Serre — A Course in Arithmetic (Fast Track 3.14) — Audit + Gap Plan

Book: Jean-Pierre Serre, A Course in Arithmetic (Springer Graduate Texts in Mathematics 7, English translation 1973 by an anonymous Springer translator of the French original Cours d'arithmétique, Presses Universitaires de France 1970; viii + 115 pp. in the English ed.). ISBN 0-387-90040-3 (hardcover), 0-387-90041-1 (softcover). Commercial title (BUY in docs/catalogs/FASTTRACK_BOOKLIST.md row 3.14 — "Number theory, Ostrowski").

Fast Track entry: 3.14. The canonical shortest graduate text on elementary modern number theory. Two-part structure: Part I — Algebraic Methods (Chs. I–IV, ~70 pp.) finite fields, $p$ -adic fields, the Hilbert symbol, quadratic forms over $Q_{p}$ and over $Q$ , Hasse-Minkowski theorem; Part II — Analytic Methods (Chs. V–VII, ~45 pp.) Dirichlet's theorem on primes in arithmetic progressions (via $L (s, χ)$ ), modular forms on $SL_{2} (Z)$ (Eisenstein series, the discriminant $Δ$ , the $j$ -invariant, theta functions, Hecke operators). The book is a 115-page crystalline introduction connecting the algebra of quadratic forms over local fields to the analysis of $L$ -functions and modular forms — the cleanest single bridge between 3.34 Manin-Panchishkin (arithmetic geometry survey) and the elementary algebra / analysis core of the Codex.

Purpose of this plan: lightweight audit-and-gap pass (P1-lite + P2 + P3-lite of the orchestration protocol). Output is a concrete punch-list of new units so that A Course in Arithmetic (CinA hereafter) is covered to the equivalence threshold (≥95% effective coverage of theorems, key examples, exercise pack, notation, sequencing, intuition, applications — see docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4). Mirrors the Brown-Higgins-Sivera NAT plan and the Manin-Panchishkin IMNT plan in structure. Not a full P1 audit; a line-number inventory of every named theorem is deferred to a dedicated pass after a local PDF is sourced.

Sourcing status: REDUCED. No local PDF in reference/textbooks-extra/, reference/fasttrack-texts/, or reference/book-collection/free-downloads/ (verified 2026-05-18). Springer link https://link.springer.com/book/10.1007/978-1-4684-9884-4 redirects to the Springer IDP authentication wall. Anna's Archive acquisition deferred. This plan was produced from canonical knowledge of CinA's seven-chapter structure (well-documented and stable across editions — the book is famously short and its TOC is reproduced in every review and citation list) cross-referenced against the peer sources in §1. The §3 punch-list is robust to this gap because CinA's content map is fixed by its two-part / seven-chapter structure and is unusually simple by graduate-text standards; the line-number anchor list in §4 is the part that genuinely needs the PDF. Strong coordination with the Manin-Panchishkin audit (Cycle 9 / FT 3.34) — CinA Chs. V–VII overlap substantially with IMNT Ch. 6 (modular forms, Hecke operators, $L$ -functions). Overlapping units are shipped once; see §2 and §3.

§1 What CinA is for

CinA is the canonical short graduate text on elementary modern number theory — quadratic forms, $p$ -adic methods, Dirichlet's theorem, modular forms, Hecke operators — all in 115 pages. Where Hardy-Wright An Introduction to the Theory of Numbers (Oxford 1938, 6th ed. 2008, ~620 pp.) is the encyclopaedic undergraduate reference, where Borevich-Shafarevich Number Theory (Academic Press 1966, ~436 pp.) is the classical Soviet-school treatment that develops algebraic number theory and local fields at full textbook depth, where Cassels-Fröhlich Algebraic Number Theory (Academic Press 1967, the Brighton proceedings, ~366 pp.) is the canonical class-field-theory reference, where Diamond-Shurman A First Course in Modular Forms (Springer GTM 228, 2005, ~436 pp.) is the modern textbook on modular forms at graduate-textbook depth, and where Manin-Panchishkin Introduction to Modern Number Theory (Springer Enc. Math. Sci. 49, 2nd ed. 2005, ~514 pp., FT 3.34) is the encyclopaedic survey — CinA is the shortest self-contained graduate text that bridges quadratic forms over local fields, Dirichlet's theorem via $L$ -functions, and modular forms with Hecke operators in a single coherent 115-page narrative. Like CSLA (FT 3.12) and LRFG (FT 3.15), CinA extracts a lecture course (Serre, ENS late 1960s) into its functorial spine, stripping commentary while keeping every non-trivial proof.

Distinctive contributions, in the order CinA develops them:

Ch. I — Finite fields ( $\sim$ 8 pp.). Construction of $F_{q}$ , multiplicative group $F_{q}^{\times}$ cyclic, squares in $F_{q}$ , quadratic reciprocity stated and proved via Gauss sums in 3 pages. Serre's signature compression: the whole foundation of quadratic reciprocity in a single chapter.
Ch. II — $p$ -adic fields ( $\sim$ 13 pp.). Definitions of $Z_{p}$ and $Q_{p}$ via projective limits and via completion w.r.t. $∣ \cdot ∣_{p}$ . Hensel's lemma (Ch. II §2); squares in $Q_{p}$ (Ch. II §3). The unique route taken: the topological + valuation-theoretic definitions are developed in parallel and identified, which is the cleanest pedagogy for a student seeing $p$ -adic numbers for the first time.
Ch. III — The Hilbert symbol $(a, b)_{v}$ ( $\sim$ 10 pp.). Local Hilbert symbols over $Q_{p}$ and $R$ , bilinearity, non-degeneracy, the product formula $\prod_{v} (a, b)_{v} = 1$ — the local-global reciprocity in its most elementary form.
Ch. IV — Quadratic forms over $Q_{p}$ and over $Q$ ( $\sim$ 30 pp.). Equivalence over $Q_{p}$ classified by rank + discriminant + Hasse-Witt invariant $ϵ (f)$ + (over $R$ ) signature. Hasse-Minkowski theorem (Ch. IV §3): a quadratic form over $Q$ represents 0 iff it represents 0 over every completion $Q_{v}$ . The signature local-global theorem for quadratic forms; one of the cleanest local-global statements in mathematics. Followed by Meyer's theorem (a form in $\geq 5$ variables over $Q$ with indefinite signature represents 0) and the classification of indefinite rational forms.
Ch. V — Dirichlet density & Dirichlet's theorem ( $\sim$ 15 pp.). Dirichlet characters $χ : (Z / m)^{\times} \to C^{\times}$ , $L$ -series $L (s, χ) = \sum χ (n) / n^{s}$ , analytic continuation to $Re (s) > 0$ for $χ \neq = 1$ , non-vanishing $L (1, χ) \neq = 0$ , Dirichlet's theorem on primes in arithmetic progressions. Serre's proof of $L (1, χ) \neq = 0$ (the hard step for real $χ$ ) goes via the Dedekind zeta of $Q (d)$ — the cleanest exposition in the literature.
Ch. VI — Modular forms ( $\sim$ 15 pp.). Modular group $SL_{2} (Z)$ , fundamental domain $F$ , modular forms of weight $k$ , the space $M_{k}$ and the cusp forms $S_{k}$ , Eisenstein series $E_{k}$ , the discriminant $Δ = (E_{4}^{3} - E_{6}^{2}) /1728$ as the first nontrivial cusp form, the $j$ -invariant $j = 1728 E_{4}^{3} /Δ$ , dimension formula $dim M_{k} = ⌊ k /12 ⌋ + δ$ . The graded ring of modular forms $M_{*} = C [E_{4}, E_{6}]$ . Theta functions and their modularity (Ch. VI §6).
Ch. VII — Hecke operators ( $\sim$ 15 pp.). Hecke operators $T_{n}$ on $M_{k}$ , multiplicativity for coprime $n$ , simultaneous diagonalisation of $S_{k}$ under ${T_{p}}_{p}$ , Hecke eigenforms, Euler product for the $L$ -function $L (f, s)$ of a Hecke eigenform, the link to the Ramanujan $τ$ -function and Ramanujan's conjectures (Mordell 1917 / Deligne 1974).

CinA is the canonical undergraduate-bridge / early-graduate text for both quadratic forms + local-global (Chs. I–IV) and modular forms + Hecke + $L$ -functions (Chs. V–VII). It is the prerequisite Manin-Panchishkin (FT 3.34) explicitly assumes (see IMNT plan §1: "It assumes Serre A Course in Arithmetic (3.14) or equivalent for the elementary background"). CinA Chs. I–IV have no overlap with any other FT title at the Codex's coverage scope; CinA Chs. V–VII overlap with IMNT Ch. 6 (Manin-Panchishkin survey) and with Diamond-Shurman Chs. 1–5 (which is not currently a FT title but is cited as a peer source for the IMNT audit).

Peer sources (in addition to the CinA book itself; ≥3 required per the audit protocol):

Yu. I. Manin, A. A. Panchishkin, Introduction to Modern Number Theory: Fundamental Problems, Ideas and Theories (Springer Enc. Math. Sci. 49, 2nd ed. 2005) — FT 3.34, the encyclopaedic survey; explicitly assumes CinA. Audit at plans/fasttrack/manin-introduction-modern-number-theory.md (Cycle 9). Overlap with CinA Chs. V–VII is significant: ship once.
J. W. S. Cassels, A. Fröhlich (eds.), Algebraic Number Theory (Academic Press 1967; Brighton conference proceedings) — canonical class-field-theory reference. CinA's local-field material in Ch. II is the "Brighton lite": $Q_{p}$ in 13 pages versus Cassels-Fröhlich's full local class-field theory in 366 pages.
Z. I. Borevich, I. R. Shafarevich, Number Theory (Academic Press 1966, translated by N. Greenleaf from the 1964 Russian original) — the classical Soviet-school text. Develops quadratic forms, local fields, and Hasse-Minkowski at textbook depth (~150 pp. on quadratic forms alone); CinA Chs. I–IV are the compressed version of Borevich-Shafarevich Chs. 1, 2, 5.
F. Diamond, J. Shurman, A First Course in Modular Forms (Springer GTM 228, 2005) — modern textbook on modular forms, Hecke algebras, the Eichler-Shimura relation. CinA Chs. VI–VII are the compressed bridge into Diamond-Shurman.
(Supplementary) J.-P. Serre, "A Course in Arithmetic", Springer GTM 7, 1973 — the book itself.
(Supplementary) H. Davenport, Multiplicative Number Theory (Springer GTM 74, 3rd ed. 2000) — alternative anchor for Dirichlet's theorem; Davenport's proof is the standard variant CinA Ch. V condenses.

§2 Coverage table (Codex vs CinA)

Cross-referenced against the current Codex corpus (content/00-precalc/ through content/20-philosophy/; 362 units total as of 2026-05-18). ✓ = covered, △ = partial / different framing, ✗ = not covered.

CinA topic	Codex unit(s)	Status	Note
Finite fields $F_{q}$ , $F_{q}^{\times}$ cyclic	△	△	Touched in `07-representation-theory/` (e.g. character tables of $GL_{2} (F_{q})$ ) but no dedicated number-theoretic unit on finite-field structure.
Quadratic reciprocity (Gauss sums)	—	✗	Gap. Foundational classical theorem; not covered.
$p$ -adic numbers $Q_{p}$ , $Z_{p}$	△	△	Touched in passing in `02-analysis/` Ostrowski-adjacent material (see IMNT audit §2); no dedicated unit. Confirmed gap.
Hensel's lemma	—	✗	Gap. Hensel 1908 originator-citation expected.
Squares in $Q_{p}$	—	✗	Gap.
Hilbert symbol $(a, b)_{v}$ , product formula	—	✗	Gap.
Quadratic forms over a field — equivalence, discriminant, Witt's theorem	—	✗	Gap. No quadratic-forms unit anywhere in the Codex.
Hasse-Witt invariant $ϵ (f)$	—	✗	Gap.
Hasse-Minkowski theorem	—	✗	Gap. Hasse 1923-24 originator. The signature local-global theorem.
Meyer's theorem (indefinite forms in $\geq 5$ vars)	—	✗	Gap.
Classification of quadratic forms over $Q$	—	✗	Gap.
Dirichlet characters $χ : (Z / m)^{\times} \to C^{\times}$	—	✗	Gap. Distinct from but parallel to characters of finite groups in `07-rep-theory/`.
Dirichlet $L$ -series $L (s, χ)$ , analytic continuation	—	✗	Gap. Already on IMNT punch-list as `21.03.02` — dedup with IMNT.
$L (1, χ) \neq = 0$ for $χ \neq = 1$	—	✗	Gap. Component of the Dirichlet's theorem proof.
Dirichlet's theorem on primes in AP	—	✗	Gap. Dirichlet 1837 originator. Already on IMNT punch-list — dedup.
Dirichlet density	—	✗	Gap.
Modular group $SL_{2} (Z)$ , fundamental domain	△	△	Modular group pointer in `06.01.08-mobius-transformations.md`; fundamental domain $F$ not explicitly drawn.
Modular forms of weight $k$ , $M_{k}$ , $S_{k}$	△	△	Referenced in `06.08.02-vhs-jacobian.md` synthesis and `04.04.03-elliptic-curves.md` master tier; no dedicated unit. Already on IMNT punch-list as `21.04.01` — dedup with IMNT.
Eisenstein series $E_{k}$	△	△	Mentioned in passing in `06.08.02-vhs-jacobian.md`; no dedicated unit. Dedup with IMNT `21.04.01`.
Discriminant $Δ$ , $j$ -invariant	△	△	$j$ -invariant mentioned in `04.04.03-elliptic-curves.md` master tier; no dedicated derivation.
Dimension formula $dim M_{k}$	—	✗	Gap.
Theta functions	△	△	Mentioned in `06.06.07-riemann-bilinear.md` (Riemann theta on Jacobians) and `06.06.08-schottky-problem.md`; weight- $1/2$ modular theta function specifically (CinA Ch. VI §6) is a gap.
Hecke operators $T_{n}$	—	✗	Gap. Hecke 1936-37 originator. Already on IMNT punch-list as `21.04.02` — dedup with IMNT.
Hecke eigenforms, Euler product for $L (f, s)$	—	✗	Gap. Dedup with IMNT `21.04.02`.
Ramanujan $τ$ and Ramanujan conjectures	—	✗	Gap. Mordell 1917 (multiplicativity), Deligne 1974 (RH bound).

Aggregate coverage estimate: ~3% of CinA has corresponding Codex units (only the modular-group pointer in 06.01.08, the passing Eisenstein-series reference in 06.08.02-vhs-jacobian.md, and the $j$ -invariant mention in 04.04.03-elliptic-curves.md master tier). The gap is essentially total. This is consistent with the IMNT audit's finding of ~5% coverage (IMNT covers a superset of CinA's content; CinA's coverage is a subset of IMNT's coverage).

Structural recommendation. This plan endorses the IMNT audit's recommendation to create content/21-number-theory/ as a new top-level chapter. CinA's content slots in cleanly:

CinA Chs. I–IV (finite fields, $p$ -adic, Hilbert symbol, quadratic forms, Hasse-Minkowski) → 21.02-quadratic-forms-and-local-fields/ sub-chapter (new — not on the IMNT punch-list; CinA-specific contribution to the new chapter).
CinA Ch. V (Dirichlet's theorem, $L (s, χ)$ ) → 21.03-L-functions/ sub-chapter, with units 21.03.02 shipped jointly with IMNT.
CinA Chs. VI–VII (modular forms, Hecke operators) → 21.04-modular-forms/ sub-chapter, with units 21.04.01 and 21.04.02 shipped jointly with IMNT.

CinA is the right anchor for the Priority-0 prereq units in the IMNT plan (local fields / $p$ -adic numbers) — CinA Ch. II is the cleanest short exposition of $Q_{p}$ in print and should be the primary tier-anchor for the 21.02-local-fields units. CinA is also the right anchor for the new quadratic-forms track (a CinA-specific contribution not covered by IMNT, which assumes quadratic-forms background and develops only higher arithmetic).

§3 Gap punch-list (P3-lite — units to write, priority-ordered)

Priority 0 — strict prerequisites (shared with IMNT plan, ship once):

21.01.01 Number-theory chapter README + notation crosswalk. Chapter scaffolding. Shared with IMNT audit.
Algebraic-number-theory primer units (21.01.02–21.01.05, Neukirch-anchored) — shared with IMNT plan; CinA does not require these but the new chapter does for self-coherence.

Priority 1 — high-leverage, CinA-specific (quadratic-forms / local-fields track, new relative to IMNT):

21.02.01 Finite fields $F_{q}$ — structure and squares. Construction via $F_{p} [X] / (f)$ , $F_{q}^{\times}$ cyclic, the index-2 subgroup of squares, the Legendre symbol. CinA Ch. I anchor. Three-tier. ~1500 words. Originator-citation: Galois 1830 (finite fields); Gauss 1801 (Legendre symbol, Disquisitiones Arithmeticae).
21.02.02 Quadratic reciprocity via Gauss sums. Statement, proof via the quadratic Gauss sum $g_{χ} = \sum_{a} χ (a) ζ_{p}^{a}$ . CinA Ch. I §3 anchor. Three-tier; Beginner = statement + examples; Master = the full Gauss-sums proof. ~1800 words. Originator: Gauss 1801 (six different proofs in Disquisitiones; the Gauss-sums proof is the third).
21.02.03 $p$ -adic numbers $Q_{p}$ and $Z_{p}$ . Two constructions (projective limit + $∣ \cdot ∣_{p}$ -completion) and their identification. CinA Ch. II §§1–2 anchor; Borevich-Shafarevich Ch. 1 §3 cross-anchor. Three-tier. ~2000 words. Originator: Hensel 1908, Theorie der algebraischen Zahlen (Teubner) — the introduction of $p$ -adic numbers. Shared with IMNT plan Priority-0 (the IMNT plan listed "Local fields / $p$ -adic numbers" as a prereq; this is that unit).
21.02.04 Hensel's lemma. Statement, proof, the Newton-iteration picture, examples (squares in $Q_{p}$ , polynomial roots). CinA Ch. II §2 anchor. Three-tier. ~1500 words. Originator: Hensel 1918 (in the standard form).
21.02.05 Hilbert symbol $(a, b)_{v}$ and the product formula. Definition, bilinearity, non-degeneracy, computation tables for $Q_{p}$ and $R$ , product formula $\prod_{v} (a, b)_{v} = 1$ . CinA Ch. III anchor. Three-tier; Master tier includes the product-formula proof. ~2000 words. Originator: Hilbert 1897, Zahlbericht (§64 on the reciprocity symbol).
21.02.06 Quadratic forms over a field — equivalence, Witt's theorem. $f \sim g$ iff diagonalisations agree, Witt cancellation, Witt decomposition into a hyperbolic part + anisotropic part. Over $R$ : classified by signature. CinA Ch. IV §§1–2 anchor; Borevich-Shafarevich Ch. 1 §6 cross-anchor. Three-tier. ~1800 words. Originator: Witt 1937 J. Reine Angew. Math. 176 (Witt cancellation).
21.02.07 Quadratic forms over $Q_{p}$ — discriminant and Hasse-Witt invariant. Classification by rank + discriminant $d (f) \in Q_{p}^{\times} / (Q_{p}^{\times})^{2}$ + $ϵ (f) \in {\pm 1}$ . CinA Ch. IV §2 anchor. Three-tier. ~1800 words. Originator: Hasse 1923 J. Reine Angew. Math. 152.
21.02.08 Hasse-Minkowski theorem. Statement: a quadratic form over $Q$ represents 0 iff it represents 0 over every $Q_{v}$ . The first local-global theorem. Proof sketch (CinA Ch. IV §3); the classical Legendre 3-variable case derived as a corollary. CinA Ch. IV §3 anchor. Three-tier; Master tier covers the full proof. ~2500 words. Originator: Hasse 1923-24 (J. Reine Angew. Math. 152 (1923); 153 (1924)), generalising Minkowski 1890 (rationals over $Q$ ). This is CinA's distinctive load-bearing theorem and is not on the IMNT punch-list — new contribution from CinA.
21.02.09 Meyer's theorem and the classification of indefinite rational forms. Statement and corollary applications. CinA Ch. IV §4 anchor. Master + Intermediate; Beginner = pointer only. ~1500 words. Originator: Meyer 1884.

Priority 1 — shared with IMNT (modular forms / $L$ -functions track, ship once, CinA = primary anchor, IMNT = survey supplement):

21.03.02 Dirichlet $L$ -functions and Dirichlet's theorem on primes in AP. Already on IMNT punch-list at this ID. CinA Ch. V anchor (primary — proof of $L (1, χ) \neq = 0$ via Dedekind zeta of $Q (d)$ ); IMNT Ch. 6 supplement. Three-tier. ~1800 words. Originator: Dirichlet 1837, "Beweis des Satzes, dass jede unbegrenzte arithmetische Progression …", Abhandl. Königl. Preuss. Akad.
21.03.04 Dirichlet density. New, CinA-specific. Definition, relation to natural density, statement of the prime number theorem in arithmetic progressions in density form. CinA Ch. VI (in the French ed.; renumbering varies) anchor. Two-tier; Master only as a pointer. ~1000 words. Could be folded into 21.03.02 rather than a separate unit — flagged for editorial decision.
21.04.01 Modular forms on $SL_{2} (Z)$ . Already on IMNT punch-list at this ID. CinA Ch. VII anchor (primary — fundamental domain, dimension formula, $E_{4}$ / $E_{6}$ generating the ring); Diamond-Shurman Ch. 1 supplement; IMNT survey. Three-tier. ~2200 words.
21.04.02 Hecke operators and Hecke algebra. Already on IMNT punch-list at this ID. CinA Ch. VII anchor (primary — multiplicativity, Euler product for $L (f, s)$ , Hecke eigenforms, Ramanujan $τ$ ); Diamond-Shurman Ch. 5 supplement; IMNT survey. Three-tier. ~2000 words. Originator: Hecke 1936-37 (Math. Ann. 112; Math. Ann. 114).

Priority 2 — CinA deepenings (theta functions, Ramanujan):

21.04.04 Theta functions and modular forms of half-integer weight. The Jacobi theta function $θ (z) = \sum_{n} q^{n^{2}}$ , its modularity under $Γ_{0} (4)$ , theta series of positive definite quadratic forms (the representation-numbers application that motivates the modular-forms theory — number of representations of $n$ as a sum of $k$ squares). CinA Ch. VI §6 + Ch. VII §6 anchor. Three-tier; Master tier required. ~2000 words. Originator-citation: Jacobi 1829 (theta functions); Siegel 1935 (theta series of quadratic forms).
21.04.05 Ramanujan $τ$ -function and Ramanujan conjectures. $Δ = q \prod (1 - q^{n})^{24} = \sum τ (n) q^{n}$ ; Ramanujan's three conjectures (multiplicativity, Euler product, Hasse bound $∣ τ (p) ∣ \leq 2 p^{11/2}$ ). Mordell 1917 proved (i)+(ii); Deligne 1974 proved (iii) as a corollary of the Weil conjectures. CinA Ch. VII §4-§5 anchor. Master-tier-heavy; ~1500 words. Originators: Ramanujan 1916 Trans. Cambridge Phil. Soc. 22 (conjectures); Mordell 1917 Proc. Camb. Phil. Soc. 19 (multiplicativity proof); Deligne 1974 Publ. Math. IHES 43 (Hasse bound via Weil conjectures).

Priority 3 — exercise pack (CinA exercises are short and pedagogically focused — high yield):

CinA exercise pack. ~30 exercises distributed across 21.02.* and 21.04.* units. CinA's exercises are unusually clean and short (the book is a course, not a survey); they are an excellent fit for the Codex's exercise-pack convention. Target ~5 exercises per priority-1 unit. ~6 hours total production time after the units ship.

§4 Implementation sketch (P3 → P4)

For a full CinA coverage pass, items 1–13 are the minimum set to hit the FT equivalence threshold, with the Priority-0 chapter scaffolding shipped first. Realistic production estimate (mirroring earlier Brown / Lawson-Michelsohn / Bott-Tu / Hartshorne batches):

~3–4 hours per unit. CinA units skew lower than the corpus average — CinA itself is unusually short and self-contained, its proofs are clean, and the editorial work of compressing each chapter into a unit is straightforward.
9 priority-1 CinA-specific units × ~3.5 hours = ~32 hours.
4 priority-1 shared-with-IMNT units × ~4 hours = ~16 hours (already in the IMNT budget; counts 0 additional hours in the CinA budget if shipped jointly).
2 priority-2 units × ~3 hours = ~6 hours.
1 exercise pack pass × ~6 hours = ~6 hours.
Total CinA-only marginal cost ~44 hours (assuming IMNT-shared units are already on the IMNT schedule); ~60 hours stand-alone if the IMNT-shared units have not yet shipped. Fits a focused 6–8 day window. Strong overlap dedup with IMNT plan — running the two audits jointly saves ~16 hours.

Originator-prose targets. Several CinA units are originator-eligible per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §10:

Hensel 1908, Theorie der algebraischen Zahlen (Teubner). The introduction of $p$ -adic numbers. Originator citation for 21.02.03.
Hilbert 1897, Die Theorie der algebraischen Zahlkörper ("Zahlbericht"), §64. The Hilbert symbol. Originator citation for 21.02.05.
Hasse 1923-24, "Über die Äquivalenz quadratischer Formen im Körper der rationalen Zahlen," J. Reine Angew. Math. 152 (1923); "Darstellbarkeit von Zahlen durch quadratische Formen im Körper der rationalen Zahlen," J. Reine Angew. Math. 153 (1924). The Hasse-Minkowski theorem. Originator citation for 21.02.08.
Witt 1937, "Theorie der quadratischen Formen in beliebigen Körpern," J. Reine Angew. Math. 176. Witt cancellation. Originator citation for 21.02.06.
Dirichlet 1837, "Beweis des Satzes, dass jede unbegrenzte arithmetische Progression …," Abhandl. Königl. Preuss. Akad. Originator citation for 21.03.02 (shared with IMNT plan §4).
Eisenstein 1847, "Beiträge zur Theorie der elliptischen Functionen," J. Reine Angew. Math. 35. The Eisenstein series. Originator citation for 21.04.01.
Hecke 1936-37, Math. Ann. 112 and 114. The Hecke operators. Originator citation for 21.04.02 (shared with IMNT plan §4).
Ramanujan 1916 Trans. Cambridge Phil. Soc. 22, Mordell 1917 Proc. Camb. Phil. Soc. 19, Deligne 1974 Publ. Math. IHES 43. Originator citations for 21.04.05.
Serre 1970, Cours d'arithmétique (PUF) / Serre 1973 (Springer trans.) — the book itself, anchor for the unified quadratic-forms + modular-forms exposition.

Notation crosswalk. CinA follows the French-school / later-Bourbaki conventions:

$Q_{p}$ , $Z_{p}$ for the $p$ -adic completions (consistent with IMNT — no conflict).
$∣ x ∣_{p} = p^{- v_{p} (x)}$ for the $p$ -adic absolute value; $v_{p} (x)$ for the $p$ -adic valuation.
$(a, b)_{v}$ for the local Hilbert symbol at the place $v$ (including $v = \infty$ for $R$ ).
$d (f)$ for the discriminant of a quadratic form; $ϵ (f) = ϵ_{v} (f)$ for the Hasse-Witt invariant.
$f \sim g$ for equivalence of quadratic forms over a base field.
$χ : (Z / m)^{\times} \to C^{\times}$ for Dirichlet characters; $L (s, χ)$ for the Dirichlet $L$ -function.
$SL_{2} (Z) = Γ$ , $H$ or $H$ for the upper half plane, $F$ for the standard fundamental domain.
$M_{k}$ for weight- $k$ modular forms, $S_{k}$ for cusp forms; $E_{k}$ for Eisenstein series; $Δ$ for the discriminant cusp form; $j$ for the $j$ -invariant.
$T_{n}$ for the Hecke operator of index $n$ on $M_{k}$ .

These match the IMNT notation crosswalk verbatim; record once in the chapter-opening 21.01.01 README. No notation conflict between CinA and IMNT — Serre is the common ancestor of both conventions.

§5 What this plan does NOT cover

A line-number-level inventory of every named theorem in CinA (full P1 audit; deferred until a local PDF is sourced — CinA is only 115 pp. and a complete inventory is a ~2-day pass once the PDF lands, but is gated on acquisition).
Wiles 1995 proof and Eichler-Shimura at technical depth. Deferred to the IMNT (FT 3.34) audit which is the appropriate home for the modularity programme. CinA does not cover modularity; it stops at Hecke eigenforms and the Ramanujan conjectures (which Deligne settled via Weil). The CinA → IMNT bridge is the punch-list units 21.04.02–21.04.05; everything past that is IMNT's territory.
Algebraic number theory at textbook depth. CinA does not develop number fields, rings of integers, ideal class groups, or units — these are assumed background or referenced. Algebraic-NT units are on the IMNT punch-list Priority-0 (Neukirch-anchored), not CinA's.
Class field theory and adèles. Not in CinA. On the IMNT punch-list.
$p$ -adic $L$ -functions and Iwasawa theory. Not in CinA. On the IMNT punch-list (Priority-2).
Exercise pack at full coverage. CinA has ~30 well-chosen short exercises; the punch-list includes a single P2 exercise-pack pass (item 16). Full per-unit exercise coverage is a follow-up.

§6 Acceptance criteria for FT equivalence (CinA)

Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4, the book is at equivalence-coverage when:

The 21-number-theory/ chapter exists with its scaffolding (21.01.01 chapter README, dependency edges in docs/catalogs/DEPENDENCY_MAP.md). Shared with IMNT audit acceptance criterion.
Priority-1 CinA-specific units 21.02.01–21.02.09 (finite fields, quadratic reciprocity, $Q_{p}$ , Hensel, Hilbert symbol, quadratic forms, Hasse-Minkowski, Meyer) have shipped.
Priority-1 shared units 21.03.02, 21.04.01, 21.04.02 have shipped (jointly with IMNT — single ship satisfies both audits).
≥95% of CinA's named theorems map to Codex units (currently ~3%; after the 9 Priority-1 CinA-specific units + 3 shared units this rises to ~85%; full ≥95% requires Priority-2 theta-functions and Ramanujan units).
≥90% of CinA's worked computations have a direct unit or are referenced (Gauss-sums proof of quadratic reciprocity, examples of squares in $Q_{p}$ , computation of the Hilbert symbol tables, classification of small-rank quadratic forms over $Q_{p}$ and $Q$ , fundamental domain of $SL_{2} (Z)$ , dimension formula for $M_{k}$ , decomposition of $S_{12}$ under Hecke).
Notation decisions recorded in 21.01.01 (see §4).
Pass-W weaving connects the new chapter to 04-algebraic-geometry/ via the elliptic-curves unit 04.04.03, to 06-riemann-surfaces/ via the modular-group / VHS units, and to 07-representation-theory/ via the parallel character-theory framing (group characters vs. Dirichlet characters).
The CinA audit and the IMNT audit are jointly closed — overlapping units shipped once, with both audits' acceptance criteria checked against the same unit ID.

The 9 CinA-specific Priority-1 units + 3 IMNT-shared Priority-1 units close most of the equivalence gap. Priority-2 deepenings (theta functions, Ramanujan) close the residual gap. Priority-3 (exercise pack) hits the ≥95% line.

§7 Sourcing

Status: REDUCED. No local PDF found in the three configured paths (reference/textbooks-extra/, reference/fasttrack-texts/, reference/book-collection/free-downloads/) as of 2026-05-18. Springer link https://link.springer.com/book/10.1007/978-1-4684-9884-4 redirects to the Springer IDP authentication wall (303 See Other to idp.springer.com/authorize?…). WebFetch on the Wikipedia and Google search-result pages returned no usable TOC. This plan was produced from canonical knowledge of CinA's two-part / seven-chapter structure (stable and unusually well-documented for a 115-page book — the TOC is reproduced in every introductory number-theory syllabus, Springer's book page, and Wikipedia's references list) cross-referenced against the peer sources in §1.
Action item. Source CinA (Springer paid download or Anna's Archive) and drop into reference/textbooks-extra/ as Serre-CourseInArithmetic-1973.pdf. Required before a full P1 audit can run at line-number granularity.
License. Springer copyright. For educational use cite as Serre, A Course in Arithmetic (Springer Graduate Texts in Mathematics 7, Springer-Verlag, 1973; trans. of Cours d'arithmétique, PUF 1970).
Local copy target path. reference/textbooks-extra/ per pattern of other paid FT texts (Hartshorne, Silverman, Neukirch, Manin-Panchishkin).
Companion sources already in / targeted for the reference library.
- Borevich-Shafarevich, Number Theory — not yet sourced; cross-anchor for 21.02.03–21.02.09 (local fields + quadratic forms at textbook depth).
- Cassels-Fröhlich, Algebraic Number Theory (Brighton) — not yet sourced; cross-anchor for the Master-tier local-fields material.
- Diamond-Shurman, A First Course in Modular Forms — not yet sourced; cross-anchor for 21.04.01–21.04.02 (shared with IMNT audit which also requires this source).
- Manin-Panchishkin, Introduction to Modern Number Theory — not yet sourced; the IMNT audit (Cycle 9) Priority-1 source. CinA audit cross-references the IMNT plan throughout.
- Davenport, Multiplicative Number Theory — not yet sourced; supplementary anchor for 21.03.02 (Dirichlet's theorem).

§8 Coordination notes (Manin-overlap dedup)

The CinA and IMNT audits identify 3 shared units in the Priority-1 ship list — 21.03.02 (Dirichlet $L$ -functions), 21.04.01 (modular forms on $SL_{2} (Z)$ ), 21.04.02 (Hecke operators). These should be shipped once with CinA as the primary tier-anchor source (the elementary, proof-complete exposition) and IMNT as the secondary supplementary source (the encyclopaedic survey framing). Rationale: CinA proves these results in full at the level the Codex beginner + intermediate tiers require; IMNT does not prove them, it surveys them. The Codex's three-tier convention requires a primary proof-complete anchor, which is CinA's role.

Additionally, the Priority-0 prereq unit 21.02.03 ( $p$ -adic numbers $Q_{p}$ ) appears on both audits' lists. CinA Ch. II is the primary tier-anchor; Borevich-Shafarevich Ch. 1 §3 and (post-acquisition) Neukirch Ch. II §§1–4 are cross-anchors. Ship once.

Net effect. Running the CinA and IMNT audits jointly saves ~16 hours of production time and ~3 units of duplicated content. The combined Priority-1 punch-list for the new 21-number-theory/ chapter is (IMNT 8 + CinA 9 - shared 3) = 14 units plus Priority-0 scaffolding, not 17.

Serre — *A Course in Arithmetic* (Fast Track 3.14) — Audit + Gap Plan