M g = {isom. classes of smooth projective curves of genus g} and

24 JENIA TEVELEV 2. Elliptic curves: j-invariant (Jan 31, Feb 4,7,9,11,14) After the projective line P 1, the easiest algebraic curve to understand is an elliptic curve (Riemann surface of genus 1). Let M 1 = {isom. classes of elliptic curves}. We are going to assign to each elliptic curve a number, called its j-invariant and prove that M 1 = A 1 j. So as a space M 1 A 1 is not very interesting. However, understanding A 1 as a moduli space of elliptic curves leads to some breath-taking mathematics. More generally, we introduce and M g = {isom. classes of smooth projective curves of genus g} M g,n = {isom. classes of curves C of genus g with points p 1,..., p n C}. We will return to these moduli spaces later in the course. But first let us recall some basic facts about algebraic curves = compact Riemann surfaces. We refer to [G] and [Mi] for a rigorous and detailed exposition. 2.1. Algebraic functions, algebraic curves, and Riemann surfaces. The theory of algebraic curves has roots in analysis of Abelian integrals. An easiest example is the elliptic integral: in 1655 Wallis began to study the arc length of an ellipse (X/a) 2 + (Y/b) 2 = 1. The equation for the ellipse can be solved for Y : Y = (b/a) (a 2 X 2 ), and this can easily be differentiated to find Y = bx a a 2 X 2. This is squared and put into the integral 1 + (Y ) 2 dx for the arc length. Now the substitution x = X/a results in 1 e s = a 2 x 2 1 x 2 dx, between the limits 0 and X/a, where e = 1 (b/a) 2 is the eccentricity. This is the result for the arc length from X = 0 to X/a in the first quadrant, beginning at the point (0, b) on the Y -axis. Notice that we can rewrite this integral as a ae 2 x 2 (1 e 2 x 2 )(1 x 2 ) dx = P (x, y) dx, where P (x, y) is a rational function and y is a solution of the equation y 2 = (1 e 2 x 2 )(1 x 2 ). This equation defines an elliptic curve! y is an example of an algebraic function. Namely, an algebraic function y = y(x) is a solution of the equation y n + a 1 (x)y n 1 +... + a n (x) = 0, (2.1.1)

MODULI SPACES AND INVARIANT THEORY 25 where a i (x) C(x) are rational functions (ratios of polynomials). Without loss of generality, we can assume that this equation is irreducible over C(x). For example, we can get nested radicals y(x) = 3 x 3 7x x, although after Abel and Galois we know that not any algebraic function is a nested radical (for n 5)! An Abelian integral is the integral of the form P (x, y) dx where P (x, y) is some rational function. All rational functions P (x, y) form a field K, which is finitely generated and of transcendence degree 1 over C (because x and y are algebraically dependent). And vice versa, given a field K such that tr.deg. C K = 1, we can let x be an element transcendent over C. Then K/C(x) is a finitely generated, algebraic (hence finite), and separable (because we are in characteristic 0) field extension. By a theorem on the primitive element, we have K = C(x, y), where y is a root of an irreducible polynomial (2.1.1). Notice that of course there are infinitely many choices for x and y, thus the equation (2.1.1) is not determined by the field extension. It is not important from the perspective of computing integrals either (we can always do u-substitutions). So on a purely algebraic level we can study isomorphism classes of f.g. field extensions K/C with tr.deg. C K = 1. Clearing denominators in (2.1.1) gives an irreducible affine plane curve C = {f(x, y) = 0} A 2 and its projective completion, an irreducible plane curve in P 2. Recall that the word curve here means of dimension 1, and dimension of an irreducible affine or projective variety is by definition the transcendence degree of the field of rational functions C(C). So we can restate our moduli problem as understanding birational equivalence classes of irreducible plane curves. Here we use the following definition 2.1.2. DEFINITION. Irreducible (affine or projective) algebraic varieties X and Y are called birationally equivalent if their fields of rational functions C(X) and C(Y ) are isomorphic. More generally, we can consider an arbitrary irreducible affine or projective curve C A n or C P n : birational equivalence classes of irreducible algebraic curves. This gives the same class of fields, so we are not gaining any new objects. Geometrically, for any such curve a general linear projection P n P 2 is birational onto its image. Let us remind some basic facts related to regular maps (morphisms) and rational maps (see lectures on the Grassmannian 1.8.2 for definitions):

26 JENIA TEVELEV 2.1.3. THEOREM ([X, 2.3.3]). If C is a smooth curve and f : C P n is a rational map then f is regular. More generally, if X is a smooth algebraic variety and f : X P n is a rational map then the indeterminancy locus of f has codimension 2. 2.1.4. THEOREM ([X, 1.5.2]). If X is a projective variety and f : X P n is a regular morphism then f(x) is closed (i.e. also a projective variety). 2.1.5. THEOREM. For any algebraic curve C, there exists a smooth projective curve C birational to C. Taken together, these facts imply that our moduli problem can be rephrased as the study of isom. classes of smooth projective algebraic curves. 2.1.6. REMARK. Theorem 2.1.5 is proved in [G] by take a plane model C P 2 (by projecting P n P 2 ). compute the normalization C C. Construction of the normalization in [G] is transcendental: one first constructs C as a compact Riemann surface and then invokes a general fact (see below) that it is in fact a projective algebraic curve. Notice however that there exist purely algebraic approaches to desingularization by either (a) algebraic normalization (integral closure in the field of fractions) [X, 2.5.3] or (b) blow-ups [X, 4.4.1]. The analytic approach is to consider Riemann surfaces instead of algebraic curves. It turns out that this gives the same moduli problem: biholomorphic isom. classes of compact Riemann surfaces. It is easy to show that a smooth algebraic curve is a compact Riemann surface. It is harder but not too hard to show that a holomorphic map between two smooth algebraic curves is in fact a regular morphism, for example ant meromorphic function is in fact a rational function. But a really difficult part of the theory is to show that any compact Riemann surface is an algebraic curve. It is hard to construct a single meromorphic function, but once this is done the rest is easy. This is done by analysis: to construct a harmonic function on a Riemann surface one (following Klein and Riemann): This is easily done by covering the Riemann surface with tin foil... Suppose the poles of a galvanic battery of a given voltage are placed at the points A 1 and A 2. A current arises whose potential u is single-valued, continuous, and satisfies the equation u = 0 across the entire surface, except for the points A 1 and A 2, which are discontinuity points of the function." A modern treatment can be found in [GH], where a much more general Kodaira embedding theorem is discussed. 2.2. Genus. The genus g of a smooth projective algebraic curve can be computed as follows: topologically: the number of handles. analytically: the dimension of the space of holomorphic differentials. algebraically: the dimension of the space of rational differentials without poles ω = a dx, where a, x are rational functions on C.

MODULI SPACES AND INVARIANT THEORY 27 One also has the following genus formula: 2g 2 = (number of zeros) (number of poles) (2.2.1) of any meromorphic (=rational) differential ω. For example, a form ω = dx on P 1 at the chart x = 1/y at infinity is dx = d(1/y) = (1/y 2 )dy. So it has no zeros and a pole of order 2 at infinity, which agrees with (2.2.1). A smooth plane curve C P 2 of degree d has genus g = (d 1)(d 2) 2 (2.2.2) (more generally, if C has only nodal singularities then g = (d 1)(d 2) 2 δ, where δ is the number of nodes). There is a nice choice of a holomorphic form on C: suppose C A 2 x,y is given by the equation f(x, y) = 0. Differentiating this equation shows that dx f y = dy f x along C, where the first (resp. second) expression is valid at points where x (resp. y) is a holomorphic coordinate. This gives a non-vanishing holomorphic form ω on C A 2. A simple calculation shows that ω has zeros at points at infinity each of multiplicity d 3. Combined with (2.2.1), this gives which is equivalent to (2.2.2). 2g 2 = d(d 3), 2.3. Divisors on curves. A divisor D is just a linear combination a i P i of points P i C with integer multiplicities. Its degree is defined as deg D = a i. If f is a rational (=meromorphic) function on C, we can define its divisor (f) = P C ord P (f)p, where ord P (f) is the order of zeros (or poles) of f at P. Analytically, if z is a holomorphic coordinate on C centered at P then near P f(z) = z n g(z), where g(z) is holomorphic and does not vanish at p. Then ord P (f) = n. Algebraically, instead of choosing a holomorphic coordinate, we choose a local parameter, i.e. a rational function z regular at P, z(p ) = 0, and such that any rational function f on C can be written (uniquely) as f = z n g,

28 JENIA TEVELEV where g is regular at P and does not vanish there (see [X, 1.1.5]) 2. For example, we can choose an affine chart where the tangent space T P C surjects onto one of the coordinate axes. The corresponding coordinate is then a local parameter at P (of course this is also exactly how one usually introduces a local holomorphic coordinate on a Riemann surface). We can define the divisor of a meromorphic form ω in a similar way: K = (ω) = P C ord P (ω)p, where if z is a holomorphic coordinate (or a local parameter) at P then we can write ω = f dz and ord P (ω) = ord P (f). This divisor is called the canonical divisor. So we can rewrite (2.2.1) as deg K = 2g 2. 2.4. Riemann Hurwitz formula. Suppose f : C D is a non-constant map of smooth projective algebraic curves. Its degree deg f can be interpreted as follows: topologically: number of points in the preimage of a general point. algebraically: degree of the induced field extension C(C)/C(D), where C(D) is embedded in C(C) by pull-back of functions f. It is easy to define a refined version with multiplicities: suppose P D and let f 1 (P ) = {Q 1,..., Q r }. If z is a local parameter at P then r deg f = ord Qi f (z) i=1 does not depend on P. In particular, a rational function f on C can be thought of as a map C P 1, Its degree is equal to the number of zeros (resp. to the number of poles) of f, and in particular deg(f) = 0 for any f k(c). A point P is called a branch point if ord Q f (z) > 1 for some Q f 1 (P ), where z is a local parameter at P. In this case Q is called a ramification point and e Q = ord Q f (z) is called a ramification index. So if t is a local parameter at Q then f (z) = t e Qg, where g is regular at Q and g(q) 0. Analytically, one can compute a branch of the e Q -th root of g and multiply t by it: this is often phrased by saying that a holomorphic map of Riemann surfaces locally in coordinates has a form t z = t e, e 1. If ω is a meromorphic form on D without zeros or poles at branch points then each zero or pole of ω contributes to deg f zeros or poles of f ω. In addition, the formula f (dz) = d(t e Q g) = e Q t e Q 1 g dt + t e Q dg shows that each ramification point will also be a zero of f ω of order e Q 1. This gives a Riemann Hurwitz formula 2 This is an instance of a very general strategy in Algebraic Geometry: if there is some useful analytic concept (e.g. a holomorphic coordinate) that does not exist algebraically, one should look for properties (e.g. a factorization f = z n g above) that we want from this concept. Often it is possible to find a purely algebraic object (e.g. a local parameter) satisfying the same properties.

2.4.1. THEOREM (Riemann Hurwitz). MODULI SPACES AND INVARIANT THEORY 29 K C = f K D + Q C(e Q 1)[Q]. and comparing the degrees and using (2.2.1), 2g(C) 2 = deg f [2g(D) 2] + Q C(e Q 1). 2.5. Riemann Roch formula and linear systems. Finally, we have the most important 2.5.1. THEOREM (Riemann Roch). For any divisor D on C, we have where and l(d) i(d) = 1 g + deg D, l(d) = dim L(D), where L(D) = {f C(C) (f) + D 0} i(d) = dim K 1 (D), where K 1 (D) = {meromorphic forms ω (ω) D}. Let s look at some examples. If D = 0 then i(d) = g: indeed K 1 (0) is the space of holomorphic differentials and one of the characterizations of the genus is that it is the dimension of the space of holomorphic differentials. On the other hand, l(d) = 1 as the only rational functions regular everywhere are constants. Analytically, this is Liouville s Theorem for Riemann surfaces (see also the maximum principle for harmonic functions). Algebraically, this is 2.5.2. THEOREM. If X is an irreducible projective variety then the only functions regular on X are constants. Proof. A regular function is also a regular morphism X A 1. Composing it with the inclusion A 1 P 1 gives a regular morphism f : X P 1 such that f(x) A 1. But by Theorem 2.1.4, f(x) must be closed in P 1, thus f(x) must be a point. One way or another, if D = 0 then we get a triviality 1 g = 1 g. If, on the other hand, D = K then the RR gives (2.2.1). 2.5.3. EXAMPLE. Suppose g(c) = 0. Let D = P be a point. Then RR gives l(p ) = i(p ) + 2 2. It follows that L(D) contains a non-constant function f with a unique pole at P. It gives an isomorphism f : C P 1. The last example shows the most common way of using Riemann Roch. We define a linear system of divisors D = {(f) + D f L(D)}. A standard terminology here is that divisors D and D are called linearly equivalent if D D = (f) for some f k(c).

30 JENIA TEVELEV A divisor D is called effective if D 0, (i.e. all coefficients of D are positive). So a linear system D consists of all effective divisors linearly equivalent to D. Choosing a basis f 0,..., f r of L(D) gives a map φ D : C P r, φ D (x) = [f 0,..., f r ]. Since C is a smooth curve, this map is regular. More generally, we can choose a basis of a linear subspace in L D and define a similar map. It called a map given by an incomplete linear system. In fact any map φ : C P r is given by an incomplete linear system as soon as C is non-degenerate, i.e. if φ(c) is not contained in a projective subspace of P r. It can be obtained as follows: Any map φ is obtained by choosing rational functions f 0,..., f r k(c). Consider their divisors (f 0 ),..., (f r ) and let D be their common denominator. Then, clearly, f 0,..., f r L(D). Moreover, in this case divisors (f 0 ) + D,..., (f r ) + D have very simple meaning: they are just pull-backs of coordinate hyperplanes in P r. Indeed, suppose h is a local parameter at a point P C and suppose that P contributes np to D. Then φ (in the neighborhood of P ) can be written as [f 0 h n :... f r h n ], where at least one of the coordinates does not vanish (otherwise we can subtract P from D, so D is not the common denominator). So pull-back of coordinate hyperplanes are (locally near P ) given by divisors (f 0 ) + D,..., (f r ) + D. If we start with any divisor D, a little complication can happen: a fixed part (or base points) of D is a maximal effective divisor E such that D E 0 for any D D. Those start to appear more often in large genus, but if they do then D = D E. In fact, this is an if and only if condition: 2.5.4. PROPOSITION. D has no base points if and only if, for any point P C, l(d P ) = l(d) 1. The last question we wish to address is when φ D gives an embedding C P r. If this happens then we call D a very ample divisor. One has the following very useful criterion: 2.5.5. THEOREM. D is very ample if and only if φ D separates points: l(d P Q) = l(d) 2 for any points P, Q C. φ D separates tangents: l(d 2P ) = l(d) 2 for any point P C. This pretty much summarizes the course on Riemann surfaces!

MODULI SPACES AND INVARIANT THEORY 31 2.6. Elliptic curves. Let us recall the following basic theorem. 2.6.1. THEOREM. The following are equivalent: (1) C A 2 x,y is given by a Weierstrass equation y 2 = 4x 3 g 2 x g 3, where = g 3 2 27g 2 3 0. (2) C is isomorphic to a smooth cubic curve in P 2. (3) C is isomorphic to a 2 : 1 cover of P 1 ramified at 4 points. (4) C is isomorphic to a complex torus C/Λ, where Λ Z Zτ, Im τ > 0. (5) C is a projective algebraic curve of genus 1. (6) C is a compact Riemann surface of genus 1. Proof. Simple implications: (1) (2) (just have to check that C is smooth), (2) (3) (project C P 2 P 1 from any point p C). (3) (5) (Riemann Hurwitz). (5) (6) (induced complex structure), (4) (6) (C is topologically a torus and has a complex structure induced from a translation-invariant complex structure on C), (2) (1) (find a flex point (by intersecting C with a Hessian curve), move it to [0 : 1 : 0] and make the line at infinity z = 0 the flex line). Logically unnecessary but fun: (2) (5) (genus of plane curve formula), (1) (3) (project A 2 x,y A 1 x, the last ramification point is at ), Now the Riemann Roch analysis. Let C be an algebraic curve of genus 1. Then L(K) is one-dimensional. Let ω be a generator. Since deg K = 0, ω has no zeros. It follows by RR that l(d) = deg D for deg D > 0. It follows that ψ D has no base points for deg D > 1 and is very ample for deg D > 2. Fix a point P C. Since 3P is very ample, we have an embedding ψ 3P : C P 2, where the image is a curve of degree 3, moreover, a point P is a flex point. This shows that (6) (2). It is logically unnecessary but still fun: since 2P has no base-points, we have a 2 : 1 map ψ 2P : C P 1, with P as one of the ramification points, which shows directly that (6) (3). Let L(2P ) be a meromorphic function with pole of order 2 at P. If C is obtained as C/Λ (and P is the image of the origin), one can pull-back to a doubly-periodic (i.e. Λ-invariant) meromorphic function on C with poles only of order 2 and only at lattice points. Moreover, this function is unique (up-to rescaling and adding a constant). It is classically known as the Weierstrass -function (z) = 1 z 2 + γ Λ, γ 0 ( 1 (z γ) 2 1 γ 2 ).

32 JENIA TEVELEV Notice that (z) has poles of order 3 at lattice points, and therefore {1, (z), (z)} is a basis of L(3P ). It follows that the embedding C P 2 as a cubic curve is given (when pull-backed to C) by map C C 2, z [ (z) : (z) : 1], and in particular and satisfy a cubic relation. It is easy to check that this relation has a Weierstrass form ( ) 2 = 4 3 g 2 g 3. This gives another proof that (4) (1). The only serious implication left is to show that (6) (4). There are several ways of thinking and generalizing this result, and we will discuss some of these results later in this course. For example, we can argue as follows: we fix a point P C and consider a multi-valued holomorphic map π : C C, z If the curve is given as a cubic in the Weierstrass normal form then those are elliptic integrals dx 4x 3 g 2 x g 3 We take the first homology group H 1 (C) = Zα + Zβ and define periods ω, ω C. α The periods generate a subgroup Λ C. If the periods are not linearly independent over R then (after multiplying ω by a constant), we can assume that Λ R. Then Im π is a single-valued harmonic function, which must be constant by the maximum principle. This is a contradiction: π is clearly a local isomorphism near P. So Λ is a lattice and π induces a holomorphic map f : C C/Λ. As we have already noticed, this map has no ramification (which also follows from Riemann Hurwitz), thus from the theory of covering spaces f corresponds to a subgroup of π 1 (C/Λ) = Λ. Thus f must have the form C homeo C/Λ C/Λ, where Λ Λ is a sublattice. Notice that the integration map is well-defined on the universal cover of C, i.e. on C and gives the map F : C C, which should be just the identity map. But then F (Λ ) = Λ, i.e. periods belong to Λ. Thus Λ = Λ. 2.6.2. REMARK. An important generalization of the last step of the proof is a beautiful Klein Poincare Uniformization Theorem: a universal cover of a compact Riemann surface is either P 1 if g = 0, or C if g = 1, or β z P ω.

MODULI SPACES AND INVARIANT THEORY 33 H (upper half-plane) if g 2. In other words, any algebraic curve of genus 2 is isomorphic to a quotient H/Γ, where Γ Aut(H) = PGL 2 (R) is a discrete subgroup acting freely on H. 2.7. J-invariant. Now we would like to classify elliptic curves up to isomorphism, i.e. to describe M 1. As we will see many times in this course, automorphisms of parametrized objects can cause problems. An elliptic curve has a lot of automorphisms: since C C/Λ, it is in fact a group itself! If thinking about an elliptic curve as a complex torus is too transcendental for you, observe also that if P, Q C then by our discussion of the Riemann Roch above, we have l(p +Q) = 2 and so we have a double cover φ P +Q : C P 1 with P + Q as one of the fibers. Any double cover has an involution permuting the two branches, which shows that any two points P, Q C can be permuted by an involution, and in particular that Aut C acts transitively on C. In a cubic curve realization, the group structure on C is a famous three points on a line group structure, but let s postpone this discussion until the lectures on Jacobians. In any case, we can eliminate many automorphisms (namely translations) by fixing a point: M 1 = M 1,1. So our final definition of an elliptic curve is: a pair (C, P ), where C is an algebraic curve of genus 1 and P C. It is very convenient to choose P to be the unity of the group structure if one cares about it. Notice that even a pointed curve (C, P ) still has at least one automorphism, namely the involution given by permuting the two branches of φ 2P. In the C/Λ model this is the involution z z (if P is chosen to be 0): this reflects the fact that the Weierstrass -function is even. Now let s work out when two elliptic curves are isomorphic and when Aut(C, P ) is larger than Z 2. 2.7.1. THEOREM. (1) Curves given by Weierstrass equations y 2 = 4x 3 g 2 x g 3 and y 2 = 4x 3 g 2 x g 3 are isomorphic if and only if there exists t C such that g 2 = t2 g 2 and g 3 = t3 g 3. There are only two curves with special automorphisms: the curve y 2 = x 3 + 1 gives Z 6 and the curve y 2 = x 3 + x gives Z 4 (draw the family of cuspidal curves in the g 2 g 3 -plane). (2) Two smooth cubic curves C and C are isomorphic if and only if they are projectively equivalent. (3) Let C (resp. C ) be a double cover of P 1 with a branch locus p 1,..., p 4 (resp. p 1,..., p 4 ). Then C C if and only if there exists g PGL 2 such that p i = g(p i) for any i. In particular, we can always assume that branch points are 0, 1,, λ. There are two cases with non-trivial automorphisms, λ = 1 (Aut C = Z 4 ) and λ = ω = e 2πi 3 (Aut C = Z 6 ). Modulo Z 2, these groups are automorphism groups of the corresponding fourtuples.

34 JENIA TEVELEV (4) C/Λ C/Λ if and only if Λ = αλ for some α C. If Λ = Z Zτ and Λ = Z Zτ with Im τ, Im τ > 0 then this is equivalent to τ = aτ + b [ ] a b cτ + d, PSL c d 2 (Z) (2.7.2) There are two elliptic curves (draw the square and the hexagonal lattice) with automorphism groups Z 4 and Z 6, respectively. Proof. (2) Suppose plane cubic realizations of C and C are given by linear systems 3P and 3P, respectively. We can assume that an isomorphism of C and C takes P to P. Then the linear system 3P is a pull-back of a linear system 3P, i.e. C and C are projectively equivalent. A similar argument proves (3). Notice that in this case Aut(C, P ) modulo the hyperelliptic involution acts on P 1 by permuting branch points. In fact, λ is simply the cross-ratio: λ = (p 4 p 1 ) (p 2 p 3 ) (p 2 p 1 ) (p 4 p 3 ), but branch points are not ordered, so we have an action of S 4 on possible cross-ratios. However, it is easy to see that the Klein s four-group V does not change the cross-ratio. The quotient S 4 /V S 3 acts non-trivially: λ {λ, 1 λ, 1/λ, (λ 1)/λ, λ/(λ 1), 1/(1 λ)} (2.7.3) Special values of λ correspond to cases when some of the numbers in this list are equal. For example, λ = 1/λ implies λ = 1 and the list of possible cross-ratios boils down to 1, 2, 1/2 and λ = 1/(1 λ) implies λ = ω, in which case the only possible cross-ratios are ω and 1/ω. (4) Consider an isomorphism f : C/Λ C/Λ. Composing it with translation automorphisms on the source and on the target, we can assume that f(0+λ ) = 0+Λ. Then f induces a holomorphic map C C/Λ with kernel Λ, and its lift to the universal cover gives an isomorphism F : C C such that F (Λ ) = Λ. But it is proved in complex analysis that all automorphisms of C preserving the origin are maps z αz for α C. So we have which gives which gives (2.7.2). Z + Zτ = α(z + Zτ ), ατ = a + bτ, α = c + dτ, So finally, we can introduce the j-invariant: j = 1728 g3 2 = 256(λ2 λ + 1) 3 λ 2 (λ 1) 2. (2.7.4) It is easy to see that the expression 256 (λ2 λ+1) 3 does not change under the λ 2 (λ 1) 2 transformations (2.7.3). For a fixed j 0, the polynomial 256(λ 2 λ + 1) 3 jλ 2 (λ 1) 2 has six roots related by the transformations (2.7.3). So the j-invariant uniquely determines an isomorphism class of an elliptic curve. The special values of the j-invariant are j = 0 (Z 6 ) and j = 1728 (Z 4 ).

MODULI SPACES AND INVARIANT THEORY 35 2.8. Monstrous Moonshine. The most interesting question here is how to compute the j-invariant in terms of the lattice parameter τ. Notice that j(τ) is invariant under the action of PSL 2 (Z) on H. This group is called the modular group. It is generated by two transformations, S : z 1/z and T : z z + 1 (pull notes from Adam s TWIGS talk). It has a fundamental domain (draw the modular figure, two special points). The j-invariant maps the fundamental domain to the plane A 1 (draw how). Since the j-invariant is invariant under z z + 1, it can be expanded in a variable q = e 2πiτ : j = q 1 + 744 + 196884q + 21493760q 2 +... What is the meaning of these coefficients? According to the classification of finite simple groups, there are several infinite families of them (like an alternating group A n ) and a few sporadic groups. The largest sporadic group is the monster group F 1 that has about 10 54 elements. Its existence was predicted by Robert Griess and Bernd Fischer in 1973 and it was eventually constructed by Griess in 1980 as the automorphism group of the Griess (commutative, non-associative) algebra whose dimension is 196884: so 196884 is to F 1 as n is to S n. The dimension of the Griess algebra is one of the coefficients of j(q)! In fact all coefficients in this q-expansion are related to representations of the Monster. This is a Monstrous Moonshine Conjecture of McKay, Conway, and Norton proved in 1992 by Borcherds (who won the Fields medal for this work). 2.9. Families of elliptic curves: coarse and fine moduli spaces. So far we were mostly concerned with moduli spaces as sets that parametrize isomorphism classes of geometric objects. The geometric structure on the moduli space came almost as an afterthought, even though it is this structure of course that is responsible for all applications. The most naive idea is that two points in the moduli space are close to each other if the objects that they represent are small deformations of each other. There exists an extremely simple and versatile language (developed by Grothendieck, Mumford, etc.) for making this rigorous. The key words are family of objects, coarse moduli space, pull-back, and fine moduli space. What is a family, for example what is a family of elliptic curves? One should think about it as a sort of fibration with fibers given by elliptic curves (appropriately called an elliptic fibration). More generally, a family of objects is a regular map f : X Y, where the fibers are geometric objects we care about. In practice, considering all maps does not work, and one has to impose some conditions on f. These required conditions in fact often depend on the moduli problem being stidied, so in the interest of drama let s call it Property X for now. Let M be the moduli set of isomorphism classes of these objects. We have a map Y M which sends y Y to the isomorphism class of the fiber f 1 (y). Geometric structure we are imposing on M should be compatible with this map Y M: basically we should just ask that this map Y M is a regular map.

36 JENIA TEVELEV This is a basic idea, but there is a minor complication: with this definition the moduli space (even when it exists) is almost never going to be unique. For example, let s suppose that A 1 is a moduli space for some problem. Let C = {y 2 = x 3 } A 2 be a cuspidal curve with the normalization map ν : A 1 C, t (t 3, t 2 ). Notice that ν is a bijection on points but not an isomorphism. Any family of objects over Y will give us a regular map Y A 1 which when composed with ν will give a regular map Y C. To guarantee uniqueness of the moduli space, we add an extra condition (3) to the following definition: 2.9.1. DEFINITION. We say that the algebraic variety M is a coarse moduli space for the moduli problem if (1) Points of M correspond to iso classes of objects in question. (2) Any family X Y (i.e. a regular map satisfying property X) induces a regular map Y M. (3) For any other algebraic variety M satisfying (1) and (2), an obvious map M M is regular. 2.9.2. REMARK. It is rare that the moduli problem studies geometric objects without any decorations. For example, an elliptic curve is not just a genus 1 curve C but also a point P C. This extra data should be built into the definition of the family. For example, we can say that an elliptic fibration is a morphism f : X Y (satisfying property X) plus a morphism σ : Y X such that f σ = Id Y. A morphism like this is called a section. 2.9.3. REMARK. It practice, it is often necessary to enlarge the category of algebraic varieties to the category of algebraic schemes. For example, in number theory one can look at an elliptic curve defined by equations with integral coefficients or with coefficients in some ring of algebraic integers. Then it is interesting to work out reductions of this elliptic curve modulo various primes. Geometrically, all primes in the ring of algebraic integers R form an algebraic scheme, called Spec R and one thinks about reductions of an elliptic curve modulo various primes as fibers of the family E Spec R where E is again a scheme called an integral model of the original complex elliptic curve. This is an arithmetic analogue of a geometric situation when we have an elliptic fibration over an algebraic curve. Some fibers won t be smooth, this happens at so called primes of bad reduction. Families can be pulled-back: if we have a morphism f : X Y (satisfying property X) and an arbitrary morphism g : Z Y then we define a pull-back (or a fibered product) X Y Z = {(x, z) X Z f(x) = g(z)} X Z. We have a morphism X Y Z Z induced by the second projection. Since we only care about maps satisfying property X, we have to make sure that Property X is stable under pull-back, i.e. if f has property X then the induced map X Y Z Z also has it (notice however that the morphism g : Z X can be arbitrary). If we include some decorations in the family, we have to modify the notion of the pull-back to include decorations. For example, a section σ : Y X will induce a section Z X Y Z, namely (σ g, Id Z ).

MODULI SPACES AND INVARIANT THEORY 37 The basic point is that X Y and X Y Z Z have the same fibers, and the map X Y Z M to the moduli space factors through Y M. This raises a tantalizing possibility that 2.9.4. DEFINITION. M is a fine moduli space if there exists a universal family U M, i.e. a regular map (with property X) such that any other family X Y is isomorphic to a pull-back along a unique regular map Y M. Let s look at various examples. 2.9.5. EXAMPLE. We know that P 1 is the only genus 0 curve (up to isomorphisms). So the moduli problem of families of genus 0 curves has an obvious course moduli space: a point. However, it is not a fine moduli space. If it were, then all families X Y (satisfying property X) with fibers isomorphic to P 1 would appear as Y pt P 1 = Y P 1. However, there exist extremely simple P 1 -fibrations not isomorphic to the product. For example, consider the Hirzebruch surface F 1. It is obtained by resolving indeterminacy locus of a projection from a point rational map f : P 2 P 1 by blowing up this point. In coordinates, the rational map is [x : y : z] [x : y], which is undefined at [0 : 0 : 1]. Its blow-up F 1 is a surface in P 2 P 1 [s:t] given by an equation xt = ys. The resolution of f is obtained by just restricting the second projection P 2 P 1 P 1 to F 1. The first projection identifies F 1 with P 2 everywhere outside of the point [0 : 0 : 1], the preimage of this point is a copy of P 1 called the exceptional divisor. All fibers of the resolved map F 1 P 1 are isomorphic to P 1 but F 1 P 1 P 1. For example, E has self-intersection 1 on F 1 but there are no ( 1)-curves in P 1 P 1. 2.9.6. EXAMPLE. How about the Grassmannian? We claim that G(2, n) is a fine moduli space for 2-dimensional subspaces of A n. What is a family here? A family over an algebraic variety X should be a varying 2- dimensional subspace of A n. In other words, a family over X is just a 2-dimensional vector sub-bundle E of the trivial vector bundle X A n. What is an r-dimensional vector bundle over an algebraic variety X? It is an algebraic variety E, a morphism π : E X, and a trivializing covering X = U α, which means that we have isomorphisms ψ α : π 1 (U α ) U α A r, p 2 ψ i = π that are given by linear maps on the overlaps, i.e. over U α U β the induced map takes U α U β A r U α U β A r p 2 A r (x, v) A(x)v, where A(x) is an invertible matrix with entries in O(U i U j ). A map of vector bundles E 1 E 2 is map of underlying varieties that is given by

38 JENIA TEVELEV linear transformations in some trivializing charts (with coefficients of these linear transformations being regular functions on charts). So let s fix a 2-dimensional vector bundle E over X and let s assume that it is a sub-bundle of a trivial bundle X A r. What is the corresponding map to the Grassmannian? Choose a trivializing affine covering {U α } of X such that E Uα U α A 2. Choose a basis u, v in A 2. An embedding E X A n in the chart gives an embedding U α A 2 U α A n. Composing it with projection to A n gives maps (x, u) i a 1i (x)e i, (x, v) i a 2i (x)e i, where a 1i (x), a 2i (x) O(U α ). Then u v = i<j p ij (x)e i e j, which after the projectivization gives a morphism U α P(Λ 2 A 2 ). Clearly the function p ij (x) satisfy the Plücker relations, thus this regular map factors through the map U α G(2, n). This map does not depend on the choice of the basis {u, v} and thus these maps glue on overlaps U α U β to give a regular map X G(2, n). What is the universal family here? Quite appropriately, it is called the universal bundle over G(2, n) and is defined as follows: the fiber over a point in G(2, n) that corresponds to a subspace U A n, is U itself. More precisely, the universal sub-bundle is E = {([U], v) v U} G(2, n) A n. It is trivialized in standard affine charts of the Grassmannian: for example let s consider the chart U 12 defined by p 12 0. For any point of this chart, rows of the matrix [ ] 1 0 a13 a A = 14... a 1n 0 1 a 23 a 24... a 2n give a basis of the corresponding subspace in A n. Choosing a basis and trivializing if clearly the same thing. Returning to the j-invariant, we have to define a family of elliptic curves. It is clear that it should be a regular map of varieties π : X Y and a section σ : Y X such that each fiber (π 1 (y), σ(y)) is an elliptic curve. Now it s time to discuss a mysterious property X. Recall that in the definition of the vector bundle above we assumed not only that each fiber of the map is a vector space but also that a vector bundle has local models, i.e. it becomes trivial in sufficiently small neighborhoods. So in particular a vector bundle is a locally trivial fibration, which is certainly something reasonable for a family of varieties. However, we can not require any interesting map to be a locally trivial fibration in Zariski topology, because a locally trivial fibration has isomorphic fibers, and this is certainly something we want to avoid in our discussion of moduli! (The corresponding

MODULI SPACES AND INVARIANT THEORY 39 notion in Algebraic Geometry is called an isotrivial fibration). So we have to come up with some property of a regular map π to ensure that it behaves like a locally trivial fibration without actually being one. In the analytic category, i.e. when X and Y are complex manifolds, the right concept is the notion of a proper submersion, i.e. a surjective holomorphic map with compact fibers and everywhere surjective differential. There is a well-known theorem of Ehresmann that, even though a submersion is almost never a locally trivial fibration in the analytic category (i.e. fibers are not isomorphic as complex analytic manifolds), it is nevertheless a locally trivial fibration in the category of real manifolds, and so for example the fibers are diffeomorphic (assuming the base is connected). So if we only want to define property X when the base is a smooth projective variety, we can ask that π is a submersion when considered as a map of complex manifolds. It is proved in any course on manifolds that the property of being a submersion is preserved by pull-backs. It turns out that Algebraic Geometry allows one to extend this notion even to the case when the base is not smooth. It is called a smooth morphism. We will not define it here, see [Ha, 3.10]. In the case of elliptic fibrations, the property of being smooth is equivalent to the following extremely useful local model result: 2.9.7. THEOREM ([MS, page 203]). π : X Y is an elliptic fibration with a section σ : Y X if and only if the following condition is satisfied. Every point y Y has an affine neighborhood such that π 1 (U) is isomorphic to a subvariety of U P 2 given by the Weierstrass normal form y 2 z = 4x 3 g 2 xz 2 g 3 z 3, where g 2, g 3 O(U) are regular function on U such that = g2 3 27g2 3 does not vanish on U. Moreover, g 2, g 3 O(U) are defined uniquely up to transformations for some invertible function t O (U). g 2 t 4 g 2, g 3 t 6 g 3 (2.9.8) 2.9.9. REMARK. Dependence on t comes from the following basic observation: multiplying y by t 3 and x by t 2 will induce (2.9.8). Notice that if t is a constant function then we can take its square root and multiply by it instead. But if t is a regular function, its square root is rarely regular, and so can not be used. 2.9.10. REMARK. This idea of presenting a fibration by using some normal form for equations of an algebraic variety where the coefficients are allowed to vary is very common. Now we can show that 2.9.11. THEOREM. The j-line is a course moduli space for elliptic curves. Proof. The proof resembles the corresponding argument for the Grassmannian. Assuming we have an elliptic fibration X Y, we have to construct a morphism Y A 1 j, i.e. we have to show that j is a regular function on Y. This is a local statement that we can check on charts of Y, thus by Theorem 2.9.7 we can assume that the fibration is in the Weierstrass normal form. But then we can just define j by the usual formula (2.7.4): since g 2 and g 3 are regular functions on the charts, j is regular as well.

40 JENIA TEVELEV Interestingly, this also shows that A 1 j is not a fine moduli space. Indeed, if A 1 carries a universal family then Theorem 2.9.7 would be applicable to A 1 j as well. This would imply that locally at any point P A1 we have g2 3 j = 1728 (j) g2 3(j) 27g2 3 (j) for some rational functions g 2 and g 3. But j has zero of order 1 at 0 where as the order of zeros of the RHS at 0 is divisible by 3. Likewise, we have 27g3 2 j 1728 = 1728 (j) g2 3(j) 27g2 3 (j). j 1728 has zero of order 1 at 1728 but the RHS has even order of vanishing! We see that special elliptic curves with automorphisms prevent the j-line from being a fine moduli space.

MODULI SPACES AND INVARIANT THEORY 41 2.10. Homework due on February 25. Write your name and sign here: Problem 1. Let M be the set of isomorphism (=conjugacy) classes of invertible complex 2 2 matrices. (a) Describe M as a set. (b) Let s define the following moduli problem: a family over a variety X is a 2 2 matrix A(x) with coefficients in O(X) such that det A(x) O (X), i.e. A(x) is invertible for any x X. Explain how the pull-back of families should be defined. (c) Show that there is no structure of an algebraic variety on M that makes it into a coarse moduli space (2 points). Problem 2. Compute j-invariants of elliptic curves (1 point): (a) y 2 + y = x 3 + x; (b) y 2 = x 4 + ax 3 + bx 2 + cx. Problem 3. Prove Theorem 2.7.1, (1) (2 points). Problem 4. Show that any elliptic curve is isomorphic to a curve of the form y 2 = (1 x 2 )(1 e 2 x 2 ) (1 point). Problem 5. Show that the two formulas in (2.7.4) agree (1 point). Problem 6. (a) Compute the j-invariant of an elliptic curve y 2 + xy = x 3 36 q 1728 x 1 q 1728, where q is some parameter. (b) Consider families of elliptic curves (defined as in Theorem 2.9.7) but with an extra condition that no fiber has a special automorphism group, i.e. assume that j 0, 1728. Show that A 1 j \ {0, 1728} carries a family of elliptic curves with j-invariant j. Is your family universal? (2 points). Problem 7. The formula j = 256 (λ2 λ+1) 3 gives a 6 : 1 cover P 1 λ 2 (λ 1) 2 λ P1 j. Thinking about P 1 as a Riemann sphere, let s color P 1 j in two colors: color the upper half-plane H white and the lower half-plane H black. Draw the pull-back of this coloring to P 1 λ (2 points). Problem 8. Let f be a rational function on an algebraic curve C such that all zeros of f have multiplicity divisible by 3 and all zeros of f 1728 have multiplicities divisible by 2. Show that C \ {f = } carries an elliptic fibration (defined as in Theorem 2.9.7) with j-invariant f. (2 points) Problem 9. Using a birational isomorphism between P 1 and the circle {x 2 + y 2 = 1} A 2 given by stereographic projection from (0, 1), describe an algorithm for computing integrals of the form P (x, 1 x 2 ) dx where P (x, y) is an arbitrary rational function (2 points). Problem 10. Let (C, P ) be an elliptic curve. (a) By considering a linear system φ 4P, show that C embeds in P 3 as a curve of degree 4. (b) Show that quadrics in P 3 containing C form a pencil P 1 with 4 singular fibers. (c) These four singular fibers define 4 points in P 1. Relate their cross-ratio to the j-invariant of C (4 points). Problem 11. Let X be an affine variety and let f O(X). A subset D(f) = {x X f(x) 0}

42 JENIA TEVELEV is called a principal open set. (a) Show that principal open sets form a basis of Zariski topology. (b) Show that any principal open set is itself an affine variety with a coordinate algebra O(X)[1/f]. (c) Show that any affine (resp. projective) variety is quasi-compact, i.e. any open cover has a finite subcover (2 points). Problem 12. Solve a cross-word puzzle (1 point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roblem 13. (a) Show that any affine (resp. projective) variety X is a union of finitely many irreducible projective varieties X 1,..., X n such that

MODULI SPACES AND INVARIANT THEORY 43 X i X j for i j (called irreducible components of X). (b) Show that irreducible components are defined uniquely (2 points). Problem 14. Let X P n be an irreducible projective variety. Show that any morphism X P m is given by m + 1 homogeneous polynomials F 0,..., F m in n + 1 variables of the same degree such that, for any point x X, at least one of the polynomials F i does not vanish (1 point). Problem 15. (a) Let X A n and Y A m be affine varieties with coordinate algebras O(X) and O(Y ). Suppose these algebras are isomorphic. Show that varieties X and Y are isomorphic. (b) Let X and Y be irreducible quasi-projective varieties with fields of rational functions C(X) and C(Y ). Show that these fields are isomorphic (i.e. X and Y are birational) if and only if there exist non-empty open subsets U X and V Y such that U is isomorphic to V (2 points). Problem 16. Let (C, P ) be an elliptic curve. Let Γ C be the ramification locus of φ 2P. (a) Show that Γ Z 2 Z 2 is precisely the 2-torsion subgroup in the group structure on C. (b) A level 2 structure on (C, P ) is a choice of an ordered basis {Q 1, Q 2 } Γ (considered as a Z 2 -vector space). Based on Theorem 2.9.7, describe families of elliptic curves with level 2 structure. Show that P 1 λ \{0, 1, } carries a family of elliptic curves with with a level 2 structure such that any curve with a level 2 structure appears (uniquely) as one of the fibers. Is your family universal? (2 points). Problem 17. Consider the family y 2 = x 3 + t of elliptic curves over A 1 \ {0}. Show that all fibers of this family have the same j-invariant but nevertheless this family is not trivial over A 1 \ {0} (2 points). Problem 18. Consider the family of cubic curves C a = {x 3 + y 3 + z 3 + axyz = 0} P 2 parametrized by a A 1. (a) Find all a such that C a is smooth and find its flex points. (b) Compute j as a function on a and find all a such that C a has a special automorphism group (2 points). Problem 19. Let (C, P ) be an elliptic curve equipped with a map C C of degree 2. By analyzing the branch locus φ 2P, show that the j-invariant of C has only 3 possible values and find these values (3 points).