# Yuta Nakayama's generals

Committee: Z: Shou-Wu Zhang (Chair), S: Christopher Skinner, K: Casey Kelleher  
Date: from 1pm to 4:15 on May 7th, 2024.

## Sources

### algebraic geometry
- R. Hartshorne, *Algebraic geometry*, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977.
- B. Poonen, *Rational points on varieties*, GSM, No. 186, AMS, Providence, 2017. For Brauer--Severi varieties.

### algebraic number theory
- A. Weil, *Basic number theory*, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the second (1973) edition.
- J. Neukirch, *Class field theory*, Springer, Heidelberg, 2013, The Bonn lectures, edited and with a foreword by Alexander Schmidt, Translated from the 1967 German original by F. Lemmermeyer and W. Snyder, Language editor: A. Rosenschon.
- 斎藤修司『整数論』(共立出版1997)
- 加藤 和也, 黒川 信重, 斎藤 毅『数論I』(岩波書店2016) available as Vol. 186 and 240 of “Translations of Mathematical Monographs.” I think this is good for review.
- 雪江整数論シリーズ
- 小野孝『数論序説』(裳華房1987)

### complex analysis
Japanese translation of Stein--Shakarchi

### real analysis
- 笠原晧司『微分積分学』(サイエンス1974)
- 吉田信夫『ルベーグ積分入門』(遊星社2006)

### algebra
Bushnell--Henniart for the representation of GL_2(F_p)

## Before we started
Saw Z and S separately on the third floor. Z “Are you getting nervous?” Just said Hello to S. I went to Z's office 10 minutes before the exam and was chased away.

Z was sick on that day, and also had a plan for a meeting later in the day. The exam was short by these factors.

## Complex Analysis
I chose complex analysis to start with. 40 minutes in total. All the questions were unexpected. My cohort told me people could ask Riemann's mapping theorem…

- **[K]** Prove the maximal modulus principle.
	- I did not review beforehand although I had thought I should. Z suggested I should argue by contradiction and asked what I know about complex analysis. I said Cauchy's integration formula and Z told me that's the only thing I can do.
	- The formula only implies the constancy of the absolute value of the function. After that I used Cauchy--Riemann equation, being afraid of having chosen the wrong strategy. At the end I saw that at each point either the derivative or the value of the function vanishes. They did not pursue what to do for the latter case as a minor problem.
	- Z told me taking the real part of log of the function is faster.

- **[K]** Is it possible to formulate the minimal modulus principle?
	- The identity on the complex plane is a counterexample.

- **[S]** But what if the function does not vanish?
	- Take the inverse.

- **[K]** I don't have many questions for complex analysis and many ones for real analysis. Do you guys want to ask something?
- **[Z]** For which functions can you say similar things to the maximal modulus principle?
	- I did not have any clue. I made a random guess that the harmonic functions could satisfy a similar property by way of … average value theorem?

- **[K]** What's that?
	- I only knew the Japanese name of the theorem because Stein--Shakarchi that I had was Japanese. Wrote 「平均値」 on the board, wishing that Z will understand.

- **[Z]** I know. The mean value theorem.
	- Ah.
	- But I had no clue for the original problem anyway. Mentioned there are classes of functions called subharmonic that I didn't know at all. For harmonic functions u,v that satisfies Cauchy--Riemann equations, “I try to calculate derivatives of sqrt(u^2+v^2) because I have no idea.”

- **[Z]** What are you doing?
	- Oh no, maybe Z did not hear me. And it's true that I am stupid… .

- **[K]** It's very analysis-minded. Don't know what to do so take derivatives!
- **[Z]** Do you know laplacian?
	- Wrote d^2/dx^2 + … on the board.

- **[Z]** That's two-dimensional. Let's do the one-dimensional analog. Write
	lim_{r->0+} (f(x+r)+f(x-r)-2f(x))/(2 r^2).
	Z told me to assume the function is smooth. It took me a while to notice that only the second or higher term in the Taylor series survives. According to Z I made a mistake up to constant. He told me to check it at home.

## Real Analysis
50 minutes in total. I didn't except any of the questions except zeta.

- **[K]** Do you know the statement of the Heine--Borel theorem and prove it?
	- ハイネ--Borel?

- **[K]** Yes, Heine--Borel.
	- After that I did not have any issues. She liked my explanation.

- **[K]** Define the measurable sets.
	- Maybe there are follow-up questions, but …

- **[K]** Usually there are always follow-up questions.
	- My intention was that the answer could get longer if I started from the definition of the outer measure, and that I was trying to omit it. Asked if we consider R.

- **[K]** Yes, let's do R.
	- Without any issues after that.

- **[K]** What's Fatou's Lemma?
	- I need some time to think.

- **[K]** You've got time.
	- I started stating the liminf version. Need MCT for sup, so I need to bound the functions from below.

- **[K]** You can assume they are nonnegative.
	- And the inequality is … >=!

- **[K]** Now which example do you have to be sure that this is correct?
	- So it's wrong!

- **[K]** I didn't mean that!
	- Took nonnegative functions that were zero outside [n,n+2].

- **[K]** Can you explain?
	- No issues in explanation.

- **[K]** I heard you (Z) were quite into Fourier analysis. Should I ask that kind of questions?
	- I think she mentioned the course that he taught that semester.

- **[Z]** Yes, it's important.
	- K checking her problem sets. (Maybe these conversations were at a different time.)

- **[K]** Any notions of convergence of functions on the unit interval that you know? Implications between them? Counterexamples if there are no implications between the given two?
	- A big sigh. And not Fourier analysis.

- **[K]** Hahaha. That was a strong reaction.
	- Listed pointwise, uniform, and L^p.

- **[K]** L^infty?
	- Ah yes, L^infty is the same as uniform.
	- Uniform implies pointwise, and L^p implies L^q where p>q. I double checked if p>q was the right direction. Don't remember if I also listed uniform implies L^1. Vague memory of some of L^p implying pointwise convergence almost everywhere.

- **[K]** Hmm.
	- By the way, forgot the word “almost everywhere” and continued saying “except on a measure zero set.” Tried to mention DCT as well but did not know the English name again. “Majorized convergence theorem?”

- **[K]** What does it say?
	- Wrote the statement. For the counterexample, I asked if the functions were assumed continuous.

- **[K]** I just asked for functions on the …
	- As an example of the pointwise convergence without the uniform one, used 1/nx. It is not continuous at 0 and that was why I asked about the continuity. Mentioned the vague memory of some of L^p implying pointwise convergence almost everywhere again. But I was not sure.

- **[Z]** How about cos 2^nx? (I am not sure if I understood him correctly anymore)
	- Anyway, the correct statement in hindsight is that the L^p-convergence implies the almost everywhere pointwise convergence of a subsequence.
	- For L^p not implying L^q for p<q, I was not sure if I could make an example for any p,q without mistakes.

- **[K]** Yes, one p and q are fine.
	- The example that I gave was not a sequence, but a single function x^{-1/2} that was L^1 but not L^2. I did not notice I was deviating from the original problem.

- **[K]** Can you explain?
	- I integrated those functions. No pursuit anymore.

- **[K]** Do you guys have any other analysis questions?
- **[Z]** The meromorphic continuation and functional equation of zeta? Just to check.
	- I made two mistakes. First, wrote Σ{n=1}^infty e^{-n^2pi x} as θ(x), which I meant to use for the sum from n=-infty to n=infty. Second, …

- **[Z]** I am confused. Why did you write x^{-s/2} on the top left of the board? Your s has real part bigger than 1, right? It does not converge!
	- Sorry.
	- This happend in the midst of the calculation, but somehow other parts were safe.

- **[Z]** This is important!
	- He stopped me in the middle later. “Do you mean I can stop?”

- **[Z]** Yes, I don't want to check your mistakes.

- **[Z]** Prove the Poisson summation formula.
	- I thought I would use R/Z, but was not sure of the details and got less confident. A bad move.

- **[Z]** What's the statement?
	- Wrote the equation.

- **[Z]** What's f?
	- Should I pursue the most general version?

- **[S]** Something that makes sense.
	- I decided my f was Schwartz.

- **[Z]** What's the defintion of Schwartz functions?
	- Wrote the definition.

- **[Z]** So the definition is not useful. Why do you want to do the Fourier analysis?
	- Decomposing L^2?

- **[Z]** What's the orthonormal basis of L^2(R/Z)?
	- e^{2pi inx}.

- **[Z]** How do you decompose L^2(R/Z)?
	- Wrote the Fourier inversion formula for Fourier series.

- **[Z]** Now prove the Poisson summation formula.
	- I did with more confidence.

- **[S]** It's been an hour and a half.
- **[Z]** We'll take a break after the algebra section.

## Algebra
15 minutes in total.

- **[S]** State the classification of finitely generated modules over a PID.
- **[S]** There should be a unicity statement as well.
- **[S]** What happens if the ring is a Dedekind domain?
	- Wrote ...oplus I^δ, where …

- **[Z]** What's that? Oh, I see.
	- I meant δ = 0, 1 as there could be no torsion free part.

- **[S]** What's the unicity statement in this case?
	- I is unique as an element in the class group.

- **[S]** Do you know anything about a similar statement for other rings?
	- Mentioned that I didn't know much about the theorems used in Iwasawa theory formulated via “almost” notions. Not sure if that is the right English name either.

- **[S]** Favorite applications of the theorem for PID?
	- Jordan canonical form

- **[S]** What's the module and the ring in that case?

- **[Z]** Show that the multiplicative group of a finite field is cyclic.
	- At some point I checked if he meant a finite subgroup of the multiplicative group of a field.

- **[S]** Classify the representations of GL_2(F_p). First how many are there?
- **[Z]** Let's do F_7.
- **[S]** Sure, that's better.
	- My memory told me 7^2-1. Rational canonical form.

- **[Z]** What's F_7[X]/(X^2+aX+b) where X^2+aX+b is irreducible (as written on the board)?
	- F_14
	- I was so stupid…

- **[Z]** Is that a field?
	- No, F_{49}, F_{49}!

	I counted some of the cases separately, and S suggested I should count X^2+aX+b, where b != 0. My guess was right in the end.

- **[S]** How do you construct them?
	- There are definitely representations that are induced from the Borel. Also, if I remember correctly, there are the cases corresponding to (super)cuspidal representations, which is the induction from center times the unipotent part minus the induction from the nonsplit torus. This is in Bushnell--Henniart.

- **[S]** What are their dimensions?
	- Wrote the definition of the induction and tried to calculated the size of the coset space B\GL_2(F_7) in a rather elementary way.

- **[S]** Do you know the geometric structure of that set?
	- P^1!
	- And maybe some of these induced cases are not irreducible because certainly there are characters of the group. For the other case Ind_{ZN}^{GL_2(F_7)}(...) - Ind_{F_{49}^x}^{GL_2(F_7)}(...), …

- **[Z]** What is that minus?
	- This means the virtual representation that I need to show is an actual representation.
	- For that case, I guess I need to do
		(48*42)/(6*7) - (48*42)/48.

- **[S]** Fair enough.
- **[Z]** Now let's take a 10 minute break.
	- Yaaay

- **[S]** How do you know that those calculations are correct?
	- My frown downside up

- **[K]** Hahaha
	- Wrote Σ(dim)^2

- **[Z]** Now let's take a 10 minute break.

## Break
Went down to the third floor to drink water from the water cooler. I reviewed some materials on Brauer--Severi varieties in front of the elevator hall. When I was back, Z was explaining my potential thesis problem to S.

“I will think about it if I pass the generals.”

S “Maybe that's your generals' problem.”

S was eating some chocolate that was a souvenir from Akshay. He offered us some of it when I noticed that it was Japanese.

## Algebraic Number Theory
I chose algebraic Number Theory to restart with.

- **[Z]** Choose your favorite cubic polynomial.
	- Wrote Y^2 = X^3...

- **[Z]** No, one variable. Will come back to that later.
	- I chose X^3-3X+1.

- **[Z]** Compute its discriminant.
	- 81.

- **[Z]** What's the Galois closure K?
	- Because the discriminant is a square, I just need one root α to be adjoined.

- **[Z]** No, the Galois closure.
- **[S]** He's correct. The discriminant is a square.
- **[Z]** Oh, you are clever.
- **[K]** Hahaha.
	- The trick was that I was asked about this polynomial in my mock generals. Many thanks to the mock committee.

- **[Z]** Compute its Galois group.
	- Z/3Z.

- **[Z]** What's the ring of integers?
	- By (O_K:Z[α])^2 * Δ_{K/Q} = 81, I have only to check over Z_3. But the cubic is Eisenstein after changing X to X-1. So O_K = Z[α].

- **[Z]** What's the class number?
	- Calculation through the Minkowski bound. The answer is 1.

- **[S]** What's the group of units?
	- The field is totally real, so it has the rank 3-1 = 2.

- **[S]** You mean there are no torsion.
	- Uh, there are +- 1, which are the only ones by the totally real property.

- **[S]** Do you know how to prove the Dirichlet's unit theorem?
	- The proof I read used geometry of numbers, but I don't remember how.

- **[Z]** Because of α^3-3α +1=0, there are obvious units. Do you know how to construct them? You don't know the proof so maybe you know examples.
	- No.

- **[S]** What is the smallest cyclotomic field that includes K?
	- Uh…

- **[S]** Look at the discriminant.
	- I wrote we should have Δ_{Q(ζ_n)/Q} = N(Δ_{Q(ζ_n)/K}) * Δ_{K/Q}^{[Q(ζ_n):K]}. But I forgot the exact formula for the LHS.

- **[S]** Let's just look at those cyclotomic fields which are ramified only at 3.
	- I wrote Q(ζ_3), Q(ζ_9), Q(ζ_27), ...

- **[S]** What are their Galois groups?
	- Maybe he first asked for the expression of the form (Z/3^m)^times. And then I rewrote them as products of cyclic groups of prime power order. Especially (Z/9)^times = Z/2 x Z/3.

- **[S]** There you go. So what is the integer ring of K?
	- I told you! It is Z[α]!

- **[S]** Right, but what is it in terms of ζ_9?
	- It is the invariant part of Z[ζ_9] under ζ_9 -> ζ_9^{-1}. So it is
		 ⊕{n=0}^4 Z(ζ_9^n + ζ_9^{9-n}),
		wait I think this is wrong. The right one is
		 Z ⊕ ⊕{n=1}^4 Z(ζ_9^n + ζ_9^{9-n}),
		It is not the direct sum either in hindsight, but no one talked about it.

- **[S]** Is it generated by one element?
	- Yes, by ζ_9+ζ_9^{-1}.

- **[S]** Is it α?
	- I don't know.

- **[Z]** Do you know any ways to provide the unit in the cyclotomic field?
	- No. I first thought about 1-ζ_9 and tried to calculate its norm.

- **[S]** Consider (1-ζ_9^2)/(1-ζ_9) and its norm.
	- Whoa! The norm is one so it is a unit!

- **[Z]** Now go back to the unit of K.
	- 1+ζ_9^2 is a unit similarly, and I can multiply another unit ζ_9^{-1} to it to make it an element of K.

- **[Z]** Do you remember the Dirichlet's L-function's formula?
	- Actually, I was confused and was thinking about Dedekind zeta function although on the board was the letter L.
	- What do I do at 3?

- **[Z]** Ignore that term. How many characters are there?
	- 3.

	For the L-function I need to remember the splitting of primes in K…

- **[S]** Actually that is a good question. How do you determine the splitting in K?
	- Frobenius at p goes to p mod 9 in Gal(Q(ζ_9)/Q), which surjects to Gal(K/Q). p splits iff it is trivial there.

- **[S]** What's the kernel?
	- I somehow got too cautious about subgroups and quotients and wrote down the elements of these groups not as those in (Z/9)^x, etc but the actual field automorphisms.
	- I noticed dealing with (Z/9)^x is fine and easier later.

- **[S]** Yes.
	- And I found out the kernel was {+-1} = Z/2 subset Z/6.

- **[S]** What are they mod 9?
	- 1 and 8.

- **[S]** So this shows p is split if and only if p is 1 or 8 mod 9.
	- Maybe this comment shows that S thought this was my first time to solve this question. But I had done this before. Trust me!!!

	So the L-function is …
	 product over p congruent to 1 or 8 mod 9 of (1/(1-p^{-s}))^3, multiplied by product over p not congruent to 1 or 8 mod 9 of 1/(1-p^{-3s}).

- **[Z]** You are writing the products of L-functions.
- **[S]** Yeah.
	- Finally noticed I was thinking about Dedekind zeta. Modified L to ζ_K.

- **[Z]** What are the special values of the L-function?
	- Class number formula tells me L(1,χ)L(1,χ^{-1}). All the numbers except the regulators have been calculated up to this point.

- **[Z]** What do you know about each of the L-values?
	- They are finite…

- **[S]** It's true. It's a true statement.
	- Maybe S thought that was a joke.
	- I mentioned there are formulas for the value of partial zeta. I thought of partial zeta in the sense of 数論I. But the thing I had in mind actually has a pole at 1 and doesn't help here.

- **[Z]** Do you know similar stuff for
	 Σ{n=1}^infty e^{2pi i n x}/n^s? 
	- Ah.

- **[Z]** Now it is the matter of Fourier analysis of finite groups.
- **[S]** This tells you the regulator and some information about the units of K.
- **[Z]** You prepared the cubic but didn't prepare the rest. Do you (S) want to ask more questions?

- **[S]** Let me ask some local questions. How do you show that given a p-adic field, there are only finitely many extensions of a given degree?
	- Maybe I could have used the class field theory, but I argued as follows. Because there are only finitely many unramified extensions, …

- **[S]** What are they? How many are there for each degree?
	- 1.
	- So we only need to consider totally ramified extensions. Somehow I restricted to the case over Q_p without noticing it but nobody cared. Those extensions are defined by Eisenstein polynomials, and if their coefficients are close the polynomials define the same extension by Krasner's Lemma. So that defines an open covering of the compact space of coefficients of Eisenstein polynomials, and we can take a finite subcover.

- **[S]** What goes wrong when we consider function field?
	- Wrote F_p((t)) on the board. Thought extensions of degree p could be a problem but didn't mention anything as I didn't figure out where the exact issue was. A bad move.

- **[S]** Separability.
	- In hindsight, this goes into Krasner.

## Algebraic Geometry

- **[Z]** So let's go back to the elliptic curve Y^2 =X^3+... that you mentioned before. Let's take your favorite cubic…, well let's do X^3-3X+1, well …, let's stop this. This way is taking me to a weird direction.
- **[K]** Hahaha.

- **[Z]** So what's your favorite definition of a curve?
	- Let k be an algebraically closed field. C/k is a curve if it is a smooth projective connected scheme over k of dimension 1.

- **[Z]** smooth projective connected… Ok. What is the invariant of a curve?
	- Genus.

- **[Z]** What are the curves of genus 0?
	- P^1.

- **[Z]** How about those over Q?
	- I started my answer in a bit weird way intentionally, treating them as a Brauer--Severi variety although I didn't mention that name.
	- Because they are forms of P^1, they are classified by
		 H^1(Gal(k^sep/k), Aut(P^1)) = H^1(Gal(k^sep/k), PGL_2(k^sep)) = H^1(Gal(k^sep/k), Aut(M_2(k^sep)))
		by Skolem--Noether. So they are classified by division algebras.

- **[S]** Quaternion algebras.
	- Yes.

- **[Z]** What happens over Q?
	- Wrote the short exact sequence involving Albert--Brauer--Hasse--Noether.

- **[Z]** No that's too complicated. Think as algebraic geometers do. On the curve you have a nice divisor.
	- You mean the anticanonical bundle.
	- I actually knew this way as well, but my intention was that I did not want to establish relationship between the resulting conic and the quaternion algebra by a direct elementary computation using norm because that would produce more mistakes…

- **[Z]** So what are the curves?
	- Conics in P^2.

- **[Z]** Classify them.
	- Started writing down the most general quadratic equation, but quickly stopped and said “I guess my point is that after a probably linear transformation, they become aX^2+bY^2=Z^2, which corresponds to (a,b / Q) by the previous correspondence.” They didn't ask why.

- **[Z]** Why can Z^2 can be transfered to the right side…, ah, you are not assuming a,b>0. Ok. Now, what about genus 1?
	- Elliptic curves.

- **[Z]** Classify them.
	- Take a rational point P.

- **[S]** Why does the curve have a rational point?
	- I thought I was working over a closed field.
	- S didn't pursue this point.

	Discussed Weierstrass equation. …, And
		 O_E(6P)(E) = O_E(5P)(E) + k y^2 = ... + k x^3.

- **[S]** Why are they direct sum?
	- I thought it was correct.

- **[S]** But what is the dimension of O_E(5P)(E)?
	- 5.

- **[S]** And what is the dimension of O_E(6P)(E)?
	- 6. Oh, I meant
		 O_E(6P)(E) = O_E(5P)(E) + k y^2 = O_E(5P)(E) + k x^3.

- **[S]** Ah sorry. I missed that equal.
	- So by this, E embeds to …

- **[Z]** Now what happens if you switch P to another point?
	- I basically referred to Chap. IV, Lemma 4.2 or 4.4 of Hartshorne, but Z was unsatisfied and after the exam, he stressed different points give different embeddings.

- **[Z]** Now, given a curve on (P^1)^2 of type (2,2), what is its genus?
	- 1.

- **[Z]** Why?
	- Because it's (a-1)(b-1). Tried to remember the adjunction formula without success. Z had told me before the exam “Solve exercises on curves and surfaces …, curves,” and I thought he would ask only about curves from his past problem sets.

- **[Z]** How would you prove it?
	- Presented the argument in Chap. II, Prop. 8.20 of Hartshorne.

- **[Z]** Why do you have the dual of I/I^2?
	- Erased the dual. This I/I^2 is …

- **[Z]** It's (-2,-2).
	- Right.

- **[Z]** What kind of elliptic curves are embedded into (P^1)^2?
	- I don't know…

- **[Z]** Oooh.
	- Around this point, Ziyang Gao entered the room and Z chased him away. It turned out he had a plan for meeting.

- **[Z]** We will finish soon.
	- Wrote E-> (P^1)^2 on the board.

- **[Z]** What are the degree of E-> P^1?
	- 2.

- **[Z]** So classify the degree 2 map.
	- Classification by way of Chap. II, Prop. 7.1 of Hartshorne. Degree 2 divisors form torsors under Pic^0(E). After fixing the origin O, O_E([P]+[Q]) = O_E([P+Q]+[O]).

- **[Z]** What is that plus in the right hand side?
	- Oh this is the addition of the elliptic curve.

- **[Z]** So now they are classified by points on E.
	- Uhhh and different points give different embeddings

- **[Z]** So all the elliptic curves are embedded in (P^1)^2. Now, do you know the Serre duality?
	- Stated it for a special case of smooth projective varieties and limited sheaves.

- **[Z]** How do you prove it in the case of curves?
	- I tried to do https://math.stanford.edu/~vakil/216blog/Baker-Csirik-serre-duality.pdf but Z directed me to Remark 7.14 of Chapter 3 in Hartshorne.

- **[Z]** OK, now I got tired.

## After the exam
The exam finished. Started leaving the room but didn't know where to go.

S “Don't go too far.”

Z “Go to the other side.”

They apparently finished the discussion and got out of the room.

Z “Where did he go?”

I went to the elevator hall. Maybe the other side meant something else.

Z “Congratulations.”

Shook hands in the order of Z, K, and S because they were in that order from left to right.

Z told me he will send materials related with my project from now on. S went back to his office. K and I left. K “Was it like phew! after the exam?” I “Yeah, I got tired more than I expected.” K “Right? It was like a marathon. See you around!”

The day after that, Z told me that generals were like dentists according to Katz in that the committee hurts you a little and then they stop.