to $X - S$. (1) The trace formula. For a non-constant rational function $g$ on $X$, let $g : X \to \mathbb{A}$ denote the corresponding finite covering of the projective line $\mathbb{A} = \mathbb{P}^1$, and let $f' = \mathrm{Tr}_g f : \mathbb{A} - \{0\} \to G$ be the trace of the map $f$, defined for every covering by $f'(a) = f\big((g)_a\big)$ where $(g)_a$ is the divisor of zeros of $g - a$ (Chapter III, no. 2, Proposition 3, and no. 6, Proposition 8). Lemma 4 of Chapter III, no. 6 states the trace formula $\mathrm{Tr}_g(\alpha_i) = f'^*(\omega_i)$, where $\mathrm{Tr}_g$ is the trace on differential forms of the covering $g : X \to \mathbb{A}$ as defined in Chapter II, no. 12 — a definition that applies exactly when $g$ is not a $p$-th power in $k(X)$, i.e. when the extension $k(X)/k(g)$ is separable. Serre's proof is valid in any characteristic under that hypothesis, and he remarks (p. 39): 'Indeed, it is likely that lemma 4 is valid without hypothesis on g, cf. Barsotti [6], thm. 4.2.' Decide this: either (i) define a trace on differential forms for coverings whose defining function is a $p$-th power — extending the trace of Chapter II, no. 12, compatible with its residue formula — and prove $\mathrm{Tr}_g(\alpha_i) = f'^*(\omega_i)$ in that setting for every such $X$, $f$, $G$, and $g$; or (ii) refute the extension, by proving that no trace definition satisfying these admissibility conditions — extending the trace of Chapter II, no. 12 and compatible with its residue formula — makes the formula hold in general: for every candidate definition with these properties, exhibit an explicit curve, map, commutative algebraic group, invariant differential, and function $g = h^p$ for which the formula fails. (2) The proof structure. Theorem 1 of Chapter III states that every regular map $f : X - S \to G$ as above admits a modulus supported on $S$: an effective divisor $\mathfrak{m}$ with support $S$ such that, by the criterion of Proposition 9, $\mathrm{Tr}_g f$ is constant for every non-constant rational $g \equiv 0 \bmod \mathfrak{m}$ (equivalently, by Chapter I, Theorem 1, $f$ factors through the generalized Jacobian $J_\mathfrak{m}$). It is proved in characteristic 0 directly (Proposition 10), but in characteristic $p$ only through the reduction of no. 7, which uses Chevalley's structure theorem (Proposition 11: a connected algebraic group contains a connected linear normal subgroup with abelian-variety quotient) and Kolchin's decomposition of connected commutative linear groups into a torus and a unipotent group (Proposition 12). Give a proof of Theorem 1 valid in characteristic $p$ that does not pass through these structure theorems, in the sense of the chapter's bibliographic note: 'It would be interesting to avoid these structure theorems, as one can do in characteristic 0.' The two questions share one objective: to free the foundations of geometric class field theory in characteristic $p$ from auxiliary hypotheses — separability of the covering in the trace formula, and the structure theory of commutative algebraic groups — that the characteristic-0 theory does not need. They are linked: the characteristic-0 proof of Proposition 10 invokes Lemma 4 for every non-constant $g \equiv 0 \bmod \mathfrak{m}$, and in characteristic $p$ such $g$ may be $p$-th powers; moreover its final step — that a map pulling back every invariant differential to zero is constant — uses characteristic 0, since in characteristic $p$ such a vanishing pullback only forces the map to factor through a Frobenius $\lambda \mapsto \lambda^p$ (Chapter III, no. 8, Lemma 7, which repairs the step for maps from the projective line only by an additional degree argument).

References

1. Jean-Pierre Serre, Maps From a Curve to a Commutative Group (Chapter III of Algebraic Groups and Class Fields, Graduate Texts in Mathematics 117), Springer, 1988, pp. 27–57, DOI 10.1007/978-1-4612-1035-1_3, https://link.springer.com/chapter/10.1007/978-1-4612-1035-1_3 (English translation of Groupes algébriques et corps de classes, Hermann, 1975; corrected second printing 1997)
2. Iacopo Barsotti, Repartitions on abelian varieties, Illinois Journal of Mathematics 2 (1958), pp. 43–70, DOI 10.1215/ijm/1255380832, https://doi.org/10.1215/ijm/1255380832 (item [6] of the bibliography of Serre's book)
3. Jean-Pierre Serre, Morphismes universels et variété d'Albanese, exposé 10, Séminaire Claude Chevalley, tome 4 (1958–1959), 1959, 22 p., no DOI, https://numdam.org/item/SCC_1958-1959__4__A10_0/
4. Jean-Pierre Serre, Morphismes universels et différentielles de troisième espèce, exposé 11, Séminaire Claude Chevalley, tome 4 (1958–1959), 1959, 8 p., no DOI, https://numdam.org/item/SCC_1958-1959__4__A11_0/
5. Brian Conrad, A modern proof of Chevalley's theorem on algebraic groups, 2002, unpublished note dated April 2, 2002, Stanford mathematics department, no DOI, https://math.stanford.edu/~conrad/papers/chev.pdf
6. Henrik Russell, Albanese varieties with modulus over a perfect field, Algebra & Number Theory 7 (2013), no. 4, pp. 853–892, DOI 10.2140/ant.2013.7.853, https://doi.org/10.2140/ant.2013.7.853 (author's preprint arXiv:0902.2533v3)
7. John B. Little, Formal Abel relations for curves in characteristic p, 2026, arXiv preprint arXiv:2607.09471v1 [math.AG], working paper of 10 July 2026, no DOI, https://arxiv.org/abs/2607.09471