24 JENIA TEVELEV
§2. Elliptic curves: j-invariant (Jan 31, Feb 4,7,9,11,14) After the projective line P1, the easiest algebraic curve to understand is an elliptic curve (Riemann surface of genus 1). Let M = isom. classes of elliptic curves . 1 { } We are going to assign to each elliptic curve a number, called its j-invariant and prove that 1 M1 = Aj . 1 1 So as a space M1 A is not very interesting. However, understanding A ! as a moduli space of elliptic curves leads to some breath-taking mathemat- ics. More generally, we introduce M = isom. classes of smooth projective curves of genus g g { } and M = isom. classes of curves C of genus g with points p , . . . , p C . g,n { 1 n ∈ } 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) (a2 X2), − and this can easily be differentiated !to find bX Y ! = − . a√a2 X2 − 2 This is squared and put into the integral 1 + (Y !) dX for the arc length. Now the substitution x = X/a results in " ! 1 e2x2 s = a − dx, 1 x2 # $ − 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 ae2x2 − dx = P (x, y) dx, (1 e2x2)(1 x2) # − − # where P (x, y) is a rational! function and y is a solution of the equation y2 = (1 e2x2)(1 x2). − − This equation defines an elliptic curve! y is an example of an algebraic func- tion. Namely, an algebraic function y = y(x) is a solution of the equation n n 1 y + a1(x)y − + . . . + an(x) = 0, (2.1.1) MODULI SPACES AND INVARIANT THEORY 25 where ai(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 x3 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.CK = 1, we can let x be an element transcendent over C. Then K/C(x) is a finitely generated, algebraic (hence finite), and sepa- rable (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 exten- sion. 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.CK = 1. Clearing denominators in (2.1.1) gives an irreducible affine plane curve 2 C = f(x, y) = 0 A { } ⊂ and its projective completion, an irreducible plane curve in P2. 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 projec- tive curve C An or C Pn: ⊂ ⊂ 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 n 2 P !!" P 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 Pn is a → rational map then f is regular. More generally, if X is a smooth algebraic vari- ety and f : X Pn 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 Pn 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 P2 (by projecting Pn P2). • ⊂ !!" compute the normalization C C. • ! → Construction of the normalization in [G] is transcendental: one first con- structs 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 alge- braic 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 sur- face. 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 al- gebraic 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 Rie- mann): “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 A1 and A2. A current arises whose potential u is single-valued, con- tinuous, and satisfies the equation ∆u = 0 across the entire surface, except for the points A1 and A2, 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 with- • out 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 P1 at the chart x = 1/y at infinity is dx = d(1/y) = (1/y2)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 P2 of degree d has genus ⊂ (d 1)(d 2) g = − − (2.2.2) 2 (d 1)(d 2) (more generally, if C has only nodal singularities then g = − − δ, 2 − where δ is the number of nodes). There is a nice choice of a holomorphic form on C: suppose C A2 is given by the equation f(x, y) = 0. Differen- ∩ x,y tiating this equation shows that dx dy = fy − fx 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 holomor- phic form ω on C A2. A simple calculation shows that ω has zeros at points ∩ at infinity each of multiplicity d 3. Combined with (2.2.1), this gives − 2g 2 = d(d 3), − − which is equivalent to (2.2.2).
§2.3. Divisors on curves. A divisor D is just a linear combination aiPi of points Pi C with integer multiplicities. Its degree is defined as ∈ %
deg D = ai.
If f is a rational (=meromorphic) function& on C, we can define its divisor
(f) = ordP (f)P, P C &∈ where ordP (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) = zng(z), where g(z) is holomorphic and does not vanish at p. Then ordP (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 = zng, 28 JENIA TEVELEV where g is regular at P and does not vanish there (see [X, 1.1.5])2. For ex- ample, we can choose an affine chart where the tangent space TP C surjects onto one of the coordinate axes. The corresponding coordinate is then a lo- cal 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) = ordP (ω)P, P C &∈ where if z is a holomorphic coordinate (or a local parameter) at P then we can write ω = f dz and ordP (ω) = ordP (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 inter- preted 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 , . . . , Q . If z is a local parameter at P then − { 1 r} r
deg f = ordQi 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 P1, 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 f (z) > 1 for some Q f 1(P ), Q ∗ ∈ − where z is a local parameter at P . In this case Q is called a ramification point and eQ = ordQ f ∗(z) is called a ramification index. So if t is a local parameter at Q then f (z) = teQ g, where g is regular at Q and g(Q) = 0. Analytically, ∗ ) one can compute a branch of the eQ-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 = te, 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
eQ eQ 1 eQ f ∗(dz) = d(t g) = eQt − g dt + t dg shows that each ramification point will also be a zero of f ω of order e 1. ∗ Q − This gives a Riemann–Hurwitz formula
2This 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 = zng 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. MODULI SPACES AND INVARIANT THEORY 29
2.4.1. THEOREM (Riemann–Hurwitz).
K = f ∗K + (e 1)[Q]. C D Q − Q C &∈ and comparing the degrees and using (2.2.1), 2g(C) 2 = deg f [2g(D) 2] + (e 1). − − Q − Q C &∈ §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 l(D) i(D) = 1 g + deg D, − − where l(D) = dim (D), where (D) = f C(C) (f) + D 0 L L { ∈ | ≥ } and i(D) = dim K1(D), where K1(D) = meromorphic forms ω (ω) D . { | ≥ } Let’s look at some examples. If D = 0 then i(D) = g: indeed K1(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 differ- entials. On the other hand, l(D) = 1 as the only rational functions reg- ular 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 A1. Composing → it with the inclusion A1 # P1 gives a regular morphism f : X P1 such → → that f(X) A1. But by Theorem 2.1.4, f(X) must be closed in P1, 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 (D) contains a non-constant function f with a unique pole L at P . It gives an isomorphism f : C P1. ! The last example shows the most common way of using Riemann–Roch. We define a linear system of divisors D = (f) + D f (D) . | | { | ∈ L } 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 posi- ! ! ≥ ! tive). So a linear system D consists of all effective divisors linearly equiv- | | alent to D. Choosing a basis f , . . . , f of (D) gives a map 0 r L r φD : C P , φD(x) = [f0, . . . , fr]. → Since C is a smooth curve, this map is regular. More generally, we can choose a basis of a linear subspace in and define a similar map. It called LD a map given by an incomplete linear system. In fact any map φ : C Pr 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 Pr. It can be obtained as follows: Any map φ is obtained by choosing rational functions f , . . . , f k(C). 0 r ∈ Consider their divisors (f0), . . . , (fr) and let D be their common denomi- nator. Then, clearly, f , . . . , f (D). 0 r ∈ L Moreover, in this case divisors (f0) + D, . . . , (fr) + D have very simple meaning: they are just pull-backs of coordinate hyperplanes in Pr. Indeed, suppose h is a local parameter at a point P C and suppose that P con- ∈ tributes nP to D. Then φ (in the neighborhood of P ) can be written as n n [f0h : . . . frh ], 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 (f0) + D, . . . , (fr) + 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 φD gives an embedding C Pr. 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 separates points: l(D P Q) = l(D) 2 for any points P, Q C. • D − − − ∈ φ separates tangents: l(D 2P ) = l(D) 2 for any point P C. • D − − ∈ 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: 2 2 3 (1) C A is given by a Weierstrass equation y = 4x g2x g3, where ∩ x,y − − ∆ = g3 27g2 = 0. 2 − 3 ) (2) C is isomorphic to a smooth cubic curve in P2. (3) C is isomorphic to a 2 : 1 cover of P1 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 P2 P1 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 A2 A1, the last ramification point is at ), ⇒ x,y → x ∞ Now the Riemann–Roch analysis. Let C be an algebraic curve of genus 1. Then (K) is one-dimensional. L 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 ψD 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 2 ψ3P : C P , → 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 1 ψ2P : C P , → with P as one of the ramification points, which shows directly that (6) ⇒ (3). Let ℘ (2P ) be a meromorphic function with pole of order 2 at P . If ∈ L 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 1 1 1 ℘(z) = + . z2 (z γ)2 − γ2 γ Λ, γ=0 ' ( ∈&% − 32 JENIA TEVELEV
Notice that ℘!(z) has poles of order 3 at lattice points, and therefore
1, ℘(z), ℘!(z) { } is a basis of (3P ). It follows that the embedding C P2 as a cubic curve L ⊂ is given (when pull-backed to C) by map 2 C C , 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 3 (℘!) = 4℘ g ℘ g . − 2 − 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 z π : C C, z ω. → *→ #P If the curve is given as a cubic in the Weierstrass normal form then those are elliptic integrals dx 4x3 g x g # − 2 − 3 We take the first homology group!H1(C) = Zα + Zβ and define periods ω, ω C. ∈ #α #β The periods generate a subgroup Λ C. If the periods are not linearly in- ⊂ dependent 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 fol- lows from Riemann–Hurwitz), thus from the theory of covering spaces f corresponds to a subgroup of π1(C/Λ) = Λ. Thus f must have the form C C/Λ! C/Λ, homeo! → 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 P1 if g = 0, or • C if g = 1, or • 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) = PGL2(R) ⊂ is a discrete subgroup acting freely on H. §2.7. J-invariant. Now we would like to classify elliptic curves up to iso- morphism, i.e. to describe M1. 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 transcen- dental 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 1 φ P +Q : C P | | → with P + Q as one of the fibers. Any double cover has an involution per- muting 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 transla- tions) by fixing a point:
M1 = M1,1. 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 automor- phism, namely the involution given by permuting the two branches of φ2P . 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 Z2.
2.7.1. THEOREM. (1) Curves given by Weierstrass equations y2 = 4x3 2 3 − g2x g3 and y = 4x g2! x g3! are isomorphic if and only if there − − 2− 3 exists t C∗ such that g2! = t g2 and g3! = t g3. There are only two ∈ 2 3 curves with special automorphisms: the curve y = x + 1 gives Z6 and 2 3 the curve y = x + x gives Z4 (draw the family of cuspidal curves in the g2g3-plane). (2) Two smooth cubic curves C and C! are isomorphic if and only if they are projectively equivalent. 1 (3) Let C (resp. C!) be a double cover of P with a branch locus p1, . . . , p4 (resp. p , . . . , p ). Then C C if and only if there exists g PGL such 1! 4! ! ! ∈ 2 that pi! = g(pi) for any i. In particular, we can always assume that branch points are 0, 1, , λ. There are two cases with non-trivial automorphisms, ∞ 2πi λ = 1 (Aut C = Z4) and λ = ω = e 3 (Aut C = Z6). Modulo Z2, − 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 ⊕ aτ + b a b τ ! = , PSL (Z) (2.7.2) cτ + d c d ∈ 2 ) * There are two elliptic curves (draw the square and the hexagonal lattice) with automorphism groups Z4 and Z6, 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 P1 by permuting branch points. In fact, λ is simply the cross-ratio: (p p ) (p p ) λ = 4 − 1 2 − 3 , (p p ) (p p ) 2 − 1 4 − 3 but branch points are not ordered, so we have an action of S4 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 /V S acts non-trivially: 4 ! 3 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 trans- → lation 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 *→ ∈ Z + Zτ = α(Z + Zτ !), which gives ατ ! = a + bτ, α = c + dτ, which gives (2.7.2). # So finally, we can introduce the j-invariant: g3 (λ2 λ + 1)3 j = 1728 2 = 256 − . (2.7.4) ∆ λ2(λ 1)2 − (λ2 λ+1)3 It is easy to see that the expression 256 λ2(−λ 1)2 does not change under the − transformations (2.7.3). For a fixed j0, the polynomial 256(λ2 λ + 1)3 jλ2(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 (Z6) and j = 1728 (Z4). 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 PSL2(Z) on . This group is called the H 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 funda- mental domain to the plane A1 (draw how). Since the j-invariant is invariant under z z + 1, it can be expanded in *→ a variable q = e2πiτ : 1 2 j = q− + 744 + 196884q + 21493760q + . . . 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 An) and a few sporadic groups. The largest sporadic group is the monster group F 1 that has about 1054 elements. Its existence was predicted by Robert Griess and Bernd Fischer in 1973 and it was even- tually 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 Sn. 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 isomor- phism 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 sti- died, so in the interest of drama let’s call it “Property X” for now. Let be M the moduli “set” of isomorphism classes of these objects. We have a map Y which sends y Y to the isomorphism class of the fiber f 1(y). → M ∈ − Geometric structure we are imposing on should be compatible with this M map Y : basically we should just ask that this map Y is a → M → M 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 A1 is a moduli space for some problem. Let C = y2 = x3 A2 be a cuspidal curve with the normalization map { } ⊂ 1 3 2 ν : A C, t (t , t ). → *→ 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 A1 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) in- → duces 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 ob- jects 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 fi- bration is a morphism f : X Y (satisfying property X) plus a morphism → σ : Y X such that f σ = Id . A morphism like this is called a section. → ◦ Y 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 (satis- → fying property X) and an arbitrary morphism g : Z Y then we define a → pull-back (or a fibered product) X Z = (x, z) X Z f(x) = g(z) X Z. ×Y { ∈ × | } ⊂ × We have a morphism X Z Z induced by the second projection. Since ×Y → 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 Z Z also has it (notice however that the morphism g : Z X ×Y → → 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 Z, namely (σ g, Id ). → → ×Y ◦ Z MODULI SPACES AND INVARIANT THEORY 37
The basic point is that X Y and X Z Z have the same fibers, → ×Y → and the map X Z to the moduli space factors through Y . ×Y → M → M This raises a tantalizing possibility that
2.9.4. DEFINITION. is a fine moduli space if there exists a universal family M U , i.e. a regular map (with property X) such that any other family → M 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 P1 is the only genus 0 curve (up to isomor- phisms). 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 P1 would appear as 1 1 Y pt P = Y P . × × However, there exist extremely simple P1-fibrations not isomorphic to the product. For example, consider the Hirzebruch surface F1. It is obtained by resolving indeterminacy locus of a “projection from a point” rational map 2 1 f : P !!" P by blowing up this point. In coordinates, the rational map is [x : y : z] [x : y], *→ 2 1 which is undefined at [0 : 0 : 1]. Its blow-up F1 is a surface in P P × [s:t] given by an equation xt = ys. The resolution of f is obtained by just restricting the second projection P2 1 1 2 × P P to F1. The first projection identifies F1 with P everywhere outside → of the point [0 : 0 : 1], the preimage of this point is a copy of P1 called the 1 exceptional divisor. All fibers of the resolved map F1 P are isomorphic 1 1 1 → to P but F1 P P . For example, E has self-intersection 1 on F1 but )! × − there are no ( 1)-curves in P1 P1. − × 2.9.6. EXAMPLE. How about the Grassmannian? We claim that G(2, n) is a fine moduli space for 2-dimensional subspaces of An. What is a fam- ily here? A family over an algebraic variety X should be a varying 2- dimensional subspace of An. In other words, a family over X is just a 2-dimensional vector sub-bundle E of the trivial vector bundle X An. × 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 , which means that we have isomorphisms ∪ α 1 r ψα : π− (Uα) Uα A , p2 ψi = π → × ◦ that are given by linear maps on the overlaps, i.e. over U U the induced α ∩ β map r r p2 Uα Uβ A Uα Uβ A Ar ∪ × → ∪ × → takes (x, v) A(x)v, *→ where A(x) is an invertible matrix with entries in (U U ). A map of O i ∩ j vector bundles E E is map of underlying varieties that is given by 1 → 2 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 Ar. What is the corresponding map × to the Grassmannian? Choose a trivializing affine covering Uα of X such 2 2 { } n that E Uα Uα A . Choose a basis u, v in A . An embedding E X A | ! × 2 n ⊂ × in the chart gives an embedding Uα A Uα A . Composing it with × ⊂ × projection to An gives maps (x, u) a (x)e , (x, v) a (x)e , *→ 1i i *→ 2i i &i &i where a (x), a (x) (U ). Then 1i 2i ∈ O α u v = p (x)e e , ∧ ij i ∧ j &i
1 Interestingly, this also shows that Aj is not a fine moduli space. Indeed, if A1 carries a universal family then Theorem 2.9.7 would be applicable to A1 as well. This would imply that locally at any point P A1 we have j ∈ g3(j) j = 1728 2 g3(j) 27g2(j) 2 − 3 for some rational functions g2 and g3. 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 27g2(j) j 1728 = 1728 3 . − g3(j) 27g2(j) 2 − 3 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 be the set of isomorphism (=conjugacy) classes of in- M vertible complex 2 2 matrices. (a) Describe as a set. (b) Let’s define the × M following moduli problem: a family over a variety X is a 2 2 matrix A(x) × with coefficients in (X) such that det A(x) (X), i.e. A(x) is invertible O ∈ O∗ 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 that makes M it into a coarse moduli space (2 points). Problem 2. Compute j-invariants of elliptic curves (1 point): (a) y2 + y = x3 + x; (b) y2 = x4 + ax3 + bx2 + 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 y2 = (1 x2)(1 e2x2) (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 36 1 y2 + xy = x3 x , − q 1728 − 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 A1 0, 1728 ) j \ { } carries a family of elliptic curves with j-invariant j. Is your family univer- sal? (2 points). (λ2 λ+1)3 1 1 Problem 7. The formula j = 256 λ2(−λ 1)2 gives a 6 : 1 cover Pλ Pj . 1 − 1 → Thinking about P as a Riemann sphere, let’s color Pj in two colors: color the upper half-plane white and the lower half-plane black. Draw the H 1 −H pull-back of this coloring to Pλ (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 P1 and the circle x2 + y2 = 1 A2 given by stereographic projection from (0, 1), describe { } ⊂ an algorithm for computing integrals of the form
P (x, 1 x2) 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 3 system φ 4P , show that C embeds in P as a curve of degree 4. (b) Show | | that quadrics in P3 containing C form a pencil P1 with 4 singular fibers. (c) These four singular fibers define 4 points in P1. Relate their cross-ratio to the j-invariant of C (4 points). Problem 11. Let X be an affine variety and let f (X). A subset ∈ O 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 (X)[1/f]. (c) Show that any affine O (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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Problem 13. (a) Show that any affine (resp. projective) variety X is a union of finitely many irreducible projective varieties X1, . . . , Xn such that MODULI SPACES AND INVARIANT THEORY 43
X X for i = j (called irreducible components of X). (b) Show that i )⊂ j ) irreducible components are defined uniquely (2 points). Problem 14. Let X Pn be an irreducible projective variety. Show that ⊂ any morphism X Pm is given by m + 1 homogeneous polynomials → F0, . . . , Fm in n + 1 variables of the same degree such that, for any point x X, at least one of the polynomials F does not vanish (1 point). ∈ i Problem 15. (a) Let X An and Y Am be affine varieties with co- ⊂ ⊂ ordinate algebras (X) and (Y ). Suppose these algebras are isomorphic. O O 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 Z2 Z2 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 Q1, Q2 Γ (considered as a Z2-vector space). Based { } ∈ on Theorem 2.9.7, describe families of elliptic curves with level 2 structure. Show that P1 0, 1, carries a family of elliptic curves with with a level 2 λ \{ ∞} 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 y2 = x3 + t of elliptic curves over A1 0 . Show that all fibers of this family have the same j-invariant but \ { } nevertheless this family is not trivial over A1 0 (2 points). \ { } Problem 18. Consider the family of cubic curves 3 3 3 2 Ca = x + y + z + axyz = 0 P { } ⊂ 1 parametrized by a A . (a) Find all a such that Ca is smooth and find its ∈ flex points. (b) Compute j as a function on a and find all a such that Ca 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 φ2P , show that the j-invariant of C has only 3 possible values and find these values (3 points).