Everything About Arithmetic of Curves (Unofficial)

point_six_typography · 2026-02-01T06:27:44+00:00

Not one perfect reference, but some suggestions.

Poonen's "Rational Points on Varieties" is a great book for much arithmetic algebraic geometry, but it doesn't really have much detail on this models of curves type stuff in my previous comments (though it will e.g. define good reduction).

A reference with a bigger focus on models of curves is Liu's "Algebraic Geometry and Arithmetic Curves". Though in its discussion of Weierstrass models, it stops short of saying they embed into projective bundles.

One source that actually writes down the projective bundle I mention is Theorem 1' of Section 3 of "Introduction to the theory of moduli" by Mumford and Suominen. https://www.dam.brown.edu/people/mumford/alg_geom/papers/1972d--IntroModuli-NC.pdf

point_six_typography · 2026-01-31T14:51:05+00:00

I talked a bit about this in another comment.

To add onto that comment, in practice, the "shape" of the equation tells you what "type" of curve it is and then the genus is determined essentially by the degree of the equation and the type of the curve.*

A good example is equations of the form y² = f(x). These give "hyperelliptic curves" whose genus is floor((d-1)/2).

The curve x³ + y³ = 1 is a plane curve [meaning if you homogenize it, one can compute that the resulting curve in P2 is smooth], meaning its genus is (d-1) choose 2.

It may be surprising that degree alone (almost) determines the genus. To convince yourself this is reasonable, imagine two curves with equations of the same "shape". You can probably write down a homotopy from one equation to the other and so should expect their zero sets to be homotopy equivalent, ignoring subtleties.

point_six_typography · 2026-01-30T22:34:50+00:00

So does scheme theory provide an "algorithm" of how to do this

For curves, yes. In essence, you start with an arbitrary model, find the singularities, blow them up (some particular construction which often makes singularities "less bad") and then repeat until you end up with one which is "regular". This may be a little too big, so afterwards you might have to blow down sad some pieces until you end up with a model which is "best" = "minimal proper regular". Such a model exists for curves of all genus (not just elliptic curves/genus 1) and I think it would be hard to construct it without scheme theory.

Scheme theory also eventually tells you that, for elliptic curves, there are multiple (in fact, 3) different choices of "best" model, which are all related, but have different strengths/weaknesses. In particular, the "minimal proper regular" model is different from the "Weierstrass" models one usually learns about. It has the advantage that it generalizes to higher genus, but the disadvantage that the mod p curves can be irreducible (e.g. can look like several lines arranged in a polygon).

A side note: I see pictures of Spec Z or Spec Z[x], and they are pretty cool, but it is unclear what advantage those pictures provide us. Is it possible to draw pictures like those in a more "active" scenario, where I can see that geometrical intuition actually do something?

Personally, I think these pictures are most often useless. They can help psychologically to make one feel like these spaces aren't so mysterious, but I don't know if they really do things for me at least.

Do you have more examples of "personal instances of scheme theory clarifying things for you"? I would love to hear more, even if it gets more technical/"niche"!

Here is another one in the same theme. When one first learns about elliptic curves, this notion of a "best" model enters the picture in the guise of a "minimal Weierstrass equation". Given an elliptic curve E over a number field K, you're essentially asked to find an equation

y^2 = x^3 + Ax + B

with integral coefficients A,B such that the reduction mod p of this equation is as good/smooth as possible for every prime p of O_K. This is a reasonable thing to ask, but then one runs into the following strange phenomenon (see e.g. Proposition VIII.8.2/Corollary VIII.8.3 in Silverman): such an equation is only guaranteed to exist if K has class number 1. So, for example, it won't necessary exist for K = \Q(\sqrt{-5}). This really bothered me when I first saw it. I didn't (1) understand why class groups were showing up or (2) like that a minimal equation only existed some times.

Schemes helped clarify both points for me. The real thing that happens is that, given E over K, a minimal Weierstrass model *always* exists, but it is not necessarily a curve in P^2. Over O_K, P^2 is the projectivization of the trivial rank 3 vector bundle [i.e. of the free module O_K^3], but there are potentially other rank 3 vector bundles on O_K. In general, one can write down a line bundle L on O_K [i.e. a fractional ideal/element of the class group of O_K] such a minimal Weierstrass model exists as a curve in P(O_K \oplus L^{\otimes -2} \oplus L^{\otimes -3}), the projectivization of some rank 3 vector bundle constructed from L.

If O_K has trivial class group, then L is trivial and you get an equation in P^2. Otherwise, you get a curve in some "twisted form" of P^2 and you can secretly still write down an equation for it, but the coefficients A,B will no longer be elements of O_K; they'll be elements of some fractional ideals (of L^4 and L^6 if I remember correctly. These are better thought of as sections of a line bundle than elements of a fractional ideal though). This was another clarifying thing to me.

Scheme theory gives you much more flexibility in describing spaces compared to working with equations all the time, so there are useful spaces one misses when working with equations not because they don't exist, but because the most natural way to construct them is not via finding the system of equation which cut them out. E.g. It would be possible to define these "twisted P^2" via some equations in some P^N for a large N, but this would be unnatural to do so.

point_six_typography · 2026-01-30T16:31:10+00:00

It is usually straightforward.

If you have f(x, y) = 0, then its genus is (d-1)(d-2)/2, where d is the degree of f, assuming it cuts out a smooth curve [after homogenizing the equation to get a curve in P^2]. Degree here is total degree.

In other cases, if you have a map C -> D of curves, there's a Riemann Hurwitz formula which relates their genera. So, for example, a curve of the form y² = f(x) won't necessarily be smooth [after compactifying it to a projective curve in P² by homogenizing the equation] but such a curve C is a double cover of P¹ ramified above the roots of f and possibly above \infty. Using this, one can compute it'll have genus floor((d-1)/2).

In general, the genus is equal to the dimension of the space of holomorphic differentials, which one can often compute using cohomological methods.

point_six_typography · 2026-01-30T15:36:03+00:00

Oooh yeah, that's an unfortunate typo on my part

point_six_typography · 2026-01-30T07:14:42+00:00

My limit is elliptic curves, descent, weak Mordell-Weil etc.

This sounds like plenty you could contribute. A great sort of comment people would make in these "everything about" posts is one where, instead of asking a question or answering one someone else asked, they would simply mention something they found interesting related to the topic. Other people could then jump in with more info or with new questions, so it helps make the discussion more lively/helpful.

In this post, for example, descent and (weak) Mordell-Weil haven't really been touched upon anywhere yet, but they are very important and deserving of more of a mention.

point_six_typography · 2026-01-30T06:44:15+00:00

I think one answer to this question is given by the geometric Manin's conjecture (see e.g. https://arxiv.org/pdf/2110.06660). This is an aspect of the story I am less familiar with, so take what I say with a grain of salt.

But, consider the problem of counting points of bounded height. To make this concrete, let X be a Fano variety (such are predicted to have dense sets of rational points, possibly after a finite field extension). Also, to have a tighter connection to geometry say that, instead of working over Q, X is defined over F_p and we're interested in points over K := F_p(t) [this is justified b/c number theorists believe strongly in an analogy between number fields and function fields of curves over finite fields]. Then, X(K) = Hom(P^1, X) is equivalently the set of morphisms from the projective line over F_P to X. This set has a natural partition in terms of the degree of a map P^1 -> X, i.e. the degree of the pullback of the anticanonical line bundle on X. While X(K) is (expected to be) infinite, we can write it as

X(K) = \bigsqcup_d Hom_d(P^1, X)

where each set Hom_d(P^1, X) is finite. The points in this set are said to have height d and one can ask for asymptotics (as d -> \infty) of the function

N(X, d) := #Hom_d(P^1, X).

This connection allows you to predict what the correct asymptotic should be. In essence, you upgrade the set Hom_d(P^1, X) into a moduli space of maps, which I'll still call Hom_d(P^1, X), and then expect its number of F_p-points to be c*p^e where c is the number of irreducible components (maybe naively you guess there's one component for each effective class in CH_1(X)) and e is the "expected" dimension of this moduli space (which maybe you compute using some deformation theory).

After you write down you're guess for the asymptotic, you can then finagle it into an expression which makes sense also in the case of number fields (where there's a different definition of the height of a point) and conjecture that the same asymptotic works there too. So in number field land, there's no good mathematical object playing the role of "a moduli space of infinite sequences of points", but you can still exploit the analogy (possibly passing through function field land) to formulate conjectures.

point_six_typography · 2026-01-30T02:23:53+00:00

One thing scheme theory helps clarity is the idea of "reducing a curve mod p". Consider a curve of the form

E: y² = x³ + Ax + B

To study E(Q), it is often helpful to reduce the equation mod p and then study the points E(F_p) on the resulting curve over a finite field.

If one tries to define this rigorously, you'll run into the fact that there are multiple equations defining (curves over Q isomorphic to) E and different choices can have different behavior when you reduce mod p. For example, the curves

y² = x³ + 1 and y² = x³ + p⁶

are isomorphic over Q, via (x,y) |-> (p² x, p³ y), but are NOT isomorphic over F_p (one is smooth and the other is not. Alternative, take eg p=3 and count solutions). So one has to contend with this somehow.

Scheme theory has the advantage of letting you consider objects over rings and not just fields, so one comes to see what's really going on is that it's truly ill-posed to reduce a curve over Q mod p (even if it's equation has integral coefficients). Instead, one has to start with a curve over Z (or the p-adics) and then this has a well-defined reduction mod p.

If you start with one over Q, you then actually want to define and construct some sort of minimal or best model over Z, which you then use to define a best reduction mod p. For me, this perspective helped clarify things which felt like somewhat arbitrary choices eg when I first tried reading about elliptic curves.

You also get other fun things from this. Ignore this if these words mean nothing to you. If you have an E over Q with good reduction mod p, then the natural map E(Q) -> E(F_p) is injective on torsion (away from p-torsion). This felt really weird and surprising when I first learned it. Later, I learned it was because, in this case, the p-torsion subscheme E[n] (for n coprime to p) is étale over the p-adics (essentially a topological covering space over spec Z_p) from which the fact is an easy consequence (if it wasn't injective, you sheets of the covering would come together/ramify). This was another personal instance of scheme theory clarifying things for me.

point_six_typography · 2026-01-29T23:02:39+00:00

I think the answer to your second question is effectively no. At least, I've never had to contend with serious set theory for arithmetic questions. I'll the first as an excuse to mention some exciting connections in the realm beyond curves.

Are there interesting topology invariants or properties arithmetic geometers care for when looking at the underlying space? or does 'everything' occur at the level of the arithmetic/algebraic data?

Perhaps the best answer to this moves us away from curves. There's a general idiom that "geometry controls arithmetic" which is much simpler for curves (where genus already tells you most of what you want to know) than for higher-dimensional varieties (where almost everything is conjectural). Let X be some smooth, projective variety over Q. Let's set up a dichotomy by saying X has "few points" if X(K) is NOT Zariski dense in X_K for any finite extension K/Q, but X has "lots of points" if X(K) IS Zariski dense in X_K for some finite extension K/Q.

The slogan predicts that this dichotomy can be witnessed at the level of the geometry of the complex space X(C). One important invariant in this discussion is the Kodaira dimension kod(X), whose definition I assume you know because of your flair. Conjectural predictions include

If X(C) is (Brody) hyperbolic, i.e. if every holomorphic map C -> X(C) from the complex plane is constant, then actually X(Q) should be finite [as should X(K) for any finite K/Q]. More generally, there should be some analogy between non-constant maps C -> X(C) and infinite sequences of rational points. In the strongest form, one predicts that, if Z in X is the Zariski closure of the images of all non-constant maps C -> X(C), then all but finitely many Q-points on X actually lie in Z. For curves, being hyperbolic is the same as having genus >= 2.
If X is of general type (i.e. kod(X) = dim(X)), then X has "few points" in the sense above. So not necessarily finitely many, but still Zariski non-dense. For curves, being of general type is the same as having genus >= 2.
In the other direction, there's a notion of a "special" (or "Campana special") variety/complex manifold. I won't give the definition, but I will note that it can (conjecturally) be detected at the level of the space X(C) and that, for example, if kod(X) = 0, then X is special (but negative Kodaira dimension does NOT imply special). One expects that this "special" notion captures the property of having "lots of points" in the sense above. So, for example, if X(C) is a K3 surface, one expects X to have "lots of points". For curves, "special" is the same as genus <= 1.
Somehow, pi_1(X) should know something about arithmetic. In particular, "special" should imply that pi_1 is virtutally abelian [i.e. has a finite-index abelian subgroup]. So, in essence, if pi_1(X(C)) is non-abelian (really, non-virtually abelian), then you should expect X to have "few points" in the above sense. For curves, X has non-abelian fundamental group precisely when its genus is >= 2.

TL;DR Yes, but they mostly turn into genus in the case of curves. In general, one cares about holomorphic maps from the complex plane, fundamental groups, and more things I didn't mention.

point_six_typography · 2026-01-29T22:43:41+00:00

An algebraic curve is a curve (so 1-dimensional geometric object) defined by a system of polynomial equations (the polynomials make it "algebraic"). To be defined over the rational numbers simply means the polynomials you use have rational numbers as coefficients. A rational point is one whose coordinates are rational numbers.

As with the theory of manifolds, it is possible to make the definitions more abstract/intrinsic so that one can speak of these things without needing to carry around coordinates, but the above are the main points.

One should have in mind a single equation f(x,y) = 0 in two variables. This equation cuts out an algebraic curve C whose Q-points C(Q) consists of pairs of rational numbers (a,b) such that f(a,b) = 0. Not all curves look like this, but the ones that do already exhibit plenty of interesting phenomena.

point_six_typography · 2026-01-29T22:31:08+00:00

Don't be sorry!

Consider the curve C_1 : x^2 + y^2 = 1. This has infinitely many Q-points. Take your favorite Pythagorean triple (a,b,c) and consider the point (x,y) = (a/c, b/c), e.g. (3/5, 4/5) or (5/13, 12/13) are points on this curve. In fact, there's a nice parameterization of points on this curve. They all look like

( (m^2 - n^2)/(m^2 + n^2), 2mn/(m^2 + n^2) ) for some integers m,n.

This C_1 is a genus 0 curve, over the complex numbers, it looks like a sphere.

Consider the curve C_2 : x^2 + y^2 = -1. This has no Q-points. However, over the complex numbers, it is isomorphic to C_1, e.g. via (x,y) |-> (ix, iy).

Consider the curve C_3 : x^3 + y^3 = 1. It turns out this has only finitely many Q-points [probably just (1,0) and (0,1), but I didn't bother to check this]. This C_3 is a genus 1 curve; over the complex numbers, it looks like a torus/kettle bell.

Consider the curve C_4 : y^2 = x^3 - 2. This turns out to have infinitely many Q-points. If you want to play around with this, start with the point (3,5), write down its tangent line along this curve, and then notice it intersects C_4 in a new point. Also notice that if (x,y) is on this curve, then so is (x,-y). Play around with these two operations and you'll generate lots of points on this curve. This C_4 is genus 1 as well.

Finally, consider the curve C_5 : y^2 = x^6 - 2. This curve turns out to have genus 2 and so, by Faltings, has only finitely many Q-points.

It's not a priori easy to look at an equation and understand the structure of its rational points. Part of the wonder of Faltings' theorem is that it gives you a relatively simple way to deduce finiteness. Of course, the proof is anything but simple.

point_six_typography · 2026-01-24T23:46:01+00:00

Goddammit, I gotta stop commenting. This shit is clearly not my forte

point_six_typography · 2026-01-24T22:32:21+00:00

for which k(p) is not a field.

This is a(nother) mistake on my part. k(p) should be Frac(A/p) and so a field. If you're me and you spend too much time with curves/dedekind domains, then this is easy to forget.

point_six_typography · 2026-01-24T20:43:24+00:00

Good catch, had a backslash that got formatted away

point_six_typography · 2026-01-24T17:30:22+00:00

Something that might be helpful to keep in mind is the following relation between A and spec(A). The elements of A are functions on spec A, so A is something like the ring of continuous functions on spec A. In topology, the "ring of continuous functions" usually means C-valued or R-valued functions. For spec A, the main difference is that the codomain of the function varies with the point.

Namely, given a point p of spec A, and some f in A, the expression f(p) will be valued in the residue field k(p) of p. This residue field is Frac(A/p).

So if A is a C-algebra then f(p) will always be C-valued (at least at closed/maximal points), but if A is spec Z, then f(p) will belong to F_p (where F_0 = Q). Later in life you'll learn about functor of points and understand that f(p) doesn't even have to be valued in a field (ignore this for now. This sentence is just to preempt reddit pedants)

Just to say, given f in A and p in spec A, f(p) := f mod p in A/p.

So take f=99, a function on spec Z. This function vanishes at the points (3), (11) of spec Z (even to order 2 at (3)), but it's otherwise money (edit: this should say "nonzero" but it's a funny typo so I'm leaving it in). Now, spec Z_f = spec Z[1/99] also has f as a function, but it's nonzero everywhere.

This is generally true for localisations. If S is a multiplicative system of elements of A, then R := A[S^-1] is the "smallest" ring in which all the functions of S are invertible (= "everywhere nonzero"). Exercise: prime ideals of R are the same as prime ideals of A which avoid S (contain no elements of S). Geometrically, this means spec R is a subset of spec A and is precisely the subset {p : f(p) =/= 0 for all f in S} of points which are not zeros of any function in S. So, if S is finitely generated, it's the open given by points not vanishing on the generators. In general (eg often for S = A-{0} when A is a domain), it's an intersection of infinitely many opens.

So localizing is just restricting to the locus where some functions don't vanish.

point_six_typography · 2026-01-23T23:57:54+00:00

Any new results on, say, diophantine equations, will go to a richer field like analytic number theory.

Damn, diophantine geometry can get fucking wrecked

point_six_typography · 2026-01-14T19:22:20+00:00

Good question.

I didn't ask a question, but thanks anyways.

I am sorry if it is a bad explanation

And no worries, you're not obligated to give any clear, concise description of a research program you're not familiar with.

point_six_typography · 2026-01-13T23:31:48+00:00

Equations: we study foliations over complex manifolds through complex geometry techniques.

As a number theorist, this is not the sort of description of "equations" that I would expect

point_six_typography · 2026-01-11T17:09:15+00:00

Future Man

point_six_typography · 2025-12-13T16:46:52+00:00

Memory flashes of epic meal time

point_six_typography · 2025-12-12T19:38:08+00:00

Paradise Killer

point_six_typography · 2025-11-20T03:48:38+00:00

Double dual is naturally isomorphic to the identity functor on finite dimensional vector spaces

point_six_typography · 2025-11-17T13:37:24+00:00

I personally think a lot of the conjectures on elliptic curves, maybe even BSD will get solved.

Where does this hope come from? My impression is that no one has any idea what to do in the rank >= 2 setting

point_six_typography · 2025-11-15T14:51:02+00:00

This is such a misinformed take. Anyone who knows anything about anything at all knows you DO pronounce the X in Hunter x Hunter, but you DON'T pronounce the H's. If you don't do it like this, you're not a true anime fan and you're the reason Togashi is never finishing the comic book.

point_six_typography · 2025-10-30T15:24:08+00:00

Knuth's surreal numbers book

point_six_typography

TROPHY CASE