Date: 5/12/2023, 1pm - 4:45pm. Location: Skinner's Office, Fine Hall 501 Topics: Algebraic Number Theory, Algebraic Geometry Committee: Chris Skinner (chair), Shou-wu Zhang, Ruobing Zhang Shou-wu attended my Generals virtually, as he was in California. I arrived at Skinner's office 10 minutes early to figure out the Zoom link and make sure that Shou-wu could see the board clearly. Throughout, everyone was extremely helpful, and the atmosphere felt much more fun and relaxed than I had anticipated, though I still got extremely nervous at times. The fact that Shou-wu was participating virtually was barely felt, as he was engaged and vocal the whole time. Here are the sources I used primarily while studying. ALGEBRA: J.S. Milne's Group Theory and J.S. Milne's Field Theory, first 7 chapters of Fulton-Harris, for review. REAL ANALYSIS: Stein-Shakarchi books 1, 3, 4. COMPLEX ANALYSIS: Wikipedia ALGEBRAIC GEOMETRY: Vakil's Rising Sea Chapters 1-24 and Chapter 29, skipping a lot of starred sections, supplemented with parts of Hartshorne Chapters I-IV in the last month before the exam. BTW there is a great proof of Serre Duality (restricted to the curves case) in Shou-wu's online AG notes which is particularly enlightening. ALGEBRAIC NUMBER THEORY: Serre's Local Fields Chapters 1-6, then Milne's Class Field Theory (except for the Lubin-Tate theory in ch1). Milne's ANT notes are also helpful. \section*{Complex Analysis} \begin{enumerate} \item Ruobing: What is a singularity? Can you classify the possible types of singularities? \item Ruobing: What is an entire function? \item Ruobing: Suppose I give you a sequence of points in the complex plane. Is there an entire function which vanishes precisely at those points? (Me: yes, but only if they don't accumulate. Should I prove that? Ruobing: Yes! I showed that in that case, the function is identically 0. If they are non-accumulating, I proved that such a function exists as an infinite product of the form \[ f(z) = z^{m}e^{p(z)}\prod_{k=1}^{\infty }\left( 1-\frac{z}{z_{k}} \right) e^{\sum_{n=1}^{m_{k}}\frac{1}{n}\left( \frac{z}{z_{k}} \right) ^{n} } ,\] and proved that you could pick the $m_{k}$ such that this converges absolutely.). \item Ruobing: Do you know anything about the orders of entire functions? What is Weierestrass infinite product? (If the function is of order $\alpha $, you can pick all $m_{k} \leq \alpha $ for each $k$, and $\deg p(z) \leq \alpha $). \item Ruobing: Can you construct an entire function with order that is not an integer? For instance, one with order $\alpha =1/2$? (I thought for a little while, and was told to consider $\cos \sqrt{z}$. I first wrote this down in terms of $e^{i \sqrt{z}}-e^{-i \sqrt{z}}$, but then realized I was missing the point entirely, since $\cos \sqrt{} $ can be defined holomorphically, but $\sin \sqrt{} $ can't. After some prodding I wrote the Laurent series and realized the same idea worked for any even entire function). \item Ruobing: What is the Schwarz Lemma? Can you prove it? \item Ruobing: What are all the automorphisms of the disc? \item Ruobing: Let $S$ be a Riemann surface, and suppose I have a meromorphic function $f$ on $S$, which has only one pole, and it is simple. What can you say about $f$? (Me: I'm not very familiar with Riemann surfaces). \item Ruobing: OK let's slow down then. Suppose $S$ is just the torus. What happens? A long discussion ensued, as I didn't know much about Riemann surfaces. I kept trying to use algebraic geometry, and then I would be corrected and told that that wasn't allowed because we were in the complex analysis section. Some sub-questions that came up: \item Shou-wu: What are the automorphisms of $\C \mathbb{P}^{1}$? Me: Well, this is a map $\mathbb{P}^{1}_{\C }\rightarrow \mathbb{P}^{1}_{\C }$, so you can pull back the hyperplane sections $x_{i}$, and show that we just have $PGL_{2}\left( \C \right) $. Shou-wu: ``No! This is complex analysis! And I said $\C \mathbb{P}^{1}$, which is a set, and not $\mathbb{P}^{1}_{\C }$, which is the same set as a projective variety!'' \item Shou-wu: Universal covering space for $\C \mathbb{P}^{1}$? \item Shou-wu: Automorphisms of $\C $? (I argued this was also an automorphism for $\C \mathbb{P}^{1}$, because the singularity at infinity had to be a pole, since if it was essential, we would be violating the injectivity by Big Picard Theorem or something less powerful; Shou-wu said this was an elegant solution, and I told him I thought he was just being nice). I was pretty flustered by this point and Skinner gave me a break from Riemann Surfaces. \item Skinner: What's your favorite theorem in complex analysis? Me: Rouche's theorem. Skinner: (Laughing) If Gunning were on your committee, you would fail immediately. The answer is the Riemann Mapping Theorem. Me: I can pretend that was my answer, because I know how to prove it! Skinner: No, what's Rouche's theorem? And what are the key ideas in the proof of that? (I said the key idea was a homotopy, which got me some weird looks so I just proved it). \end{enumerate} \section*{Real Analysis} \begin{enumerate} \item Ruobing: What does it mean for a function to be Riemann integrable? \item Ruobing: Is there a necessary and sufficient condition for this to hold? (I wrongly said $f$ is almost everywhere continuous, before they started to correct me and I said it had to be bounded, too). \item Ruobing: Okay, can you define the Lebesgue measure from the beginning? \item Ruobing: Can you construct an open set $E$, for which the characteristic function of that set is not Riemann integrable? \item Ruobing: Okay, define Lipschitz functions. (Me: writes something completely wrong. Ruobing: what? I don't even know what that is. The correct definition is...) \item Ruobing: Prove that Lipschitz functions are almost everywhere differentiable. (I asked for a hint). \item Ruobing: Do you know what functions of bounded variation are? (Me: Yes! and they're almost everywhere differentiable. I defined functions of bounded variation, gave a short explanation of why they're almost everywhere differentiable without proof, and gave the proof that Lipscitz functions satisfy that condition). \item Ruobing: Okay, suppose I have a null set $E \subseteq \R $, and a Lipschitz function $f$. What can you say about the Lebesgue measure of $f\left( E \right) $? (It's null, by the explicit definition of the outer Lebesgue measure and the Lipschitz condition). \item Ruobing: Okay, now a hard question. Let $f$ be a function on the interval satisfying $\left| f(z)-f(w) \right|\leq \alpha \left| z-w \right|^{1/2}$. Can you construct such an $f$ which is not of bounded variation? (I struggled for a second, and then he suggested taking something which looked like a bunch of triangles, and I drew out the desired shape. I said the triangles could have base length $1/n$ (everyone: What??) me: no, $1/n^2$! everyone: OK), and it worked out by the divergence of the harmonic series). Ruobing: Okay, I am satisfied on real analysis. Any more questions? \item Shou-wu: Prove the functional equation for the Riemann $\zeta $ function. (I went through the whole computation, using Poisson summation to write $\theta \left( 1/t \right) =t^{-1/2}\theta \left( t \right)$. I crucially forgot that you need to split the integral into two pieces, $\int_{0}^{1} $ and $\int_{1}^{\infty } $, due to the pole at $s=1$, which I corrected after some prodding. Also, I kept forgetting how to change variables in an integral; Skinner joked that perhaps after generals I could learn calculus.). Skinner prompted us to move onto algebra, in the interest of time. \end{enumerate} \section*{Algebra} \begin{enumerate} \item Shou-wu: What is a Dedekind Domain? Can you classify all finitely generated modules over a Dedekind domain? (I said Noetherian, dimension 1, integral domain which is integrally closed, is sufficient. More intuitively, we are imposing sufficient conditions to make an affine scheme which is a curve whose stalks are DVRs. If $M$ is a finitely generated module and $A$ the dedekind domain, then \[ A \simeq \oplus A/\mathfrak{p_{i}^{a_{i}}} \oplus A^{s-1} \oplus \mathfrak{a} ,\] where $\mathfrak{a}$ is a fractional ideal, and $s$ the rank of the torsion-free part). \item Shou-wu: Is this unique? (The torsion part is unique up to ordering by Chinese Remainder Theorem, and the fractional ideal has to be a representative of the right element of the class group). (SIDE-NOTE: a full proof of this fact can be found split between two sources: the end of Chapter 1 of Lang's Algebraic Number Theory, and Curtis and Reiner, Representation Theory of Finite Groups and Associative Algebras, Wiley (1962), Section 22. The key idea is that all finitely-generated torsion-free $A$-modules must be projective, so that after you get rid of the torsion stuff, all the exact sequences you can possibly write all have to split. After this it is not hard to take your module and start extracting rank 1 pieces, all of which you can show are isomorphic to a fractional ideal. Then you just have to show that for two sets $\{ \mathfrak{a}_{i} \}_{i=1}^{s}$, $\{ \mathfrak{b}_{i} \}_{i=1}^{s}$ of fractional ideals, $\oplus \mathfrak{a}_{i}\simeq \oplus \mathfrak{b}_{i}$ if and only if $\prod_{}^{}\mathfrak{a}_{i}\equiv \prod_{}^{}\mathfrak{b}_{i}$ in the class group.) \item Shou-wu: What's the class group? Can you define it for a Dedekind domain? \item Skinner: Can you construct something with nontrivial Class group? Why is the class group finite over number fields? Can you give a Dedekind domain with infinite class group? (Me: any elliptic curve with infinitely many points). \item Skinner: Can you give an example of such an elliptic curve? (Me: $\C [x,y]/\left( y^2-x^3-x \right) $). Skinner: Okay, now make that $y^2-x^2-x$. Can you compute the class group of this ring? Do it both for $\R $ and for $\C $. (I showed using excision that it was a UFD in both cases, as long as the polynomial was irreducible over both $\R $ and $\C $. I thought maybe it would have to be reducible over $\C $ but irreducible over $\R $ since otherwise there was no difference in the two cases, and mumbled for a while before Skinner told me it was irreducible over both and to move on). \item Shou-wu: Okay that's enough algebra. Skinner: Wait, we should ask some more. Do you guys have more questions? Me: You could ask me some group theory, representation theory, or Galois theory... \item Shou-wu: Oooh, representation theory! Of what kind! Lie groups? Me: finite groups! Please finite groups! \item Skinner: OK. How many irreducible representations are there for $GL_2$ over a finite field? (Me: as many as there are conjugacy classes. I computed the number of conjugacy classes, using rational canonical form). \item Can you construct these representations explicitly? (We talked for a little bit about inducting on abelian subgroups like the Borel, but as representation theory is not one of my special topics, we quickly moved on). \item Skinner: Consider formal power series over the $p$-adics, $\Z _{p}[[T]]$. Is this a UFD? \item Shou-wu, before I could answer: Is it a DVR? (Me: It's definitely not a PID, since $\left( p, T \right) $ is not principal). (Skinner walked me through a proof of Weierstrass Preparation Theorem). Skinner: Okay, shall we move onto number theory? \end{enumerate} \section*{Algebraic Number Theory} \begin{enumerate} \item Skinner: Give me your favorite two primes bigger than 2 and 3. I wrote down $p,q \neq 2,3$ on the board. Skinner: No, I meant it literally. Give me two primes. (5,7). Okay, consider $\Q \left( \sqrt{5\cdot 7} \right) $. What can you say about it? Can you compute the class group? (I showed that $\left( 2 \right) =\mathfrak{p}_{2}^2$ and $\left( 5 \right) =\mathfrak{p}_{5}^2$ and that these two elements should generate the class group, and that $\mathfrak{p}_{2}$ was definitely not principal. I struggled with showing that $\mathfrak{p}_{5}$ was not principal; it was only as we were about to move on that I yelled, consider $\left( 5+\sqrt{35} \right) $, which has norm $-10$; then, $\left( 10 \right) =\mathfrak{p}_{2}\mathfrak{p}_{5}$, which tells you that $\mathfrak{p}_2=\mathfrak{p}_{5}^{-1}$ in the class group, and thus the class group is $\Z /2$). Shou-wu then gave some big-brain reason for why the class number was actually 4. Skinner disagreed and said my solution showing that it's 2 was correct. Since Shou-wu suggested that the primes above 7 should give something new, I pointed out that considering the norm of $\left( 7+\sqrt{35} \right) $ would show that $\mathfrak{p}_{7}=\mathfrak{p}_{5}=\mathfrak{p}_{2}$ in the class group. I think the conclusion was that me/Skinner were right. \item Skinner: Okay, now that you know the class group for $\Q \left( \sqrt{35} \right) $, what does class field theory tell you about its extensions? (I gave the Hilbert Class field explicitly, and justified that it was the right thing). \item Skinner: Okay, now a hard question. Suppose $K$ is a number field, and $L/K$ an infinite extension for which $\Gal \left( L/K \right) \simeq \Z _{p}^{d}$, where $\Z _{p}$ are the $p$-adics. What can you say about $K$? (The ensuing computation took a long time, and throughout it I was asked: Can you state local class field theory and see where the inertia group goes? How does $L$ ramify at different primes, including archimedean ones? Can you describe the artin map explicitly on each component of the ideles? We concluded by talking about bounding $[K:\Q ] \geq d$; Skinner asked me if this was optimal, and we found that it should actually be higher, using the fact that the Artin map vanishes on the global units. Skinner was definitely guiding me through this section, as I hadn't seen anything like this; I later realized the content is all in Washington's Cyclotomic Fields). \end{enumerate} By the end of number theory, we finally took a break, and Ruobing and I went for cookies together and brought Skinner some cookies back. In the elevator ride, Ruobing asked me where I was from (Costa Rica!). I asked him if he used to be a student here, and he said yes. Ruobing asked me how I felt, I said I felt OK, and he said my generals were going better than his did. I told him I was pretty sure he was lying. He laughed, and didn't confirm or deny this. \section*{Algebraic Geometry} \begin{enumerate} \item Shou-wu: What is your definition of a curve? (Me: dimension 1, projective, smooth, geometrically integral $k$-scheme, for $k$ a field). \item Shou-wu: What is the most important invariant associated to a curve? (Genus). Can you define it in a few different ways? At this point Ruobing had to leave, and said, ``I have to go to a meeting but, he's good on the analysis. Bye!'' \item Shou-wu: Okay, can you classify all curves of genus 0? (Me: If you have a closed point, it's $\mathbb{P}^{1}$. Otherwise, it's a conic in $\mathbb{P}^2$ [proof]). \item Shou-wu: Can you classify all conics in $\mathbb{P}^2$? Me: Are you asking me to classify all quadratic forms? Shou-wu: Yes. Do it over $\C $, then $\R $, then $\Q $. (Over $\Q $, I essentially laid out the argument in the relevant exercise in Cassels-Frohlich; the idea is that two forms are equivalent iff they are equivalent over each $\Q _{p}$ iff they have the same rank (in this case, always 3) and Hilbert symbol for each prime. I defined the Hilbert symbol (both using the explicit form, if $x^2-ay^2-bz^2$ has a solution, and as the action of the local Artin map). There were several follow-up questions, such as: why is $\prod_{}^{}\left( a,b \right) _{v}=1$? Why does the Hilbert symbol have to be 1 at all but finitely many primes? What is the number of primes at which the symbol is $-1$? (even). Given a finite (even) set of primes for which the symbol is $-1$, can you construct a quadratic form which has Hilbert symbol $-1$ at exactly these places? I said it should be possible in general but I didn't know how. They asked me to make one for which $\left( a,b \right) _{\infty }=\left( a,b \right) _{2}=-1$, and all others were 1; I realized that $x^2+y^2+z^2$ works). \item Shou-wu: Okay, let's do genus 1. Me: Let's pretend it has a rational point, so that it's an elliptic curve. Then you can describe it as a cubic in $\mathbb{P}^2$, or as a hyperelliptic mapping to $\mathbb{P}^{1}$. I proved this, and we talked about the formula for the Weierstrass form. We talked about the $j$-invariant. \item Shou-wu: Can you write down some explicit differentials for the curve, using the Weierstrass form? (I did this with a bit of help, and realized along the way that the curve being smooth is what makes this possible). \item Shou-wu: What can you say about curves of genus 1 without a rational point? Me: I didn't know what to do. Shou-wu: Can you construct the Jacobian? I defined it in general and we talked about it briefly, but we moved on instead. \item Shou-wu: Okay, let's move onto genus 2. What can you say about these? I stated some general facts about the canonical bundle for genus $g \geq 2$ (following Hartshorne), then showed that all genus 2 curves are hyperelliptic, and showed that the mapping corresponds to a polynomial $y^2=f(x)$, where $\deg f = 2g+2$, and the roots of $f$ correspond to the branch points. \item Shou-wu: Can you write down some explicit differentials for the curve? \end{enumerate} After this, the exam was over. I was asked to leave the room, and when I came back, Skinner congratulated me and said I passed. After we talked for a while, Skinner took me upstairs (where there was a Wine and Cheese going on), and he announced to the room that I passed, and everyone clapped; it was a very nice moment!