ELEMENTARY MODULAR IWASAWA THEORY HARUZO HIDA Contents 1. Curves over a field 3 1.1. Plane curves 3 1.2. Tangent space and local rings 5 1.3. Projective space 8 1.4. Projective plane curve 9 1.5. Divisors 11 1.6. The theorem of Riemann–Roch 12 1.7. Regular maps from a curve into projective space 13 2. Elliptic curves 14 2.1. Abel’s theorem 14 2.2. Weierstrass Equations of Elliptic Curves 15 2.3. Moduli of Weierstrass Type 17 3. Modular forms and functions 20 3.1. Geometric modular forms 20 3.2. Topological Fundamental Groups 21 3.3. Classical Weierstrass ℘-function 23 3.4. Complex Modular Forms 24 3.5. Weierstarss ζ and σ functions 26 3.6. Product q-expansion 28 3.7. Klein forms 29 4. Modular units 32 4.1. Siegel units 33 4.2. Distribution on p-divisible groups 34 4.3. Stickelberger distribution 35 4.4. Rank of distribution 36 4.5. Cusps of X(N) 37 4.6. Finiteness of ClX(N) 38 4.7. Siegel units generate p×m 38 4.8. Fricke–Wohlfahrt theoremA 42 4.9. Siegel units and Stickelberger’s ideal 44 4.10. Cuspidal class number formula 46 4.11. Cuspidal class number formula for X1(N). 49 Date: May 25, 2018. 1 ELEMENTARY MODULAR IWASAWA THEORY 2 References 50 We discuss the first three topics of the following list: (1) Explicit construction of modular forms and modular functions; (2) Determination of units in the elliptic modular function fields (modular units); (3) The cuspidal class group of modular curves including a proof of the cuspidal class number formula; (4) Construction of units (called elliptic units) of the Hilbert ring class field of imaginary quadratic fields as specialization of modular units; (5) Iwasawa theory for imaginary quadratic fields via elliptic units. If time allows, we further go into the topics (4) and (5). A modular curve is an affine plane curve (i.e., an open Riemann surface) classifying elliptic curves with certain additional structure (called level structure) naturally defined over Q. Asa Riemann surface, it is a quotient of the upper half complex plane by SL2(Z) (and its subgroups). Adding finite number of points (called cusps), we can complete the curve into a projective curve (i.e., a compact Riemann surface). Most of recent progress in number theory and arithmetic geometry is based on the study of modular curves and modular forms defined on them; e.g., proof of Iwasawa’s conjectures (Mazur–Wiles) and Fermat’s last theorem (Wiles). Modular units are the units in the ring of holomorphic functions of the affine modular curve. Divisors supported on cusps modulo principal divisors (divisors of modular units) give the cuspidal class group (which is the torsion subgroup of rational points of the Jacobian of the modular curve). We can determine explicitly the group of modular units via classical results of Weierstrass and Siegel (like the determination of cyclotomic units in the field generated by roots of unity by Dirichlet–Kummer). Striking points are that the class group is finite and that we have an explicit class number formula of Kubert–Lang in terms of the Dirichlet L-values at s = 2 (while the classical class number formula of Dirichlet/Kummer of the cyclotomic field is in terms of the values at s = 1). This is done by generalizing Stickelberger’s theory of cyclotomic class groups to the setting of modular curves. This theory gives a base of the proof of the Iwasawa main conjecture by Mazur–Wiles. For each prime p> 2, sin(πa/p)/ sin(π/p) 0<a<p/2 gives independent units in the integer ring of the field of{p-th root of unity for} each prime p (the cyclotomic units). Analogously, the value of modular units at a point on the upper half complex plane belonging to an imaginary quadratic field K gives independent units in the (certain) Hilbert ring class field over K, and this is a base of the generalization of the cyclotomic Iwasawa theory to the elliptic Iwasawa theory of imaginary quadratic fields. If time allows, we describe these finer results (possibly including the class number formulas of the ring class field as an index of elliptic units inside the entire units). We start with a sketch of the theory of affine plane curves and how to compactify it in a projective space. Then we turn analytic and construct rational functions over modular curves in analytic means. This explicit construction helps us to compute cucpidal class groups inside the Jacobian of the projective modular curves. ELEMENTARY MODULAR IWASAWA THEORY 3 1. Curves over a field Any algebraic curve over an algebraically closed field can be embedded into the 3-dimensional projective space P3 (e.g., [ALG, IV.3.6]) and any closed curve in P3 is birationally isomorphic to a curve inside P2 (a plane curve; see [ALG, IV.3.10]), we give some details of the theory of plain curve defined over a field k C in this section to accustom with the theory of curves. We also write k for the algebraic⊂ closure k of k inside C. 1.1. Plane curves. Let a be a principal ideal of the polynomial ring k[X, Y ]. Note that polynomial rings over a field is a unique factorization domain. We thus have e(p) prime factorization a = p p with principal primes p. We call a square free if 0 e(p) 1 for all principal primes p. Fix a square-free a. The set of A-rational points≤ for≤ any k-algebra AQof a plane curve is given by the zero set 2 Va(A)= (x,y) A f(x,y)=0 for all f(X, Y ) a . ∈ ∈ Obviously, for a generator f(X, Y ) of a, we could have defined 2 Va(A)= V (A)= (x,y) A f(x) = 0 , f ∈ but this does not depend on the choice of generators and depen ds only on the ideal a; 2 so, it is more appropriate to write Va. As an exceptional case, we note V(0)(A)= A . 2 Geometrically, we think of Va(C) as a curve in C = V(0)(C) (the 2-dimensional “plane”). This view point is more geometric. In this sense, for any algebraically closed field K over k, a point x Va(K) is called a geometric point with coefficients ∈ 2 in K, and V(f)(K) V(0)(K) is called the geometric curve in V(0)(K) = K defined by the equation f(X,⊂ Y )=0. By Hilbert’s zero theorem (Nullstellensatz; see [CRT] Theorem 5.4 and [ALG] Theorem I.1.3A), writing a the principal ideal of k[X, Y ] generated by a, we have (1.1) a = f(X, Y ) k[X, Y ] f(x,y)=0 forall(x,y) Va(k) . ∈ ∈ Thus we have a bijection square-free ideals of k[X, Y ] plane curves Va(k) V (k) . { }↔{ ⊂ (0) } The association Va : A Va(A) is a covariant functor from the category of k-algebras to the category of sets (denoted7→ by SETS). Indeed, for any k-algebra homomorphism σ : A A0, Va(A) (x,y) (σ(x),σ(y)) Va(A0)as 0 = σ(0) = σ(f(x,y)) = f(σ(x),σ→(y)). Thus a3= a k7→[X, Y ] is determined∈ uniquely by this functor, but the ∩ value Va(A) for an individual A may not determine a. From number theoretic view point, studying Va(A) for a small field is important. Thus it would be better regard Va as a functor in some number theoretic setting. If a = p p for principal prime ideals p, by definition, we have Q Va = Vp. p [ The plane curve Vp (for each prime p a) is called an irreducible component of Va. Since | p is a principal prime, we cannot further have non-trivial decomposition Vp = V W with plane curves V and W . A prime ideal p k[X, Y ] may decompose into∪ a ⊂ ELEMENTARY MODULAR IWASAWA THEORY 4 product of primes in k[X, Y ]. If p remains prime in k[X, Y ], we call Vp geometrically irreducible. Suppose that we have a map FA = F (φ)A : Va(A) Vb(A) given by two poly- nomials φ (X, Y ),φ (X, Y ) k[X, Y ] (independent→ of A) such that F (x,y) = X Y ∈ A (φ (x),φ (y)) for all (x,y) Va(A) and all k-algebras A. Such a map is called X Y ∈ a regular k-map or a k-morphism from a plane k-curve Va into Vb. Here Va and Vb 1 are plane curve defined over k. If A = Vb is the affine line, i.e., Vb(A) = A for all A 1 ∼ (taking for example b =(y)), a regular k-map Va A is called a regular k-function. → Regular k-functions are just functions induced by the polynomials in k[x,y] on Va; so, Ra is the ring of regular k-functions of Va defined over k. We write Homk-curves(Va, Vb) for the set of regular k-maps from Va into Vb. Obvi- ously, only φ? mod a can possibly be unique. We have a commutative diagram for any k-algebra homomorphism σ : A A0: → FA Va(A) Vb(A) −−−→ σ σ Va(A0) Vb(A0). y −−−→FA0 y Indeed, σ(FA((x,y))) = (σ(φX(x,y)),σ(φY (x,y))) =(φX(σ(x),σ(y)),φY (σ(x),σ(y)) = FA0 (σ(x),σ(y)).