Picard Minicourse Plan

Legend. A number in the # column is a slide in the deck as it stands today; + marks a proposed addition not present in any deck. Sources are files under artifacts/picard_minicourse/_sources/ (frobenius-dist-ctnt.tex, vantage.tex, nyc-jnts.tex, K3workshop.tex, mukai_leiden.tex), cited as file:lines. TeX in slide titles is left as plain text.

Lecture 1: Reduction methods

50 minutes budgeted; the table below is the proposed order, 24 spoken slides against a budget of about 21. Sections: 1.0 (new slides 1-8), 1.1 (9-10), 1.2 (11-13, old 1.3 folded in), 1.4 (14-17), 1.5 (18-24).

Author, 2026-09-13: "this is a course, not a research talk, so here, we should perhaps start with the definitions. Let's introduce the picard lattice early on, and several of its properties." The order below is that reordering, carrying the three jury verdicts (artifacts/plan/structure-gpt-5.6-sol.md, artifacts/plan/structure-gpt-6-astra.md, and Fable's, recorded here only); the outline it grew from is artifacts/plan/definitions-draft.md. Not applied to the deck. Old # is the slide's number in the deck as it stands today, - if it has none.

New #	Old #	Slide	Source	Status	Carries
+	-	Title slide proposed	NEW	PROPOSED	course title, lecture number and title, speaker, venue
+	-	Divider: 1.0 The Picard lattice: definitions and properties proposed	NEW	PROPOSED	section number and title
1	3 (first half)	The Picard lattice	vantage.tex:358-380; mukai_leiden.tex:444-449	MOVE	the definition: curves mod linear equivalence, Pic = NS, rank rho
2	-	The lattice structure	no frame yet	NEW	intersection pairing, even, signature (1, rho-1), discriminant and its change by a square under finite index; Pic is a natural Galois module
3	3 (second half)	Pic inside H^2	vantage.tex:381-394	MOVE	Lefschetz (1,1), rho <= 20, the primitive embedding, T(X) as the minimal Hodge structure containing H^{2,0}
4	-	Geometric versus ground-field Picard group	K3workshop.tex:151-173; mukai_leiden.tex:472-496	NEW	rho(X) against rho(Xbar), and Pic Xbar as a Galois module
5	4	Picard lattice, over finite fields	vantage.tex:396-419, 421-440	KEEP	even rank between 2 and 22, read off the zeta function via Tate
6	5	What the characteristic polynomial gives you	K3workshop.tex:203-224	KEEP	rank and discriminant from P_2, via Tate and Artin-Tate
7	6	Reduction to finite characteristic	nyc-jnts.tex:1144-1165	KEEP	specialization is injective, so the rank can only grow mod p
8	-	Pic plays the role of End(A)	vantage.tex:367-369	NEW	the Rosati analogue, as the bridge into the elliptic example
+	-	Divider: 1.1 Two primes determine the answer proposed	NEW	PROPOSED	section number and title
9	1	How to distinguish between the two types?	frobenius-dist-ctnt.tex:233-265	MOVE	read End from a_p at one prime; the non-CM/CM table
10	2	Examples: 11.a2 and 27.a2	frobenius-dist-ctnt.tex:268-289	MOVE	two LMFDB curves, End computed at two primes each
+	-	Divider: 1.2 van Luijk for K3, with the Elsenhans-Jahnel refinement folded in proposed	NEW	PROPOSED	section number and title
11	8	Improving upper bounds: two specializations	vantage.tex:466-477; nyc-jnts.tex:1167-1182	MOVE	van Luijk, with discriminants in place of endomorphism algebras
12	9	The quartic, worked	NEW	MOVE	the criterion run once with numbers: rho and disc at 11 and 13
13	10	Torsion-free cokernel	vantage.tex:479-498	MOVE, AUTHOR'S CALL on the fold	Elsenhans-Jahnel: integral information specializes for p not 2; 1.3 folded into 1.2 per Fable, sol and astra keep 1.3 as its own section
+	-	Divider: 1.4 Kummer: jumping is frequent, prime choice matters proposed	NEW	PROPOSED	section number and title
14	12	Jumping Picard ranks	vantage.tex:559-576	MOVE	Charles: eta, Pi_jump, gamma(X,B), with eta explained where it appears
15	22	K3 surfaces	vantage.tex:533-557	MOVE	what jumping buys: Li-Liedtke, Bogomolov-Zarhin, the odd-rank corollary
16	14, absorbing 13, 15 and 16	Product of elliptic curves	K3workshop.tex:298-326, 328-352, 512-518; vantage.tex:577-596	MOVE	rho = 18 + rk Hom(E1,E2) and rho(Km A) = 16 + rho(A); the CM/non-CM table with its jump-probability column; Pi_jump(X) = Pi_jump(A)
17	17	Jumping Picard ranks for Kummer surfaces	vantage.tex:597-630	MOVE	supersingular primes, Lang-Trotter, Elkies, Charles, the open End = Z case
+	-	Divider: 1.5 Charles and the jump character: the destination proposed	NEW	PROPOSED	section number and title
18	18	O or SO?	NEW	MOVE	tau: Gal to O(T_l(1)), and det tau as a quadratic character
19	19	What det = -1 costs you	NEW	AUTHOR'S CALL: Fable keeps, sol and astra drop	the four-line eigenvalue argument forcing the +2; Fable calls it the one step the room can verify, sol and astra hand it to 18
20	20	Discriminant of a K3 surface	K3workshop.tex:626-646	MOVE	the sign theorem, and the +2 jump at non-square D_X
21	21	We can explain the 1/2	vantage.tex:672-702	MOVE	the density payoff, with a 100-digit d_X printed
22	23	Computing rho(Xbar)	K3workshop.tex:523-550	MOVE	Charles's theorem in the E_X-dependent form
23	24	When every prime overshoots	NEW	AUTHOR'S CALL: astra keeps, sol and Fable drop with a callback on 22 or 24	two cases with eta > 0, and the quartic of 12 revisited
24	25	A surface where that happens	NEW (saard PDF p.35)	MOVE	rho = 16, RM by Q(sqrt 2), case 2; hands over to Lecture 2

Dropped, five slides:

Old 7, Computing the Picard lattice over Q^al (vantage.tex:442-464): an algorithm survey, none practical; dropped to make room for the new 2 and 4, and spoken over 11.
Old 11, The sign in the functional equation (K3workshop.tex:626-646): the same frame as old 20, stated twice; the sign theorem stays on 20.
Old 13, Kummer surface (K3workshop.tex:512-518): its one formula, rho(Km A) = 16 + rho(A), opens 16.
Old 15, The simplest case (K3workshop.tex:328-352): a near-duplicate of old 14's table; its jump-probability column moves to 16.
Old 16, Jumping Picard ranks for Kummer surfaces, first frame (vantage.tex:577-596): its transfer lines, Pi_jump(X) = Pi_jump(A), move to 16.

Count: the recommended version is 24 spoken slides; dropping both author's-call slides, 19 and 23, brings it to 22. The budget is about 21, so even the shorter version runs one slide over.

Lecture 2: P-adic Hodge-theoretic obstructions

45 minutes budgeted, 20 slides now (about 1 over). Sections: 2.1 (slides 1-4), 2.2 (5-20).

#	Slide	Source	What is on it	Purpose
+	Title slide proposed	NEW	course title, lecture number and title, speaker, venue	Opens the lecture after the break; no deck has one
+	Divider: 2.1 From reduction to a lifting problem proposed	NEW	section number and title	Poses the geometric question before any cohomological test
1	Where we got to yesterday	NEW	display of the reduction bound; four bullets on jumping versus forced excess	Re-enters after the break and re-draws the distinction the lecture rests on
2	Picard lattice, over finite fields	nyc-jnts.tex:1108-1142	Tate conjecture box; three displays: kernel, zeta function, chi(t)	Recalls Tate and the zeta function before the obstruction
3	Reduction to finite characteristic	nyc-jnts.tex:1205-1222	theorem box with the Q_p refinement; thickenings display; goal and idea lines	Restates specialization and introduces the thickenings
4	The lifting question	NEW	two bullets: which classes lift, not which prime; no formulas	Turns yesterday's dead end into today's question
+	Divider: 2.2 P-adic Hodge-theoretic obstructions proposed	NEW	section number and title	The culmination: a computable obstruction and a rigorous upper bound
5	1st ingredient: cohomology	nyc-jnts.tex:1224-1239	four bullets; display of the rank assumptions; filtration with dims 22, 21, 1	Introduces the Hodge filtration and the crystalline comparison
6	Berthelot-Ogus-Raynaud	nyc-jnts.tex:1240-1242	one theorem box; a geometric-version note underneath	The room should leave knowing this statement and not its proof
7	What Frobenius acts on	nyc-jnts.tex:1245-1247	one display: Frob_p acting on H^2_dR(X/Q_p)	Answers the obvious objection before it is raised
8	Tate over a finite field	nyc-jnts.tex:1249-1251	one theorem box: the Tate kernel, now on H^2_dR	Puts the Tate kernel where the filtration lives
9	The obstruction map	vantage.tex:500-516; nyc-jnts.tex:1263	display of pi; four algorithm steps; pi(C) nonzero means no lift	The sentence the lecture is built around
10	What you actually compute	NEW	four bullets: Frobenius known only mod p^N, Kedlaya-style machinery not taught	How the inputs are obtained, without teaching the engines
11	Why finite precision still proves something	NEW	four bullets: nonvanishing is an open condition, so a finite N proves a bound	Converts the method from evidence into proof
12	Abelian surface	nyc-jnts.tex:1267-1290	4-by-4 Frobenius matrix mod 31^3; L(t); the deduced H^2 polynomial	The only example whose whole computation fits on a slide
13	Abelian surface, continued	nyc-jnts.tex:1291-1315	two eigenvectors mod 31^2; the last coordinate kills one; bound drops to 1	The payoff: a dimension removed by hand, once
14	K3 surface	nyc-jnts.tex:1377-1401	sage transcript at p = 89; five bullets; rho = 4, matched by four lines	The K3 example where the obstruction reaches the sharp value
15	Quartic surface	nyc-jnts.tex:1402-1423	sage transcript at p = 31; four bullets; no obstruction found	The honest middle case, and why the method is stated per prime
16	Quintic surface	nyc-jnts.tex:1428-1458	two sage transcripts, p = 23 and 29; CM by Q(zeta_5) blocks p = 29	Looking harder cannot help at this prime
17	What the three examples say	NEW	three bullets contrasting the p = 89, 31 and 29 outcomes	Stops the three examples reading as a list
18	What is being computed, exactly	nyc-jnts.tex:1462-1480	three bullets: the Q_p approximation against the rational structure needed	The distinction between rational classes and their p-adic span
19	Theoretical example	NEW (saard PDF p.35-37)	the double cover again; six bullets; chi_1 at p = 83; rho = 16 or 18	One concrete surface carries both lectures
20	The question	nyc-jnts.tex:1462-1480	three bullets: no tight prime in general, the RM counterexample, hope over Q	Ends the lecture, and the break falls here

Lecture 3: From periods to explicit geometry

45 minutes budgeted, 20 slides now (about 1 over). Sections: 3.1 (slides 1-3), 3.2 (4-7), 3.3 (8-20).

#	Slide	Source	What is on it	Purpose
+	Title slide proposed	NEW	course title, lecture number and title, speaker, venue	Opens the lecture after the break; no deck has one
+	Divider: 3.1 Explicit objects and lower bounds proposed	NEW	section number and title	Short opening: what explicit curves establish and what remains
1	The other direction	NEW	two bullets: upper bounds stop at a number; today, produce the classes	The break means this lecture cannot continue a sentence
2	An analytic approach	mukai_leiden.tex:524-554	Lefschetz (1,1) box; period display Pi R = 0; five bullets on computing Pi	Converts geometry into numerical linear algebra in one line
3	How far numerics can be trusted	mukai_leiden.tex:550-552	one bullet and one display: the Lairez-Sertoz bound B	Bounds how much of the lattice the numerical candidates cover
+	Divider: 3.2 Jacobian endomorphisms as a concrete model proposed	NEW	section number and title	Make the periods-to-geometry passage tangible on one example
4	Our setup	mukai_leiden.tex:174-191	three prose lines and a goal box: compute End of the Jacobian from equations	Sets up the endomorphism problem in one goal box
5	Heuristic solution	mukai_leiden.tex:216-239	two displays: End(J) as a lattice stabiliser, and T Pi = Pi R; LLL caveat	Exactly parallel to slide 2: numerics give candidates and no proof
6	Representing endomorphisms via correspondences	mukai_leiden.tex:241-279	correspondence display on C x C; Costa-Mascot-Sijsling-Voight theorem box	The template: numerical candidate, then an exact witness
7	Examples	mukai_leiden.tex:352-380	four bullets: 66158 LMFDB genus-2 curves; a genus-4 RM curve displayed	The certified algorithm at scale, and one hard instance
+	Divider: 3.3 Picard lattices of K3 surfaces proposed	NEW	section number and title	The destination: candidate classes to exact curves to a certified lattice
8	Picard lattice of a K3 surface	mukai_leiden.tex:472-496	goal box; three displays; the Zariski quote; the Pic + T(X) splitting	Names what computing the Picard lattice has meant all along
9	A running example inspired by Klein-Mukai	mukai_leiden.tex:556-607	quartic display; eight bullets; rank 19 and 133056 predicted quartics	The number 133056 is the hook
10	Reconstructing isolated curves from their Hodge classes	mukai_leiden.tex:609-628	period pairing display; Movasati-Sertoz and Cifani-Pirola-Schlesinger boxes	The theorems that turn a Hodge class into equations for a curve
11	Reconstructing quadric surfaces	mukai_leiden.tex:630-657	goal box; the 133056 orbit decomposition; quadric display; 168 quadrics	Picks the small orbit and the quadrics to reconstruct
12	Reconstructing quadric surfaces	mukai_leiden.tex:658-692	goal box; a truncated degree-168 minimal polynomial; three bullets on cost	Shows the cost: 9k-character heights, one field in nine guises
13	Isomorphism problem	mukai_leiden.tex:694-720	goal box; two displays: compatible embeddings and the linear system for v	Solves the isomorphism problem via compatible embeddings
14	Intersecting the quadric surfaces with the K3 surface	mukai_leiden.tex:723-753	goal box; quadric display; four bullets: 10 reduced points, Gotzmann, F_p check	The candidates become curves
15	Certifying $\operatorname{Pic} \overline{X} = \Lambda$	mukai_leiden.tex:755-791	two displays: the chain of inclusions, then saturation and rank 19	The destination of the course: not a number, but the lattice
16	Computing the Galois action	mukai_leiden.tex:794-819	goal box; quadric display; period reconstruction over K; open questions	Poses the Galois action and what cannot yet be certified
17	Computing the Galois action	mukai_leiden.tex:821-849	goal box; same quadric display; the guess K = F(14th root of u)	Three questions handed to a room that could answer them
18	Summary	mukai_leiden.tex:851-870	one theorem box (Costa-Sertoz); a wanna-be theorem underneath	States the theorem the lecture proves for the running example
19	Do you have a challenge K3 surface for us?	mukai_leiden.tex:868	one line, the closing question to the room	The deck ends by asking the room for work
20	What is open, in one place	NEW	six bullets, each tagged with the section it came from	Collects the open questions of all three lectures

Open on the plan

No title slides and no divider slides exist: all three decks open on slide 1, and the ten section headings live only in the speaker notes; the 13 rows marked + above are the proposed additions (3 title slides, 10 dividers).
Eleven slides have no source at all: Lecture 1 slides 9, 18, 19, 24; Lecture 2 slides 1, 4, 10, 11, 17; Lecture 3 slides 1 and 20. Two further new slides cite a PDF only, not a deck: Lecture 1 slide 25 and Lecture 2 slide 19, both saard PDF p.35.
Repeated titles within a lecture: Lecture 1 slides 16 and 17 are both "Jumping Picard ranks for Kummer surfaces" (vantage.tex:577-596 and 597-630); Lecture 3 slides 11 and 12 are both "Reconstructing quadric surfaces"; Lecture 3 slides 16 and 17 are both "Computing the Galois action".
Two titles also repeat across lectures: "Picard lattice, over finite fields" (L1 slide 4, L2 slide 2) and "Reduction to finite characteristic" (L1 slide 6, L2 slide 3), from different sources, as the post-break re-entry.
Lecture 3 slide 20, the collected open-problems slide, is unapproved: its notes read "Proposed, not taken" and "needs your yes" (bead minima-uv3.2). It is the only slide in the course that adds content rather than rearranging it, and it sits after slide 19, the course's own last line.
Section 3.3's destination is not last in the section: slide 15, "Certifying Pic Xbar = Lambda", carries the note "the destination of the whole course" and is followed by five slides (16 to 20).
Lecture 1 states the functional-equation sign theorem twice, on slides 11 and 20, both split from the same frame (K3workshop.tex:626-646), attributed to Costa-Elsenhans-Jahnel on slide 11 and to Elsenhans-Jahnel on slide 20 (bead minima-pln).
Lecture 1 slides 14 and 15 repeat the same display and a near-identical six-row table, slide 15 adding one column; the scaffold marked that second frame (K3workshop.tex:328-352) "compress or omit".
The SECTION 1.5 note sits in the speaker notes of Lecture 1 slide 24, not slide 18, which opens the section; the other nine section notes are on their section's first slide.
Lecture 1 carries two visible cross-references by slide number: "the quartic on slide 9" (slide 24) and "the certified RM of slide 24" (slide 25). The second points at a Spoken note on slide 24, not at anything visible there.
Twenty-five of the 65 slides carry no "What it does." line in their notes, so the Purpose cell above is derived from the slide and its other notes: Lecture 1 slides 1, 4, 6, 7, 10, 13, 14, 16, 20, 23; Lecture 2 slides 2, 3, 5, 8, 14; Lecture 3 slides 3, 4, 7, 10, 11, 12, 13, 16, 18, 20.
The scaffold's titles for 1.1, 1.3 and 1.4 are not the ones the decks and the written pages use ("Two primes determine the answer", "Refinements: Elsenhans-Jahnel", "Jumping, and choosing your primes"); the divider rows above use the scaffold's, per the brief.