Lecture Notes: An invitation to modular forms

Lecture Notes: An invitation to modular forms October 3, 20 Tate s thesis in the context of class field theory. Recirocity laws Let K be a number field or function field. Let C K = A K /K be the idele class grou. One of the main theorems of class field theory is the existence of a recirocity ma rec K : C K Gal(K ab /K). This ma is continuous with dense image. Whenever L/K is a finite abelian extension, there is a relative version of the above ma, call it rec L/K, that makes the diagram commute: rec K C K Gal(K ab /K) C K /N L/K (C L ) rec L/K Gal(L/K) (There is also a diagram which relates rec K to rec L.) For each lace v of K there is a local recirocity ma: rec Kv : K v Gal(K ab v /K v ). There is a relative version of rec Kv as well. In addition, when v is a nonarchimedean lace, we recognize that either side of the above has a distinguished quotient: Z on the left side, and Gal(kv ab /k v ) = Ẑ on the right. The

local recirocity ma has the roerty that the diagram K v v Z rec Kv Gal(K ab v /K v ) Ẑ commutes. Here we are identifying Ẑ with Gal(kab v /k v ) by matching u Ẑ to the arithmetic Frobenius (the automorhism sending x to x q, where q = #O v / v ). In other words, rec Kv sends uniformizers to arithmetic Frobenius elements. (There is another, and erfectly good, normalization of local class field theory, where uniformizers get sent to geometric Frobenius elements, in which case the definitions of the recs have to be inverted.) The recirocity mas exhibit local-global comatibility, meaning that K v rec Kv Gal(K ab v /K v ) C K rec K Gal(K ab /K) commutes..2 Artin L-functions and Hecke L-functions Consider the consequences for Artin L-functions. Let K be a number field and let σ : Gal(K ab /K) C be a (necessarily finite-order) one-dimensional Artin character. The associated Artin L-function is L(σ, s) = ( σ(frob ) ), N s where the roduct runs over the rimes of K at which σ is unramified (meaning that σ is trivial on the inertia grou of ). Let χ = σ rec; this is a Hecke character. Then χ and σ are unramified at the same laces. For an unramified rime with uniformizer π, we have χ(π ) = σ(frob ) (because of local-global comatibility, and because rec K takes uniformizers to arithmetic Frobenius elements. Therefore the Artin L-function is a Hecke L-function: L(σ, s) = L(χ, s) 2

As a consequence, L(σ, s) admits an analytic continuation to all s C (which is even entire if σ is nontrivial). If Λ(χ, s) is the comleted L-function (throw in the Γ-factors for each infinite lace), then Λ(χ, s) is related to Λ(χ, s) via the functional equation: L(χ, s) = (fudge) v S ρ v (χ, s)l(χ, s) Here fudge only deends on the ramified rimes in K/Q and essentially not on χ. The local factor ρ v (χ, s) only deends on the restriction χ v of χ to K v. (This is the innovation which ushed Tate s results beyond Hecke s.) Therefore ρ v (s) only deends on the restriction of σ to the decomosition grou at v. What sorts of (analytic) objects π could be associated to n-dimensional Artin L-functions? Whatever they are, they should have:. An L function L(π, s) which is a meromorhic function of all s C, such that L(π, s) = L(σ, s) for s with large enough real art: 2. Local comonents π v, which only deend on the restriction of σ to the decomosition grou at v, 3. For each v, a local factor ρ(π v, s), which only deends on π v, such that L(π, s) has a functional equation of the form L(ˇπ, k s) = v ρ(π v, s)l(π, s) u to a fudge factor. 2 Modular Forms: Definitions For K = Q and n = 2, the answer turns out to be modular forms and their cousins the Maass forms, although we will have to ut in a significant amount of work to see why these objects have the desired roerties. (A modular form is a function on the uer half-lane. What could it mean to take its local comonent at 7?) The classical theory of modular forms is due to Hecke. It has all to do with the action of discrete subgrous of SL 2 (R) (known as Fuchsian grous) 3

on the uer half lane H. We start with the rincial congruence subgrou {( ) } a b Γ(N) = SL c d 2 (Z) a, d (mod N), b, c 0 (mod N) and say that a subgrou Γ SL 2 (Z) is a congruence subgrou if it contains Γ(N) for some N. Two imortant examles are Γ 0 (N) (res., Γ (N)), which are those subgrous of SL 2 (Z) where c 0 (mod N) (res., c 0 (mod N) and a b (mod N)). Let Γ be a congruence subgrou, and let k be an integer. A modular form of weight k for Γ is a comlex-valued function f on the uer half-lane H with the following roerties: ( ) a b. For all γ = Γ, c d 2. f is holomorhic on H. f(γz) = (cz + d) k f(z). 3. f is holomorhic at the cuss of Γ\H. ( ) a b For any g = GL c d 2 (R) with ositive determinant, we write f g,k (z) = (cz + d) k (det g) k/2 f(gz); this actually defines an action of GL + 2 (R) on functions on H. (The center acts trivially, so that in fact we have an action of PGL + 2 (R).) The first condition can be restated as f γ,k = f for all γ Γ. The last oint requires some exlanation. A cus of Γ is an equivalence class in an element of R { } which is fixed by a arabolic subgrou of Γ (a subgrou which has exactly one fixed oint on the ( Riemann ) shere). For Z instance, is always fixed by the intersection of with Γ, so is always a cus of a congruence subgrou. We now define what it means for f to be ( holomorhic ) at. The subgrou Γ Γ which fixes is of the Zw form for some w, known as the width of the cus at. We 4

have f(z + w) = f(z) for z H, so that it makes sense to define ˆf(q) on the domain 0 < q < by ˆf(e 2πiz/w ) = f(z) Then the condition of holomorhicity at is the condition that ˆf extend to a holomorhic function on q <. For general cuss c, one can always find a γ SL 2 (Z) which translates onto c. Then the condition that f be holomorhic at c is understood to be the same as the condition that f γ,k (which is attemting to be a modular form for γ Γγ) be holomorhic at. One must check that this definition does not deend on the choice of γ. We very often consider the Taylor series exansion of ˆf(q) around q = 0, which converges for all z H: f(z) = n 0 a n e 2πinz/w. Note that in the cases of Γ 0 (N) and Γ (N), the width of is w =. Write M k (Γ) for the comlex vector sace of modular forms of weight k on Γ. Write S k (Γ) for the sace of modular forms which are cus forms, which means they are 0 at every cus of Γ. It turns out that M k (Γ) is a finite-dimensional vector sace, so that S k (Γ) is a finite-dimensional Hilbert sace. There is a bilinear airing S k (Γ) S k (Γ) C known as the Petersson inner roduct: (f, g) k,γ = f(z)g(z)y k dxdy, y 2 F where F is a fundamental domain for Γ\H. (Check that the integrand really is Γ-invariant!) It is generally far easier to roduce examles of elements of M k (Γ) than it is to roduce cus forms in S k (Γ). For k even, we have the Eisenstein series E k (z) = 2ζ(k) (cz + d), k (c,d) where the sum is over all airs of integers (c, d) (0, 0). E k is very easily seen to ba modular form for SL 2 (Z) of weight k. A little maniulation shows that its Fourier exansion is E k (z) = + 2 ζ( k) σ k (n)q n, q = e 2πiz. n 5

In can be shown that the graded ring k even M k (SL 2 (Z)) is generated over C by the elements E 4 and E 6. As a result M 2 (SL 2 (Z)) is sanned by E4 3 and E6, 2 and = 728 (E3 4 E6) 2 S 2 (SL 2 (Z)) sans the (one-dimensional) sace of cus forms for SL 2 (Z). The Dirichlet series attached to a modular form f = n 0 a nq n on Γ 0 (N) is L(f, s) = a n n s n This is related to the Mellin transform of f: (2π) s Γ(s)L(f, s) = 0 (f(iy) a 0 )y s dy Now for the Eisenstein series E k we have (u to a constant) L(E k, s) = n σ k (n) n s = ζ(s)ζ(s k + ), and this function extends via analytic continuation to all comlex s, with a functional equation relating L(E k, s) to L(E k, k s). More rovocatively, we can consider the Galois reresentation ρ l : Gal(Q/Q) GL 2 (Q l ) ( ) χ k cycl σ, where χ cycl : Gal(Q/Q) Z l is the l-adic cyclotomic character, defined by the condition ζ σ = ζ χcycl(σ) whenever ζ is an l-ower root of. (We have χ cycl (Frob ) = whenever l.) Then ρ l isn t an Artin reresentation (it isn t finite-order). Still, it seems we can go ahead and define an L-function anyway. Seeing as the exression P (T ) = det ρ k ( T Frob ) always lies in Z[T ] for l, with the result not deending on l, the L-function of ρ l is ( ) ) ( k = L(E s k, s) L(ρ l, s) = P ( s ) = s (again u to a constant). Much, much less obvious is the existence (due to Deligne) of a family of l-adic Galois reresentations ρ l : Gal(Q/Q) GL 2 (Q l ) which has the same roerties with resect to cus forms such as. 6

3 Hecke s theory, art I: the case of level 3. Formalism in terms of double cosets Suose Γ and Γ are two congruence subgrous, and g GL + 2 (Q). Then the double coset sace ΓgΓ can be written as a disjoint union of finitely many left cosets: ΓgΓ = Γg i, g i GL + 2 (Q) j If f belongs to M k (Γ), then let f ΓgΓ = i f gi. (Here we are writing f gi for f gi,k, because the weight k is fixed in this discussion.) A riori this is just a function on H. Already we see, though, that f ΓgΓ only deends on the double coset ΓgΓ and not on the reresentatives g i. Let γ Γ, then according to our decomosition of ΓgΓ, for all i we have g i γ = γ i g j, for some γ i Γ and some index j. The ma i j is a ermutation of the indices. We find (f ΓgΓ ) γ = i f gi γ = j f γi g j = j f gj = f ΓgΓ Therefore f ΓgΓ belongs to M k (Γ ). The ma M k (Γ) M k (Γ ) f f ΓgΓ is a Hecke corresondence. In fact you get such an oerator for every function h on GL + 2 (Q) which is left-invariant by Γ and right-invariant by Γ, and which has finite suort on GL + 2 (Q)/Γ. Then h oerators on modular forms f via convolution: h f = h(g)f g g GL + 2 (Q)/Γ Note that f g only deends on the coset of g in GL + 2 (Q)/Γ, so that this 7

exression makes sense, and that for γ Γ, we have (h f) γ = h(g)f g γ g GL + 2 (Q)/Γ = g GL + 2 (Q)/Γ h(γg)f g = h f because h is left Γ -invariant. The oerator f f ΓgΓ is the secial case of when h is the characteristic function of Γ g Γ. Suose Γ is a third congruence subgrou. If h is a function on Γ \ GL + 2 (Q)/Γ and h 2 is a function on Γ \ GL + 2 (Q)/Γ, then we can define the convolution roduct h 2 h by (h 2 h )(x) = h 2 (g)h (g x), g GL + 2 (Q)/Γ rovided the sum is finite. Then (h 2 h ) f = h 2 (h f). If there is only one congruence subgrou Γ involved, then the C-vector sace of comlex-valued functions H(GL + 2 (Q), Γ) which are left and right Γ- invariant and which are finitely suorted on GL + 2 (Q)/Γ. Then H(GL + 2 (Q)) is an algebra under, called the Hecke algebra; the identity is the characteristic function of Γ itself. The convolution roduct h f gives a reresentation of H(GL + 2 (Q), Γ) on the comlex vector sace M k (Γ). The Hecke algebra also reserves S k (Γ). We examine the case of Γ = SL 2 (Z). For a rime (, we ) write T for the Hecke oerator corresonding to the double coset Γ Γ. That is, for f M k (Γ) we write T f for f Γdiag(,)Γ. We decomose the double coset sace as ( Γ ) ( Γ = Γ ) ( ) i Γ i=0 8

( ) Why does it work this way? An g Γ Γ translates the standard basis of Z 2 onto the basis of a sublattice L Z 2 of index. Then L deends only on the image of g in Γ\ GL + 2 (Q). Conversely, L determines g. Thus the above double coset decomosition is equivalent to classifying the sublattices L Z 2 of index. There are + of these: the one generated by (, 0) and (0, ), and the ones generated by (, 0) and (a, ) for a = 0,...,. For f M k (Γ) this works out to ( ) z + a T f(z) = k/2 f(z) + k/2 f a=0 If f is a normalized cus form with Fourier exansion f(z) = n a n q n, a = then T f(z) = k/2 n a n q n + k/2 n a n b=0 ζ bn q n/ The sum over b is zero unless n is divisible by, in which case it is : T f(z) = k/2 n a n q n + k/2 n a n q n It is convenient to rescale the Hecke oerator so that there aren t any denominators: T = k 2 T. Then T f(z) = k n a n q n + n a n q n (3..) The oerators T n for n comosite can be defined the same way, although the double coset decomosition will be more comlicated. One has the rules T m T n = T mn, gcd(m, n) = T T n = T n+ + k T n In articular all the T m commute with one another. One also checks that (T n f, g) k,γ = (f, T n g) k,γ 9

so that the T n are a collection of commuting self-adjoint oerators on a finite-dimensional Hilbert sace. Therefore they can be simultaneously diagonalized. 3.2 The L-function of a normalized eigenform Let f = a n q n be a normalized eigenform with T f = λ f. It follows immediately from our exlicit formula for the action of T on q-exansions that a = λ. (In fact the eigenvalue of T n is a n for all n.) Thus f is comletely determined by its sequence of Hecke eigenvalues. We also have the estimate a n = O(n k/2 ) (see Gelbart Cor..6). This imlies that f(z) aroaches zero very raidly as Iz ; indeed f(z) = O(e 2πy. Since T m T n = T mn for relatively rime m, n, we have a m a n = a mn. Thus the L-function of f admits an Euler factorization: L(f, s) = L (f, s) with L (f, s) = n 0 a n ns. The recursion relating the T n imlies that a a n = a n+ + k a n, and with this we can determine L (f, s): ( a ) + k L s 2s (f, s) = ( ) a ns n a a n + k a n 2 n 0 = (We set a r = 0 for non-integral r. The exression in arentheses is 0 unless n = 0.) Thus ( L (f, s) = a ) + k s 2s 0

3.3 The Ramanujan-Petersson Conjecture The cus form of lowest weight for SL 2 (Z) is (z) = q n ( q n ) 24 = n τ(n)q n. This belongs to S 2 (SL 2 (Z)), which is one-dimensional. Thus is automatically an eigenform. It follows that L(, s) = ( τ() ) +, s 2s a fact which was conjectured by Ramanujan in 96 and roved by Mordell in 920. Ramanujan also conjectured that τ() 2 /2. Petersson s generalization of this is as follows: if f = n a nq n S k (Γ 0 (N)) is a normalized eigenform, and is a rime not dividing N, then a 2 (k )/2 This inequality has a reresentation-theoretic significance which will become clear only later. For now we observe that if L (f, s) = ( α s ) ( β s ), so that a ± a 2 4 k α, β =, 2 then the Ramanujan-Petersson conjecture holds if and only if the exression under the radical is non-ositive, which is true if and only if α = β = k 2 Another reformulation is that the oles of L (f, s) all have real art equal to (k )/2, which is a kind of local Riemann hyothesis for f. It isn t a suerficial analogy: Deligne s 974 roof of the Weil conjectures, which include the Riemann hyothesis for varieties over finite fields, has the Ramanujan- Petersson conjecture for cus forms as a (difficult) corollary. Finally, we mention that the RP conjecture says that the eigenvalues of the nonnormalized Hecke oerator T are bounded by 2, a formulation which has the virtue of not deending on the weight k.

3.4 The L-function Let f be a normalized eigenform in S k (SL 2 (Z)) (so that k is even). Since a n = O(n k/2 ), the Dirichlet series L(f, s) = n a nn s only converges a riori for Rs > k/2 +. Put Λ(f, s) = (2π) s Γ(s)L(f, s) = Noting that f( /y) = y k f(y), we have Λ(f, s) = = = f(iy)y s dy y + f(iy)y s dy y + f(iy)y s dy y + 0 0 f(iy)y s dy y. f(iy)y s dy y s dy f(i/y)y y i k k s dy f(iy)y y ; this exression shows that Λ(f, s) extends to an entire function of s satisfying Λ(f, s) = i k Λ(f, k s). The analytic number theorists refer to normalize the L-function like this: ( L an (f, s) = L f, s + k ) 2 This way, the functional equation relates L an (f, s) to L an (f, s). 3.5 Hecke s converse theorem Suose n a nn s is a Dirichlet series which converges in some right halflane to a function L(s). Suose that L(s) behaves like the L-function attached to a modular form of weight k. That is, L(s) is entire and Λ(s) = (2π) s Γ(s)L(s) satisfies Λ(s) = i k Λ(k s). Hecke showed that there exists a cus form f S k (SL 2 (Z)) for which L(s) = L(f, s). The roof is simle: the function f(z) = n a nq n is easily shown to satisfy f(z + ) = f(z) (because it is a function of q = e 2πiz ), and the functional equation ( of ) L(s) shows that f( /z) = z k f(z). One then alies the fact that and ( ) generate SL 2 (Z). 2

4 Hecke s theory II: Modifications to the theory in higher level 4. Γ 0 (N) and Γ (N) The sace of modular forms S k (Γ 0 (N)) arises very often, esecially in the context of ellitic curves and modular abelian varieties. Sometimes it is just written S k (N). It won t be comletely clear why the grous Γ 0 (N) and Γ (N) deserve such status above other congruence subgrous, at least until we develo the reresentation-theoretic oint of view. In the meantime it is imortant to know the basic modifications to Hecke s theory required in higher level, esecially the henomenon of newforms. Note that Γ (N) Γ 0 (N) is a normal subgrou, with quotient isomorhic to (Z/NZ). Therefore the sace S k (Γ (N)) admits an action of Γ 0 (N)/Γ (N) = (Z/NZ), via f f γ. This action allows us to decomose S k (Γ (N)) into eigensaces S k (N, χ) indexed by the Dirichlet characters modulo N: S k (Γ (N)) = S k (N, χ) χ There are Hecke oerators T n acting on S k (N, χ) for all n, but the formula for the action of (say) T on q-exansions is different from the one aearing in Eq. (3..). The Hecke oerators also act differently with resect to the Petersson inner roduct on S k (N, χ). When does not divide N, the rule is (T f, g) = χ()(f, T g) So there still exists a basis of S k (N, χ) consisting of eigenforms for the T, N. However, the situation really is different for N. For such, T doesn t behave well at all with resect to (f, g), and it is no longer true that T can be diagonalized on S k (N, χ). As a result, it is ossible to have two distinct normalized forms with the same Hecke eigenvalues for all T n. This isn t ossible when N =! 4.2 Old and newforms The situation can be remedied by dividing the new forms from the old. First consider the case of trivial character. For each roer divisor m N, and each 3

divisor d of N/m, we have a ma S k (m) S k (N) (4.2.) f(z) f(dz) (4.2.2) The images of all such mas generate a subsace S k (N) S k (N), the old subsace. Eigenforms in S k (N) have levels which are strictly smaller than N. Let S k (N) + be the subsace of S k (N) which is comlementary to S k (N) under the Petersson inner roduct. This is the new subsace. It is immediate that S k (N) + is reserved by T, all N, and that one can find a basis for S k (N) consisting of eigenforms f for these T. Call such an f a newform of level N. It is a nontrivial task to determine how the T for N act on newforms. It turns out that for N, T f = λ f, where { ± k/2, N λ = 0, 2 N In articular, newforms are eigenforms for all T. Something very similar can be done to define newforms in S k (N, χ), but here one observes that you only get old forms for those levels m for which χ descends to a Dirichlet character modulo m. (As an extreme case, if χ is a rimitive Dirichlet character modulo N, then all eigenforms in S k (N, χ) are automatically new.) 4.3 The L-functions of newforms: the Euler factors Let f = n a nq n be a newform in S k (N, χ). For N, the -Euler factor is ( L (f, s) = a ) + k χ(). s 2s If divides N once, then L (f, s) = ( a ). s And if 2 divides N, then L (f, s) =. 4

4.4 The L-functions of newforms: the functional equation Start with Γ 0 (N). We can t exactly relicate ( the) derivation of the functional equation in the case of level, because does not normalize Γ 0 (N) ( ) for N >. But does normalize Γ N 0 (N), so we can define an oerator W N on S k (N) by W N (f) = f. Then W N 2 = ; W N is N called the Atkin-Lehner involution. It can be shown that W N commutes with the Hecke oerators T for N, so that W N takes a normalized newform f onto another newform with the same rime-to-n Hecke eigenvalues. But this must be a multile of f itself: W N f = εf, for ε = ±. One arrives at a functional equation of the form Λ(f, s) = εi k Λ(f, k s) Things are a bit more comlicated for forms with nontrivial character. If f S k (N, χ) is a normalized newform with rime-to-n eigenvalues a, then W N f has rime-to-n eigenvalues χ() a ; there exists a normalized newform g S k (N, χ ) and a constant w such that W N f = wg. Then Λ(f, s) = wi k Λ(g, k s). This constant w, which Atkin and Li call a seudo-eigenvalue, is shown to factor as a roduct of local constants, just as in Tate s theory. These constants really are quite subtle, esecially in the cases where 2 divides N. We won t be able to give a meaning to the local constants until we develo the adelic formulation of the theory of modular forms. 5