Arithmetic, Homotopy, and Geometry

The Homology–Homotopy Frontier in Arithmetic Geometry

Course level

Advanced graduate / beginning research seminar in arithmetic geometry.

Format

14-week course.

Course level: advanced graduate / beginning research seminar in arithmetic geometry. Format: fourteen weeks, two 90-minute lectures per week, with a weekly 60–90 minute problem, computation, or reading seminar. The course assumes basic algebraic number theory, schemes and morphisms, smooth projective curves and Jacobians, Galois theory and profinite groups, homological algebra, basic sheaf cohomology, and elementary (p)-adic analysis. Étale cohomology, algebraic groups, category theory, motives, deformation theory, (p)-adic Hodge theory, and homotopical algebra are useful but are developed as needed.

The governing problem is to determine how much arithmetic information is visible after successive linearizations and how much requires retention of genuinely non-abelian structure. The recurring comparison is among (H^1), Tate modules, Jacobians, étale fundamental groups, unipotent completions, Selmer schemes, arithmetic local systems, monodromy, motivic fundamental groups, and period maps. The course is not organized around the slogan that “higher is better.” At every stage students must identify a concrete arithmetic obstruction that survives the current representation and determine whether passing to a richer structure actually removes that obstruction. This retains the original course’s emphasis on anabelian reconstruction, non-abelian Chabauty, Tannakian reconstruction, motivic fundamental groups, Selmer varieties, iterated (p)-adic integration, arithmetic monodromy, period maps, and special loci.

By the end of the course students should be able to distinguish cohomological, fundamental-group, and higher-homotopical arithmetic invariants; explain the passage (\pi_1\to\pi_1^{ab}\to H^1) and identify what is lost; formulate Chabauty–Coleman through Jacobians, (p)-adic logarithms, and abelian Kummer maps; explain Kim’s non-abelian lift through unipotent fundamental groups and non-abelian Kummer maps; work with pro-unipotent étale and de Rham groupoids; use basic Tannakian reconstruction; construct local and global Selmer conditions and the Chabauty–Kim diagram; understand iterated Coleman integrals as coordinates on non-abelian period/Kummer maps; analyze arithmetic monodromy; compare anabelian and motivic reconstruction; and determine when arithmetic information appears only beyond first cohomology.

Week 1 — Homology, homotopy, and arithmetic information.

The first lecture develops the basic hierarchy (\pi_1(X)\to\pi_1(X)^{ab}\to H_1(X)), then moves immediately to its arithmetic analogues. Students compare (\pi_1^{\acute et}(X_{\bar K})), (H^1_{\acute et}(X_{\bar K},\mathbf Q_\ell)), and (T_\ell J_X), asking not merely how they are related but which distinctions disappear under each map. Étale fundamental groups, geometric versus arithmetic fundamental groups, and Galois actions are introduced through the exact sequence (1\to\pi_1(X_{\bar K})\to\pi_1(X)\to G_K\to1).

The second lecture develops sections and rational points, linear representations of (G_K), abelianization, and the difference between information encoded by a group and by its representation theory. The problem session takes one smooth curve (X/K) and asks students to construct an explicit information-loss table for (\pi_1^{\acute et}(X_{\bar K})), (\pi_1^{ab}), (H^1_{\acute et}), and (T_\ell J_X). The guiding question is the one already built into the source syllabus: what survives abelianization, and what genuinely requires non-abelian structure?

Week 2 — Curves, Jacobians, and the first arithmetic compression.

The first lecture covers curves of genus (g), divisors, Picard groups, (\operatorname{Pic}^0), the Jacobian (J), and the Abel–Jacobi map (X(K)\to J(K)). The goal is to make precise why the Jacobian is simultaneously powerful and lossy: it converts the nonlinear geometry of a curve into the arithmetic of an abelian variety.

The second lecture develops Mordell–Weil, rank versus dimension, reduction modulo primes, local-global comparison, and the (p)-adic Lie structure of (J(K_p)). Students derive the codimension phenomenon produced by (\operatorname{rank}J(K)<\dim J). The problem seminar works through genus-two and genus-three examples and asks where rational-point information is retained by (J) and where it has been forgotten. This provides the geometric substrate for Chabauty without introducing Chabauty itself prematurely.

Week 3 — Kummer maps, Galois cohomology, and Selmer conditions.

The first lecture develops classical Kummer theory, Tate modules (T_pJ), and the passage from arithmetic points to Galois classes. Students construct (J(K)\to H^1(G_K,T_pJ)) and the corresponding local maps (J(K_v)\to H^1(G_v,T_pJ)). The purpose is to reinterpret arithmetic geometry through cohomological parameter spaces.

The second lecture introduces Selmer groups, localization maps, local conditions, and the Bloch–Kato viewpoint. Classical rational-point problems are rewritten schematically as comparisons between global and local cohomology. This week is moved before Chabauty because the Kummer reinterpretation is the actual interface through which classical Chabauty admits a non-abelian lift. The problem session follows a point through (X(K)\to J(K)\to H^1(G_K,T_pJ)) and asks which step is injective, which is representational, and which introduces extra ambient classes.

Week 4 — Étale fundamental groups, sections, and anabelian reconstruction.

The first lecture develops Grothendieck’s anabelian philosophy. Students study hyperbolic curves, arithmetic fundamental exact sequences, outer Galois actions, decomposition groups, and the relation between rational points and sections (G_K\to\pi_1(X)). The section conjecture is treated as a reconstruction problem, not as an isolated conjecture.

The second lecture studies finite approximations to the full group: metabelian quotients, abelian-by-central quotients, lower-central series, and (\pi_1/\Gamma^{m+1}\pi_1). The central research question is how much geometry is already visible at finite depth. Students must distinguish “full non-abelian structure is necessary” from the weaker statement that first cohomology is insufficient. The problem session compares reconstruction from (H^1), from (\pi_1/\Gamma^3), and from the full (\pi_1). This material is moved forward from the original Week 12 because it supplies the conceptual reason for retaining non-abelian arithmetic information at all.

Week 5 — Classical Chabauty and Coleman’s analytic realization.

The first lecture presents Chabauty’s theorem and its dimension mechanism. Students analyze the embeddings (X(K)\to J(K)) and (X(K_p)\to J(K_p)), the closure of (J(K)) in (J(K_p)), and the role of the inequality (\operatorname{rank}J(K)<g).

The second lecture develops Coleman’s effective formulation: (p)-adic logarithms, abelian integrals (\int_P^Q\omega), residue disks, Coleman functions, and annihilating differentials. Students see how a global rank condition becomes a local analytic vanishing condition. The seminar requires explicit Chabauty dimension counts for curves of genus two and three and asks students to identify the exact point where the method fails when the Mordell–Weil rank becomes too large.

Week 6 — Non-abelian fundamental groups and the unipotent tower.

The first lecture introduces groupoids rather than groups, basepoint dependence, étale paths, unipotent completion, and the lower-central series (U=U^1\supset U^2\supset U^3\supset\cdots). Finite quotients (U_n=U/U^{n+1}) are treated as controlled approximations to the full path space.

The second lecture introduces non-abelian cohomology (H^1(G,U_n)) and non-abelian Kummer maps. Classical Chabauty is reinterpreted as the depth-one approximation; Kim’s construction replaces (H^1(G,V)) with (H^1(G,U_n)). Students compare the algebraic nature of a vector-space-valued cohomology group with the nonlinear moduli structure of non-abelian (H^1). The problem session computes the first two levels of the lower-central filtration in an elementary free or punctured-curve example.

Week 7 — Arithmetic local systems and monodromy.

The first lecture develops local systems, monodromy representations (\rho:\pi_1(X)\to GL(V)), geometric versus arithmetic monodromy, and invariant tensors. Students study how a family of varieties gives a family of representations rather than a single fixed representation.

The second lecture introduces variation in families, Zariski closures of monodromy, degeneration loci, and special subvarieties. The main question is whether exceptional arithmetic behavior can be detected through a drop or constraint in the monodromy group. This material is moved before Tannakian reconstruction because the category of local systems and its monodromy representations are the data from which the Tannaka group is reconstructed. The seminar asks students to recover invariant information from a simple monodromy representation and to distinguish geometric from arithmetic monodromy. The source syllabus originally groups these themes with period maps and mixed Hodge structures.

Week 8 — Tannakian reconstruction.

The first lecture develops rigid tensor categories, duals, fiber functors, tensor automorphisms, and neutral Tannakian categories. The central reconstruction is (\mathcal C\mapsto\operatorname{Aut}^{\otimes}(\omega)). Students learn why the correct object may be encoded not by presenting the group directly but by presenting its entire representation category.

The second lecture specializes to unipotent local systems and pro-unipotent algebraic groups. Étale, de Rham, and rigid fundamental groupoids are constructed as Tannakian automorphism groups. Students compare direct group-theoretic and Tannakian descriptions of the same fundamental object. The reading seminar reconstructs one fundamental group from its category of representations and asks precisely what depends on the choice of fiber functor.

Week 9 — Motivic fundamental groups and realization systems.

The first lecture develops Deligne’s motivic philosophy, mixed Tate motives, weight filtrations, and motivic fundamental groups. The guiding question is whether the various cohomological fundamental groups should be regarded as realizations of a single deeper arithmetic object.

The second lecture compares étale, de Rham, Hodge, and crystalline realizations and introduces comparison morphisms. The model object throughout is (X=\mathbf P^1-{0,1,\infty}). Multiple polylogarithms and motivic iterated integrals are introduced as explicit functions carrying motivic and path-theoretic information. The problem session tracks one low-depth iterated integral through different realizations and asks which relations are motivic and which depend on a particular realization.

Week 10 — Selmer schemes, local conditions, and period geometry.

The first lecture begins with Bloch–Kato Selmer groups and then passes to local and global Selmer schemes relative to systems of local conditions. Students study representability of non-abelian cohomology functors and construct the comparison between (H_f^1(G_K,U_n)) and (H_f^1(G_p,U_n)).

The second lecture develops the Chabauty–Kim square and the Chabauty–Kim locus. Arithmetic finiteness is formulated as an intersection between global Selmer geometry and local (p)-adic period geometry. The source syllabus already identifies this global/local intersection as the conceptual heart of the method.

The same week introduces enough period geometry to make the local condition intelligible: weight filtrations, de Rham structures, (p)-adic period maps, and the role of mixed Hodge or filtered structures as comparison data. Students are asked to identify exactly which pieces of the Selmer condition are Galois-cohomological and which are period-theoretic.

Week 11 — Iterated integrals and non-commutative periods.

The first lecture begins with ordinary versus iterated integration, Chen iterated integrals, shuffle relations, path-ordered exponentials, unipotent connections, parallel transport, and holonomy. A typical coordinate is (\int_P^Q\omega_1\omega_2\cdots\omega_n). Students derive why depth one recovers ordinary abelian integration while higher depth retains ordering information lost under abelianization.

The second lecture interprets iterated integrals as functions on unipotent path torsors. The main objective is to connect analysis back to the algebraic structures developed earlier: unipotent fundamental groupoids, Tannakian categories, and non-abelian Kummer maps. The problem session derives the shuffle relations explicitly and computes low-depth iterated integrals on the thrice-punctured line.

Week 12 — Coleman integration, Frobenius continuation, and non-abelian Kummer coordinates.

The first lecture explains why ordinary analytic continuation is problematic in the (p)-adic setting and develops Coleman’s solution using analytic continuation along Frobenius. Students study Coleman functions and the transition from ordinary (p)-adic integrals to iterated Coleman integrals.

The second lecture develops Besser’s Tannakian approach, unipotent vector bundles with integrable connection, parallel transport, and the identification of the local non-abelian Kummer map with iterated Coleman integrals. The computation session works through low-depth examples on (\mathbf P^1-{0,1,\infty}), emphasizing how abstract non-abelian classes acquire explicit analytic coordinates.

Week 13 — Explicit non-abelian Chabauty, quadratic Chabauty, and rational points.

The first lecture develops depth filtrations, finite-level Selmer varieties, explicit Chabauty–Kim equations, polylogarithmic quotients, and (S)-integral points. The main case study is (X=\mathbf P^1-{0,1,\infty}) and (X(\mathbf Z[1/2])).

The second lecture introduces quadratic Chabauty, (p)-adic heights, and the relationship between quadratic conditions and deeper non-abelian information. Students compare classical Chabauty, quadratic Chabauty, and depth-two Chabauty–Kim on one explicit curve. The computational laboratory uses SageMath or equivalent tools for Coleman integration, local coordinates, (p)-adic root finding, iterated integral evaluation, and Chabauty–Kim locus computation. These computational targets are all present in the source syllabus.

Week 14 — The homology–homotopy frontier.

The final week returns to the central comparison, now with enough machinery to make it precise. Students compare (H^1(X)), (\pi_1^{ab}(X)), finite nilpotent quotients, (\pi_1^{un}(X)), (\pi_1^{\acute et}(X)), motivic fundamental groups, Tannaka groups, Selmer schemes, and period maps on the same arithmetic examples.

The first seminar question is what linearization actually destroys. The second is whether non-abelian complexity produces genuinely new arithmetic restrictions rather than merely more elaborate notation. Students must give concrete examples in which depth greater than one cuts down a local locus that classical Chabauty cannot cut down. The third question asks where the next theory should live: derived Tannakian geometry, higher Galois cohomology, motivic homotopy theory, higher stacks of Selmer conditions, non-abelian (p)-adic Hodge theory, or higher categorical arithmetic monodromy. The criterion is not sophistication but necessity: identify the obstruction that forces the next lift.

The weekly problem seminar is continuous across the semester rather than a sequence of unrelated exercises. Students choose one hyperbolic curve, preferably (\mathbf P^1-{0,1,\infty}) or a low-genus curve with known rational points, and follow it through (X(K)\to J(K)\to H^1(G_K,T_pJ)\to H^1(G_K,U_2)\to H^1(G_K,U_3)\to) Selmer locus (\to) iterated Coleman equations. The point is to observe the same arithmetic problem through increasingly non-abelian representations and identify exactly what new constraint appears at each level.

Assessment retains the source structure. Five substantial problem sets count for 25% and emphasize derivation, reconstruction, and obstruction analysis rather than routine exercises. A 30-minute blackboard presentation counts for 15% and requires reconstruction of an important theorem or construction from its prerequisites. A computational notebook counts for 15% and must contain a reproducible (p)-adic computation involving Coleman integration, a Chabauty locus, or an iterated integral. A research-paper audit counts for 15% and must map input, representation, obstruction, new invariant, and arithmetic conclusion. The final project counts for 30% and consists of either a 15–25 page research-style paper or a substantial computational project.

Suggested final projects retain the full source range. Students may compare classical, quadratic, and depth-two Chabauty–Kim on one explicit curve; study iterated Coleman integrals and unipotent parallel transport; reconstruct the Chabauty–Kim locus from the non-abelian Kummer map; compare Tannakian and group-theoretic constructions of a fundamental group; analyze information loss under (\pi_1\to\pi_1^{ab}); study finite-step anabelian reconstruction using metabelian or nilpotent quotients; investigate weight filtrations on Selmer schemes; implement low-depth iterated integrals on the thrice-punctured line; compare (p)-adic and complex/Hodge period maps; or determine whether a chosen arithmetic finiteness theorem is fundamentally cohomological or fundamentally homotopical.

The reading sequence should be treated as a dependency chain rather than a bibliography. McCallum–Poonen supplies classical Chabauty–Coleman. Corwin and Betts bridge classical and non-abelian Chabauty. Kim’s papers supply the unipotent Albanese map, motivic fundamental groups, and Selmer varieties. Deligne supplies the fundamental-group model for the projective line minus three points. Hadian connects motivic fundamental groups with integral points. Bloch–Kato and Bellaïche supply the Selmer and local-condition background. Betts develops weight filtrations and effective Chabauty–Kim. Besser supplies the Tannakian formulation of Coleman integration. Dan-Cohen–Wewers provides explicit depth-two Chabauty–Kim on the thrice-punctured line. Balakrishnan–Dogra supplies quadratic Chabauty and (p)-adic heights. These are the references listed in the source syllabus.

The finished course should leave students able to move through the same arithmetic object at several structural resolutions: rational points, Jacobian points, Kummer classes, Selmer classes, finite-depth non-abelian classes, iterated-integral coordinates, Tannakian representations, and motivic or period-theoretic realizations. The final competency is not merely familiarity with non-abelian Chabauty. It is the ability to diagnose where a particular arithmetic problem sits on the homology–homotopy frontier and to justify the minimal structural lift needed to make the obstruction visible and executable.