# 20 The modular equation

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1 Elliptic Curves Lecture #20 Spring /26/ The modular equation In the previous lecture we defined modular curves as quotients of the extended upper half plane under the action of a congruence subgroup (a subgroup of SL 2 (Z) that contains Γ(N) := { ( ) c d SL2 (Z) : ( ) c d ( )} for some integer N 1). Of particular interest is the modular curve X 0 (N) := H /Γ 0 (N), where Γ 0 (N) = {( c d ) SL2 (Z) : c 0 mod N }. This modular curve plays a central role in the theory of elliptic curves. One form of the modularity theorem (a special case of which implies Fermat s last theorem) is that every elliptic curve E/Q admits a morphism X 0 (N) E for some N Z 1. It is also a key ingredient for algorithms that use isogenies of elliptic curves over finite fields, including the Schoof-Elkies-Atkin algorithm, an improved version of Schoof s algorithm that is the method of choice for counting points on elliptic curves over a finite fields of large characteristic. Our immediate interest in the modular curve X 0 (N) is that we will use it to prove the first main theorem of complex multiplication; among other things, this theorem implies that the j-invariants of elliptic curve E/C with complex multiplication are algebraic integers. There are two properties of X 0 (N) that make it so useful. The first, which we will prove in this lecture, is that it has a canonical model over Q with integer coefficients; this allows us to interpret X 0 (N) as a curve over any field, including finite fields. The second is that it parameterizes isogenies between elliptic curves (in a sense that we will make precise in the next lecture). In particular, given the j-invariant of an elliptic curve E and an integer N, we can use our explicit model of X 0 (N) to determine the j-invariants of all elliptic curves that are related to E by an isogeny whose kernel is a cyclic group of order N. In order to better understand modular curves, we introduce modular functions Modular functions Modular functions are meromorphic functions on a modular curve. To make this statement precise, we first need to discuss q-expansions. The map q : H D defined by q(τ) = e 2πiτ = e 2π im τ (cos(2π re τ) + i sin(2π re τ)) bijectively maps each vertical strip H n := {τ H : n re τ < n + 1} (for any n Z) to the punctured unit disk D 0 := D {0}. We also note that lim q(τ) = 0. im τ If f : H C is a meromorphic function that satisfies f(τ + 1) = f(τ) for all τ H, then we can write f in the form f(τ) = f (q(τ)), where f : D 0 C is a meromorphic function that we can define by fixing a vertical strip H n and putting f := f (q Hn ) 1. The q-expansion (or q-series) of f(τ) is obtained by composing the Laurent-series expansion of f at 0 with the function q(τ): f(τ) = f (q(τ)) = + n= a n q(τ) n = + n= a n q n. As on the RHS above, it is customary to simply write q for q(τ) = e 2πiτ, as we shall do henceforth; but keep in mind that the symbol q denotes a function of τ H. Lecture by Andrew Sutherland

2 If f is meromorphic at 0 (meaning that z k f (z) has an analytic continuation to an open neighborhood of 0 D for some k Z 0 ) then the q-expansion of f has only finitely many nonzero a n with n < 0 and we can write f(τ) = a n q n, n=n 0 with a n0 0, where n 0 is the order of f at 0. We then say that f is meromorphic at, and call n 0 the order of f at. More generally, if f satisfies f(τ + N) = f(τ) for all τ H, then we can write f as f(τ) = f (q(τ) 1/N ) = n= a n q n/n, (1) and we say that f is meromorphic at if f is meromorphic at 0. If Γ is a congruence subgroup of level N, then for any Γ-invariant function f we have f(τ + N) = f(τ) (for γ = ( 1 0 N 1 ) Γ we have γτ = τ + N), so f can be written as in (1), and it makes sense to say that f is (or is not) meromorphic at. Definition Let f : H C be a meromorphic function that is Γ-invariant for some congruence subgroup Γ. The function f(τ) is said to be meromorphic at the cusps if for every γ SL 2 (Z) the function f(γτ) is meromorphic at. It follows immediately from the definition that if f(τ) is meromorphic at the cusps, then for any γ SL 2 (Z) the function f(γτ) is also meromorphic at the cusps. In terms of the extended upper half-plane H, notice that for any γ SL 2 (Z), lim γτ im τ P1 (Q), and recall that P 1 (Q) is the SL 2 (Z)-orbit of H, whose elements are called cusps. To say that f(γτ) is meromorphic at is to say that f(τ) is meromorphic at γ. To check whether f is meromorphic at the cusps, it suffices to consider a set of Γ-inequivalent cusp representatives γ 1, γ 1,..., γ n, one for each Γ-orbit of ; this is a finite set because the congruence subgroup Γ has finite index in SL 2 (Z). If f is a Γ-invariant meromorphic function, then for any γ Γ we must have lim f(γτ) = lim f(τ) im τ im τ whenever either limit exists, and if neither limit exits then f must still have the same order at and γ. Thus if f is meromorphic at the cusps then it determines a meromorphic function g : X Γ C on the modular curve X Γ := H /Γ (as a complex manifold). Definition Let Γ be a congruence subgroup. A modular function for Γ is a Γ- invariant meromorphic function f : H C that is meromorphic at the cusps; equivalently, it is a meromorphic function g : X Γ C. Sums, products, and quotients of modular functions for Γ are also modular functions for Γ, as are constant functions, thus the set of all modular functions for Γ forms a field C(Γ) that we view as a transcendental extension of C. As we will shortly prove for X 0 (N), modular curves X Γ are not only Riemann surfaces, they are also algebraic curves, and the field C(Γ) of modular functions for Γ is isomorphic to the function field C(X Γ ) of X Γ as an algebraic curve over C Spring 2017, Lecture #20, Page 2

3 Remark In fact, every compact Riemann surface corresponds to a smooth projective (algebraic) curve over C that is uniquely determined up to isomorphism. Conversely, if X/C is a smooth projective curve then the set X(C) can be given a topology and a complex structure that makes it a compact Riemann surface S. The function field of X and the field of meromorphic functions on S are both finite extensions of a purely transcendental extension of C (of transcendence degree one), and the two fields are isomorphic. We will make this isomorphism completely explicit for X(1) and X 0 (N). Remark If f is a modular function for a congruence subgroup Γ, then it is also a modular function for any congruence subgroup Γ Γ, since Γ-invariance obviously implies Γ -invariance, and the property of being meromorphic at the cusps does not depend on Γ. Thus for all congruence subgroups Γ and Γ we have Γ Γ = C(Γ) C(Γ ), and the corresponding inclusion of function fields C(X Γ ) C(X Γ ) induces a morphism X Γ X Γ of algebraic curves, a fact that has many useful applications Modular Functions for Γ(1) We first consider the modular functions for Γ(1) = SL 2 (Z). In Lecture 16 we proved that the j-function is Γ(1)-invariant and holomorphic (hence meromorphic) on H. To show that the j(τ) is a modular function for Γ(1) we just need to show that it is meromorphic at the cusps. The cusps are all Γ(1)-equivalent, so it suffices to show that the j(τ) is meromorphic at, which we do by computing its q-expansion. We first record the following lemma, which was used in Problem Set 8. Lemma Let σ k (n) = d n dk, and let q = e 2πiτ. We have ( ) g 2 (τ) = 4π σ 3 (n)q n, 3 ( ) g 3 (τ) = 8π σ 5 (n)q n, 27 (τ) = g 2 (τ) 3 27g 3 (τ) 2 = (2π) 12 q Proof. See Washington [1, pp ]. Corollary With q = e 2πiτ we have where the a n are integers. (1 q n ) 24. j(τ) = 1 q a n q n, Spring 2017, Lecture #20, Page 3

4 Proof. Applying Lemma 20.5 yields g 2 (τ) 3 = π12 ( q q 2 + ) 3 = π12 ( q q 2 + ), 27g 3 (τ) 2 = 64π12 27 (1 504q 16632q2 ) 2 = π12 (1 1008q q 2 + ), (τ) = π12 q(1728q 41472q 2 + ) = π q(1 24q + ), and we then have with a n Z, as desired. j(τ) = 1728g 2(τ) 3 (τ) = 1 q a n q n, Remark The proof of Corollary 20.6 explains the factor 1728 that appears in the definition of the j-function: it is the least positive integer that ensures that the q-expansion of j(τ) has integral coefficients. The corollary implies that the j-function is a modular function for Γ(1), with a simple pole at. We proved in Theorem 18.5 that the j-function defines a holomorphic bijection from Y (1) = H/Γ(1) to C. If we extend the domain of j to H by defining j( ) =, then the j-function defines an isomorphism from X(1) to the Riemann sphere S := P 1 (C) that is holomorphic everywhere except for a simple pole at. In fact, if we fix j(ρ) = 0, j(i) = 1728, and j( ) =, then the j-function is uniquely determined by this property (as noted above, we put j(i) = 1728 to obtain an integral q-expansion). It is for this reason that the j-function is sometimes referred to as the modular function. Indeed, every modular function for Γ(1) = SL 2 (Z) can be written in terms of the j-function. Theorem Every modular function for Γ(1) is a rational function of j(τ); in other words C(Γ(1)) = C(j). Proof. As noted above, the j-function is a modular function for Γ(1), so C(j) C(Γ(1)). If g : X(1) C is a modular function for Γ(1) then f := g j 1 : S C is meromorphic, and Lemma 20.9 below implies that f is a rational function. Thus g = f j C(j). Lemma Every meromorphic function f : S C on the Riemann sphere S := P 1 (C) is a rational function. Proof. Let f : S C be a nonzero meromorphic function. We may assume without loss of generality that f has no zeros or poles at := (1 : 0), since we can always apply a linear fractional transformation γ SL 2 (C) to move a point where f does not have a pole or a zero to and replace f by f γ (note that γ and γ 1 are rational functions, and if f γ is a rational function, so is f = f γ γ 1 ). Let {p i } be the set of poles of f(z), with orders m i := ord pi (f), and let {q j } be the set of zeros of f, with orders n j := ord qj (f). We claim that m i = i j n j Spring 2017, Lecture #20, Page 4

5 To see this, triangulate S so that all the poles and zeros of f(z) lie in the interior of a triangle. It follows from Cauchy s argument principle (Theorem 15.17) that the contour integral f (z) f(z) dz about each triangle (oriented counter clockwise) is the difference between the number of zeros and poles that f(z) in its interior. The sum of these integrals must be zero, since each edge in the triangulation is traversed twice, once in each direction. The function h: S C defined by h(z) = f(z) i(z p i) m i j (z q j) n j has no zeros or poles on S. It follows from Liouville s theorem (Theorem 15.30) that h is a constant function, and therefore f(z) is a rational function of z. Corollary Every modular function f(τ) for Γ(1) that is holomorphic on H is a polynomial in j(τ). Proof. Theorem 20.8 implies that f can be written as a rational function of j, so f(τ) = c i(j(τ) α i) k (j(τ) β k), for some c, α i, β j C. Now the restriction of j to any fundamental region for Γ(1) is a bijection, so f(τ) must have a pole at j 1 (β k ) for each β k. But f(τ) is holomorphic and therefore has no poles, so the set {β j } is empty and f(τ) is a polynomial in j(τ). We proved in the previous lecture that the j-function j : X(1) S determines an isomorphism of Riemann surfaces. As an algebraic curve over C, the function field of X(1) S = P 1 (C) is the rational function field C(t), and we have just shown that the field of modular functions for Γ(1) is the field C(j) of rational functions of j. Thus, as claimed in Remark 20.3, the function field C(X(1)) = C(t) and the field of modular functions C(Γ(1)) = C(j) are isomorphic, with the isomorphism given by t j. More generally, for every congruence subgroup Γ, the field C(X Γ ) C(Γ) is a finite extension of C(t) C(j) whose degree is equal to the index of Γ in Γ(1). Theorem Let Γ be a congruence subgroup. The field C(Γ) of modular functions for Γ is a finite extension of C(j) of degree at most n := [Γ(1) : Γ]. Proof. Let γ 1 be the identity in Γ(1) and let {γ 1,, γ n } Γ(1) be a set of right coset representatives for Γ as a subgroup of Γ(1) (so Γ(1) = Γγ 1 Γγ n ). Let f C(Γ) and for 1 i n define f i (τ) := f(γ i τ). For any γ i Γγ i the functions f(γ i τ) and f(γ iτ) are the same, since f is Γ-invariant. For any γ Γ(1), the set of functions {f(γ i γτ)} is therefore equal to the set of functions {f(γ i τ)}, since multiplication on the right by γ permutes the cosets {Γγ i }. Any symmetric polynomial in the functions f i is thus Γ(1)- invariant, and meromorphic at the cusps (since f, and therefore each f i, is), hence an element of C(j), by Theorem Now let P (Y ) = (Y f i ). i {1,,n} Spring 2017, Lecture #20, Page 5

6 Then f = f 1 is a root of P (since γ 1 is the identity), and the coefficients of P (Y ) lie in C(j), since they are all symmetric polynomials in the f i. It follows that every f C(Γ) is the root of a monic polynomial in C(j)[Y ] of degree n; this implies that C(Γ)/C(j) is an algebraic extension, and it is separable, since we are in characteristic zero. We now claim that C(Γ) is finitely generated: if not we could pick functions g 1,..., g n+1 C(Γ) such that C(j) C(j)(g 1 ) C(j)(g 1, g 2 ) C(j)(g 1,..., g n+1 ). But then C(j)(g 1,..., g n+1 ) is a finite separable extension of C(j) of degree at least n + 1, and the primitive element theorem implies it is generated by some g C(Γ) whose minimal polynomial most have degree greater than n, which is a contradiction. The same argument then shows that [C(Γ) : C(j)] n. Remark If I Γ then in fact [C(Γ(1)) : C(Γ)] = [Γ(1) : Γ]; we will prove this for Γ = Γ 0 (N) in the next section. In general [C(Γ(1)) : C(Γ)] = [Γ(1) : Γ], where Γ denotes the image of Γ in PSL 2 (Z) := SL 2 (Z)/{±I} Modular functions for Γ 0 (N) We now consider modular functions for the congruence subgroup Γ 0 (N). Theorem The function j N (τ) := j(nτ) is a modular function for Γ 0 (N). Proof. The function j N (τ) is obviously meromorphic (in fact holomorphic) on H, since j(τ) is, and it is meromorphic at the cusps for the same reason (note that τ is a cusp if and only if Nτ is). We just need to show that j N (τ) is Γ 0 (N)-invariant. Let γ = ( ) c d Γ0 (N). Then c 0 mod N and ( ) ( ) N(aτ + b) anτ + bn where j N (γτ) = j(nγτ) = j cτ + d = j c N Nτ + d ( ) γ N = SL c/n d 2 (Z), = j(γ Nτ) = j(nτ) = j N (τ), since c/n is an integer and det(γ ) = det(γ) = 1. Thus j N (τ) is Γ 0 (N)-invariant. Theorem The field of modular functions for Γ 0 (N) is an extension of C(j) of degree n := [Γ(1) : Γ 0 (N)] generated by j N (τ). Proof. By the previous theorem, we have j N C(Γ 0 (N)), and from Theorem we know that C(Γ 0 (N)) is a finite extension of C(j) of degree at most n, so it suffices to show that the minimal polynomial of j N over C(j) has degree at least n. As in the proof of Theorem 20.11, let us fix a set of right coset representatives {γ 1,, γ n } for Γ 0 (N) Γ(1), and let P C(j)[Y ] be the minimal polynomial of j N over C(j). We may view P (j(τ), j N (τ)) as a function of τ, which must be the zero function. If we replace τ by γ i τ then for each γ i we have 0 = P (j(γ i τ), j N (γ i τ)) = P (j(τ), j N (γ i τ)), so the function j N (γ i τ) is also a root of P (Y ) Spring 2017, Lecture #20, Page 6

7 To prove that deg P n it suffices to show that the n functions j N (γ i τ) are distinct. Suppose not. Then j(nγ i τ) = j(nγ k τ) for some i k and τ H that we can choose to have stabilizer ±I. Fix a fundamental region F for H/Γ(1) and pick α, β Γ(1) so that αnγ i τ and βnγ j τ lie in F. The j-function is injective on F, so j(αnγ i τ) = j(βnγ k τ) αnγ i τ = ±βnγ k τ αnγ i = ±βnγ k, where we may view N as the matrix ( ) N 0 0 1, since Nτ = Nτ+0 0τ+1. Now let γ = α 1 β = ( ) c d. We have ( ) ( ) ( ) N 0 N 0 γ 0 1 i = ± γ c d 0 1 k, and therefore ( ) ( ) ( ) γ i γ 1 1/N 0 N 0 k = ± 0 1 c d 0 1 ( ) /N = ±. cn d We have γ i γ 1 k SL 2 (Z), so b/n is an integer, and cn 0 mod N, so γ i γ 1 k Γ 0 (N). But then γ i and γ k lie in the same right coset of Γ 0 (N), which is a contradiction The modular polynomial Definition The modular polynomial Φ N is the minimal polynomial of j N over C(j). It follows from the proof of Theorem 20.14, we may write Φ N C(j)[Y ] as Φ N (Y ) = n (Y j N (γ i τ)), i=1 where {γ 1,... γ n } is a set of right coset representatives for Γ 0 (N). The coefficients of Φ N (Y ) are symmetric polynomials in j N (γ i τ), so as in the proof of Theorem they are Γ(1)- invariant. They are holomorphic on H, so they are polynomials in j, by Corollary Thus Φ N C[j, Y ]. If we replace every occurrence of j in Φ N with a new variable X we obtain a polynomial in C[X, Y ] that we write as Φ N (X, Y ). Our next task is to prove that the coefficients of Φ N (X, Y ) are actually integers, not just complex numbers. To simplify the presentation, we will prove this only prove for prime N, which is all that is needed in most practical applications (such as the SEA algorithm), and suffices to prove the main theorem of complex multiplication. The proof for composite N is essentially the same, but explicitly writing down a set of right coset representatives γ i and computing the q-expansions of the functions j N (γ i τ) is more complicated. We begin by fixing a specific set of right coset representatives for Γ 0 (N). Lemma For prime N we can write the right cosets of Γ 0 (N) in Γ(1) as { Γ0 (N) } { Γ 0 (N)ST k : 0 k < N }, where S = ( ) and T = ( ) Spring 2017, Lecture #20, Page 7

8 Proof. We first show that the these cosets cover Γ(1). Let γ = ( ) c d Γ. If c 0 mod N, then γ Γ 0 (N) lies in the first coset. Otherwise, pick k [0, N 1] so that kc d mod N (c is nonzero modulo the prime N, so this is possible), and let ( ) ka b a γ 0 := Γ kc d c 0 (N). Then ( ) ( ) ( ) γ 0 ST k ka b a 0 1 = = = γ, kc d c 1 k c d lies in Γ 0 (N)ST k. We now show the cosets are distinct. Suppose not. Then there must exist γ 1, γ 2 Γ 0 (N) such that either (a) γ 1 = γ 2 ST k for some 0 k < N, or (b) γ 1 ST j = γ 2 ST k with 0 j < k < N. Let γ 2 = ( ) c d. In case (a) we have ( ) ( ) ( ) 0 1 b bk a γ 1 = = Γ c d 1 k d dk c 0 (N), which implies d 0 mod N and det γ 2 = ad bc 0 mod N, a contradiction. In case (b), with m = k j we have ( ) ( ) ( ) ( ) γ 1 = γ 2 ST m S a bm b = = Γ c d 1 m 1 0 c dm d 0 (N). Thus c dm 0 mod N, and since c 0 mod N and m 0 mod N, we must have d 0 mod N, which again implies det γ 2 = ad bc 0 mod N, a contradiction. Theorem Φ N Z[X, Y ]. Proof (for N prime). Let γ k = ST k. By Lemma we have N 1 Φ N (Y ) = (Y j N (τ)) (Y j N (γ k τ)). Let f(τ) be a coefficient of Φ N (Y ). Then f(τ) is holomorphic function on H, since j(τ) is, and f(τ) is Γ(1)-invariant, since it is symmetric polynomial in j N (τ) and the functions j N (γ k τ), corresponding to a complete set of right coset representatives for Γ 0 (N); and f(τ) is meromorphic at the cusps, since it is a polynomial in functions that are meromorphic at the cusps. Thus f(τ) is a modular function for Γ(1) holomorphic on H and therefore a polynomial in j(τ), by Corollary By Lemma below, if we can show that the q-expansion of f(τ) has integer coefficients, then it will follow that f(τ) is an integer polynomial in j(τ) and therefore Φ N Z[X, Y ]. We first show that the q-expansion of f(τ) has rational coefficients. We have k=0 j N (τ) = j(nτ) = 1 q N a n q nn, where the a n are integers, thus j N Z((q)). For j N (γ k τ), we have ( ( ) ) j N (γ k τ) = j(nγ k τ) = j N ST k τ = j ( S ( N ) ( ) ) (( 1 k 0 1 τ = j N ) ( 1 k 0 1 ( ) ) ) τ + k τ = j, N Spring 2017, Lecture #20, Page 8

9 where we are able to drop the S because j(τ) is Γ-invariant. If we let ζ N = e 2πi N, then and q ( (τ+k) τ+k /N) = e 2πi( N ) = e 2πi k N q 1/N = ζnq k 1/N, j N (γ k τ) = ζ k N q 1/N + n=0 a n ζ kn N q n/n, thus j N (γ k τ) Q(ζ N )((q 1/N )). The action of the Galois group Gal(Q(ζ N )/Q) on the coefficients of the q-expansions of each j N (γ k τ) induces a permutation of the set {j N (γkτ)} and fixes j N (τ). It follows that the coefficients of the q-expansion of f are fixed by Gal(Q(ζ N ))/Q) and must lie in Q. Thus f Q((q 1/N )), and f(τ) is a polynomial in j(τ), so its q-expansion contains only integral powers of q and f Q((q)). We now note that the coefficients of the q-expansion of f(τ) are algebraic integers, since the coefficients of the q-expansions of j N (τ) and the j N (γ k ) are algebraic integers, as is any polynomial combination of them. This implies f(τ) Z((q)). Lemma (Hasse q-expansion principal). Let f(τ) be a modular function for Γ(1) that is holomorphic on H and whose q-expansion has coefficients that lie in an additive subgroup A of C. Then f(τ) = P (j(τ)), for some polynomial P A[X]. Proof. By Corollary 20.10, we know that f(τ) = P (j(τ)) for some P C[X], we just need to show that P A[X]. We proceed by induction on d = deg P. The lemma clearly holds for d = 0, so assume d > 0. The q-expansion of the j-function begins with q 1, so the q-expansion of f(τ) must have the form n= d a nq n, with a n A and a d 0. Let P 1 (X) = P (X) a d X d, and let f 1 (τ) = P 1 (j(τ)) = f(τ) a d j(τ) d. The q-expansion of the function f 1 (τ) has coefficients in A, and by the inductive hypothesis, so does P 1 (X), and therefore P (X) = P 1 (X) + a d X d also has coefficients in A. References [1] Lawrence C. Washington, Elliptic curves: number theory and cryptography, second edition, Chapman & Hall/CRC, Spring 2017, Lecture #20, Page 9

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