# Riemann surface, elliptic curves

1. Jun 6, 2007

### Ratzinger

Are there notes on the net or books that give a gentle introduction on Riemann surfaces ( say undergraduate math or math for physicists type level)?

Always read of the importance and beauty of Riemann surfaces but can't find surveys or intros for outsiders. Same for elliptic curves.

Maybe a complex analysis book that gives in his last chapters a nice overview on these two topics without too much rigour.

thanks

2. Jun 6, 2007

### mathwonk

survey of resuts on riemann surfaces

Math 8320 Spring 2004, Riemann's view of plane curves

? seems to be either curly d or not equals sign.

Riemann's idea was to classify all complex holomorphic functions of one variable.

1) Method: Consider a convergent power series as representing a holomorphic function in an open disc, and consider two power series as representing the same function if one is an analytic continuation of the other.

2) Problem: Two power series may be analytic continuations of each other and yet not determine the same function on the same open disc in the complex plane, so a family of such power series does not actually define a function on a maximal open subset of the complex plane C.

Solution: Construct the Riemann surface S on which they do give a well defined holomorphic function, by considering all pairs (D,f) where D is an open disc, f is a convergent power series in D, and f is an analytic continuation of some fixed power series f0. Then take the disjoint union of all the discs D, subject to the identification that on their overlaps the discs are identified if and only if the (overlap is non empty and the) functions they define agree there.

Then S is a connected real 2 manifold, with a holomorphic structure and a holomorphic projection S-->C mapping S to the union (not disjoint union) of the discs D in C, and f is a well defined holomorphic function on S.

3) Enlarging the surface: If we include also points where f is meromorphic, and allow discs D which are open neighborhoods of the point at infinity on the complex line, then we get a holomorphic projection ?:S-->P^1 = C U {*}, and f is also a holomorphic function f:S-->P^1.

4) This poses a new 2 part problem:
(i) Classify all the holomorphic surfaces S (moduli problem)
(ii) Given a surface S, classify all the meromorphic functions on S (Riemann Roch problem).

5) The fundamental example: the Riemann surface of a plane curve.
Given a polynomial F(z,w) of two complex variables, for each solution pair F(p,q) = 0, such that ?F/?w (p,q) ? 0, there is by the implicit function theorem, a neighborhood Dp of p, and a nbhd Dq of q, and a holomorphic function w = f(z) defined in Dp such that for all z in Dp, we have f(z) = w if and only if w is in Dq and F(z,w) = 0. I.e. we say F determines w = f(z) as an "implicit" function. If F is irreducible, then any two different implicit functions determined by F are analytic continuations of each other. For instance if F(z,w) = z-w^2, then for each p ? 0, F defines two holomorphic functions w(z) near p, the two square roots of z.

In this example, the surface S determined by F is almost equal to the closure X in the projective plane P^2 of the plane curve {F(z,w) = 0}. More precisely, a point (p,q) of F = 0 is called "singular" if either ?F/?z (p,q) ? 0, or ?F/?w (p,q) ? 0. Just as P^2 is covered by three copies of C^2, the projective curve X has an open cover by three curves of form Fi = 0, and a point is singular in reference to one equation F iff it is singular in reference to the others. If X has no singular points, then X is a complex submanifold of P^2, and S simply equals X. If X has singular points, then (assuming F is irreducible) it has only a finite number. Then S is constructed by removing the singular points of X, and adding back a finite number of points in a "non singular" way as follows.

Consider the open set of X where either ?F/?w ? 0 or ?F/?z ? 0. These are the non singular points of X. To these we wish to add some points in place of the singular points of X. I.e. the set of non singular points is a non compact manifold and we wish to compactify it without introducing singularities. E.g. the complement of the origin on the union of the z and w axes is a non compact manifold, and one local compactification is just the union of the z and w axes. But this introduces a non manifold, i.e. singular, point at the origin. So we wish to introduce two points at the origin, so the z and w axes can pass over each other without intersecting, sort of like a highway overpass to prevent cars from colliding.

Consider an omitted i.e. a singular point p of C. Then projection of C onto at least one axis, either the z or w axis, is a finite covering space from a punctured neighborhood of p to a punctured disc D* centered at the z or w coordinate of p. Such a connected covering space has form t-->t^r for some r ? 1, hence the domain of the covering map, which need not be connected, is a finite disjoint union of copies of D*. We can enlarge this space by adding a separate center for each disc, making a larger real 2 - manifold. Thus on S, p is replaced by possibly several new points, one for each connected component of a small punctured neighborhood of p on X.

Doing this on an open cover of X in P^2, we eventually get the surface S, which is compact, and comes equipped with a holomorphic map S-->X in P^2 which is an isomorphism over the non singular points of X. S is thus a "desingularization" of X. For example if X crosses itself with two transverse branches at p, then S has two points lying over p, one for each branch or direction. If X has a cusp, or pinch point at p, but a punctured neighborhood of p is still connected, there is only one point of S over p, but the disc of which it is the center is no longer pinched. An example of this phenomenon is the map t-->(t^2,t^3) which is a homeomorphism from the large t disc D, onto the curve z^3-w^2 = 0 in ^2 which is topologically a disc, but has a pinch at the center (0,0).

Theorem: (i) The Riemann surface S constructed above from an irreducible polynomial F is compact and connected, and conversely, any compact connected Riemann surface arises in this way.
(ii) The field of meromorphic functions M(S) on S is isomorphic to the field of rational functions k(X) on the plane curve X, i.e. to the field generated by the rational functions z and w on X.

Thus our example exhausts all the compact Riemann surfaces.

Corollary: The study of compact Riemann surfaces and meromorphic functions on them is equivalent to the study of algebraic plane curves and rational functions on them.

6) Analyzing the meromorphic function field M(S).

If S is any compact R.S. then M(S) = C(f,g) is a finitely generated field extension of C of transcendence degree one, hence by the primitive element theorem, can be generated by two elements, and any two such elements define a holomorphic map S-->X in P^2 of degree one onto an irreducible plane algebraic curve X, such that k(X) = M(S).

Question: (i) Is it possible to embed S isomorphically onto an algebraic curve, either in P^2 or in some larger space P^n?
(ii) More generally, try to classify all holomorphic mappings S-->P^n and decide which ones are embeddings.

Riemann's intrinsic approach:

Given a holomorphic map ƒ:S-->P^n, with homogeneous coordinates z0,...,zn on P^n, the fractions zi/z0 pull back to meromorphic functions ƒ1,...,ƒn on S, which are holomorphic on S0 = ƒ-1(z0?0), and conversely these ƒi determine the map ƒ. Indeed the ƒi determine the holomorhic map S0-->C^n = {z0=0} in P^n.

Analyzing ƒ by the poles of the ƒi
Note that since the ƒi are holomorphic in ƒ-1(z0?0), their poles are contained in the finite set ƒ-1(z0=0),and on that set the pole order cannot exceed the order of the zeroes of the coordinate z0 at these points. I.e. the hyperplane divisor {z0 = 0}0 in P^n, pulls back to a divisor ? nipi on S, and if ƒi = zi/z0 then the meromorphic function ƒi has divisor div(ƒi) = div(zi/z0) = div(zi) - div(z0) = ƒ*(Hi)-ƒ*(H0).
Hence div(ƒi) + ƒ*(H0) = ƒ*(Hi) ? 0, and this is also true for every linear combination of the functions ƒ1,...,ƒn.

I.e. the pole divisor of every ƒi is dominated by ƒ*(H0) = D0. Lets give a name to functions whose pole divisor is dominated by D0.

Definition: L(D0) = {f in M(S): div(f) +D0 ? 0}.

Thus we see that a holomorphic map ƒ:S-->P^n with ƒ(S) not contained in H0, is determined by a finite sequence of functions in L(D0) where D0 = ƒ*(H0) is the divisor of the hyperplane section H0.

Theorem(Riemann): For any divisor D on S, the space L(D) is a finite dimensional vector space over C. Moreover, if g = genus(S) as a toplogical surface, then:
(i) deg(D) + 1 ? dimL(D) ? deg(D) + 1 - g.
(ii) If some positive divisor D has dimL(D) = deg(D)+1, then S is isomorphic to P^1.
(iii) If deg(D) > 2g-2, then dimL(D) = deg(D)+1-g.

Corollary of (i): If deg(D) ? g then dim(L(D)) ? 1, and deg(D)?g+1 implies dimL(D) ? 2, hence, there always exists a holomorphic branched cover S-->P^1 of degree ? g+1.

Q: When does there exist such a cover of lower degree?

Definition: S is called hyperelliptic if there is such a cover of degree 2, if and only if M(S) is a quadratic extension of C(z).

Corollary of (iii): If deg(D) ? 2g+1, then L(D) defines an embedding S-->P^(d-g), in particular S always embeds in P^(g+1).

In fact S always embeds in P^3.
Question: Which S embed in P^2?

Remark: The stronger Riemann Roch theorem implies that if K is the divisor of zeroes of a holomorphic differential on S, then L(K) defines an embedding, the "canonical embedding", if and only if S is not hyperelliptic.

7) Classifying projective mappings
To classify all algebraic curves with Riemann surface S, we need to classify all holomorphic mappings S-->X in P^n to curves X in projective space. We have asociated to each map ƒ:S-->P^n a divisor D0 that determines ƒ, but the association is not a natural one, being an arbitrary choice of the hyperplane section by H0. We want to consider all hyperplane sections and ask what they have in common. If h is any linear polynomial defining a hyperplane H, then h/z0 is a rational function f with div(f) = ƒ*(H)-ƒ*(H0) = D-D0, so we say:

Definition: two divisors D,D0 on S are linearly equivalentand write D ? D0, if and only if there is a meromorphic function f on S with D-D0 = div(f), iff D = div(f) + D0.

In particular, D ? D0 implies that L(D) ? L(D0) via multiplication by f. and L(D) defines an embedding iff L(D0) does so. Indeed from the isomorphism taking g to fg, we see that a basis in one space corresponds to a basis of the other defining the same map to P^n, i.e. (ƒ0,...,ƒn) and (fƒ0,...,fƒn) define the same map.

Thus to classify projective mappings of S, we want to classify divisors on S up to linear equivalence.

Definition: Pic(S) = set of linear equivalence classes of divisors on S.

Fact: The divisor of a meromorphic function on S has degree zero.

Corollary: Pic(S) = ?d Pic^d(S) where d is the degree of the divisors classes in Pic^d(S).

Definition: Pic^0(S) = Jac(S) is called the Jacobian variety of S.

Definition: S(d) = (Sx...xS)/Sym(d) = dth symmetric product of S
= set of positive divisors of degree d on S.

Then there is a natural map S(d)-->Picd(S), taking a positive divisor D to its linear equivalence class O(D), called the Abel map. [Actually the notation O(D) usually denotes another equivalent notion the locally free rank one sheaf determined by D.]

Remark: If L is a point of Picd(S) with d > 0, L = O(D) for some D>0 if and only if dimL(D) > 0.
Proof: If D>0, then C is in L(D). And if dimL(D)>0, then there is an f ? 0 in L(D) hence D+div(f) ? 0, hence > 0.QED.

Corollary: The map S(g)-->Pic^g (S) is surjective.
Proof: Riemann's theorem implies dimL(D)>0 if deg(D) ? g. QED.

It can be shown that Pic^g hence every Pic^d can be given the structure of algebraic variety of dimension g. In fact:

Theorem: (i) Picd(S) isom to C^g/L, where L is a rank 2g lattice subgroup of C^g.
(ii) The image of the map S(g-1)-->Pic^(g-1)(S) is a subvariety T of codimension one, i.e. dimension g-1, called the "theta divisor".
(iii) There is an embedding Pic(g-1)-->P^N such that 3T is the divisor of a hyperplane section.
(iv) If O(D) = L in Pic^(g-1)(S) is any point, then dimL(D) = multL(T).
(v) If g(S) ? 4, then g-3 ? dim(singT) ? g-4, and dim(singT) = g-3 iff S is hyperelliptic.
(vi) If g(S) ? 5 and S is not hyperelliptic, then rank 4 double points are dense in singT, and the intersection in P(T0Pic^(g-1)(S)) isom to P^(g-1), of the quadric tangent cones to T at all such points, equals the canonically embedded model of S.
(vii) Given g,r,d ? 0, every S of genus g has a divisor D of degree d with dimL(D) ? r+1 iff g-(r+1)(g-d+r) ? 0.

Roy Smith

3. Jun 6, 2007

### mathwonk

i also have notes on elliptic curves but no time to render them accessible right now.

basic fact: an elliptioc curve over C, is either a plane cubic curve with no singular points, or a quotient of C by a lattice, or an abstract compact conected one dimensional complex manifold of genus one, i.e. with exactly one independent holomorphic differential form.

4. Jun 7, 2007

### Ratzinger

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5. Jun 7, 2007

### neutrino

http://users.ictp.it/~stefanov/mylist.html#Geom [Broken]
http://users.ictp.it/~stefanov/mylist.html#AlgAlgGeo [Broken] (scroll down a bit)

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6. Jun 7, 2007

### mathwonk

the book by mckean looks very interesting and appealing although i was not familiar with it.

other classics on the topic are the wonderful lectures by tate, now made into a book by silverman and tate, listed on amazon as by silverman. and the more advanced wonderful volume 1 of the series on complex function theory by carl ludwig siegel. (also volume 2 is outstanding, but volume 1 is just on elliptic curves.)

7. Jun 8, 2007

### mathwonk

there is a brief but excelent introduction to riemann surfaces and elliptic curves on pages 188-206 of the beginning complex book by Henri Cartan, Theory of analytic functions.