p-adic L-function of an ordinary weight 2 cusp form Measures on locally compact totally disconnected spaces

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1 p-adic L-unction o an ordinary weight 2 cusp orm The ollowing exposition essentially summarizes one section o the excellent article [MS]. For any ixed cusp orm S 2 (Γ 0 (N), Q), we will associate a p-adic modular orm on weight space. The idea is to produce a measure on weight space Hom cont (Z p, Z p ) which interpolates on the characters < k > to give the special values 1 τ(χ, 1)L(, χ, 1) αn Ω χ( 1) This constructs one hal o the 2-variable p-adic analytic L-unction associated to any Hida ordinary amily. We will not discuss Hida amilies here. Measures on locally compact totally disconnected spaces Let T be a locally compact totally disconnected space, e.g. G(Q p ) (resp. G(Z p )) or any linear algebraic group over Q p (resp. Z p ). Let W be an abelian group. Deinition. A W -valued distribution is a initely additive homomorphism µ : Step(T ) W, where Step(T ) denotes the locally constant Z-valued unctions. This is equivalent to a initely additive W -valued unction on compact open subsets o T. We denote the set o such distributions Dist(T, W ). Suppose T = lim T n or some inite (discrete) spaces T n with surjective transition maps T n+1 T n. Deine the norm map N : Dist(T n+1, W ) Dist(T n, W ) : (Nµ)(t) = t t µ(t ). We can express any distribution µ Dist(T, W ) as an inverse limit o distributions µ n : T n W satisying the norm compatibility condition. Remarks. (Nµ n+1 )(t) = µ n (t). It may help to think o the above compatibility condition in the context o the example T = Z p, T n = Z/p n Z, W = R and µ the distribution corresponding to Haar measure on Z p. 1

2 Note how reminiscent the norm map N is o Hecke operators, which are other instances o summing over the iber. More on that later. Distributions become more interesting when we endow W with an interesting topology and demand that our distributions be continuous. Namely, suppose that W is a p-adic Banach space; most useul to us will actually be the case where W is a inite dimensional vector space over a inite extension o Q p. Deinition. A W -valued measure on T is a continuous linear map rom the space o W -valued continuous unctions C 0 (T, W ) to W. By the above deinition, we see that i µ is a measure, then or any compact open U T, µ(1 U ) µ <. On the other hand, i µ Dist(T, W ) is any distribution or which µ(1 U ) is bounded or every compact open U, an approximating an arbitrary continuous W -valued unction on T by a step unction will show that µ extends uniquely to a measure. Example. The Dirac measures δ t () = (t) are indeed W -valued measures. Measures on groups Suppose G is a topological group which can be expressed as an inverse limit o quotients by inite index, normal subgroups G = lim G/G n where G 1 G 2 G 3... and G n = 0, which ensures that G is separated. There are canonical isomorphisms Dist(G/G n, O K ) i O K [G/G n ] µ g G/G n µ(g)g which are compatible with the earlier norm maps, i.e. the diagram Dist(G/G n+1, O K ) O K [G/G n+1 ] N Dist(G/G n, O K ) i i O K [G/G n ] commutes. The (O K -valued) Iwasawa algebra o G is the inverse limit Λ = O K [[G]] := lim O K [G/G n ]. It is a compact topological O K -algebra and the above maps i deine an isomorphism Dist(G, O K ) i Λ 2

3 where by deinition, we let µ α be the preimage o an element α o the Iwasawa algebra. For example, it is easy to check that or g G, the Dirac measure δ g = µ g, i.e. it corresponds to the element g O K [[G]]. The topology on the Iwasawa algebra is such that Lemma. The evaluation map is continuous. C(G, K) Λ K (, α) µ α () Continuity o the above evaluation map implies that any continuous homomorphism χ : G O K extends to a continuous homomorphism ρ : Λ O K o the Iwasawa algebra, by the ormula ρ(α) = µ α (χ). The Iwasawa algebra on groups essentially = Z n p We will deine p-adic L-unctions on weight space = Hom cont (Z p, Z p ). This group equals (Z/pZ) (1 + pz p ), and 1 + pz p is isomorphic to Z p by the logarithm map. This provides weight space with a linear p-adic structure. In general, Deinition. A linear p-adic structure on a compact group G is a decomposition G = C H together with an isomorphism o topological groups Z n p γ H. This linear structure gives an explicit coordinatization o the Iwasawa algebra o G. We let h i = γ(e i ) where e i is the tuple with a 1 in the i th coordinate and zeros in all other coordinates. Lemma 1. The linear structure γ provides a unique isomorphism Λ G O K [C] OK O K [[T 1,..., T n ]] h i T i + 1. Thus, we appear to be identiying the Iwasawa algebra with an algebra o analytic unctions. Though we won t deine them here, elements o the Iwasawa algebra are rigid analytic unctions on the unit polydisk (or a inite union o polydiscs). Let X = Hom cont (G, O K ) be weight space. Let F (X, O K) denote the O K -algebra o (rigidt) analytic unctions on X. 3

4 Proposition 1. The map is injective with image contained in F (X, O Cp ). Λ G C 0 (X, O Cp ) α ( µ α = χ µ α (χ)) Remark. The above map α µ α is a kind o Fourier transorm, associating a unction/measure on an abelian topological group to a unction on the dual group. Proo. This amounts to identiying the group algebra O Cp [[Z n p]] with O Cp [[T 1,..., T n ]]. The injectivity statement ollows rom a p-adic approximation theorem, the key point being the ollowing: Fact: Any unction on G/G n O K can be written as an O K -linear combination o characters on G/G n. From this, it is easy to show that we can uniormly approximate any unction by (locally constant) characters. So i α Λ G maps to 0, then µ α () = 0 or any C 0 (G, O K ), i.e.µ α = 0. But since the association α µ α is an isomorphism, α must be zero. Call the image o the homomorphism rom this proposition, a subalgebra o the analytic unctions F (X, O K ), the Iwasawa algebra. Serre s congruences Not every unction in F (X, O K ) lies in the image o the Iwasawa algebra. For example, take the case o X = Hom cont (Z p, O K ). Serre wrote down congruence conditions which completely characterize the image o the Iwasawa algebra inside F (X, Z p ) or the case X = Hom cont (1 + pzp, Z p ). We proceed to describe these congruences below. Let be a Z p -valued unction on Hom cont (1 + pz p, Z p ). Via the exponential map, we may view it as an analytic unction on Z p. We can express uniquely as (s) = ( ) s δ n. n n Let c in be the coeicient o Y i in Y (Y 1)...(Y n + 1). Then lies in the image o the Iwasawa algebra i the ollowing congruences are satisied: δ n 0 (mod p n ) or all n. val p ( n i=0 c inδ i p i ) val p (n!). Ignoring the speciics, we should expect some result o this type in analogy with real analysis. There, Paley-Winer type theorems relate the decay o Fourier transorm o a distribution to smoothness o the distribution itsel. The above congruence conditions pin down the necessary decay on the Fourier transorm in order or the distribution µ to be a measure. 4

5 Computation o special L-values Let (q) = a n q n be a weight 2 cusp orm or some congruence subgroup γ. Let χ be a Dirichlet character o conductor N. We have the ollowing lemma rom inite group theory. Lemma 1. For any character χ o (Z/mZ), we have the realtion χ(a)e 2πian/m = χ(n)τ(χ, 1). Proo. Let V be the space o unctions G = (Z/mZ) C. Consider the unction 0 (x) = e 2πix/m V. Then a (a)e 2πian/m = 0 (n). a Any convolution operator on a inite abelian group is semisimple. This right convolution operator commutes with let translations, and so its eigenspaces are representations o G. Since all representations o G are a sum o characters, it ollows that the eigenunctions o 0 are given by characters and so each character is an eigenunction. But then χ 0 (n) = c χ χ(n) or some constant c χ. Plugging in n = 1 gives c χ = χ 0 (1) = τ(χ, 1). Now let χ (q) = a n χ(n)q n be the twist o any cusp orm. Then by the above lemma, χ (z) = 1 τ(χ, 1) ollows easily. We can express L(, s) as the Mellin transorm a L(, s) = (2π)s Γ(s) mod m 0 χ(a)(z + a/m) (it)t s dt t. Thereore, or weight 2 cusp orms, the above computation shows that L(, χ, 1) = 1 τ(χ, 1) a (Z/mZ) χ(a) 2π a/m (it)dt. Thus, remembering that the modular symbols I ± were deined as periods o between ixed cusps and normalized by ixed (transcendental) periods Ω ± (see Brandon s notes), we get that τ(χ, 1)L(, χ, 1)/Ω χ( 1) = χ(a)i χ( 1) (a/m, ). a mod m Now let m = p n. I we squint our eyes, the above ormula looks like taking a Riemann sum o χ with respect to some measure I ± on Z p, namely the measure µ ± ({x Z p : x = a mod p n }) = I ± (a/pn, ). 5

6 This almost works. For µ ± through the equation or equivalently µ ± ({x Z p : x = a mod p n }) = to be a distribution, it must be initely additive. We can express that I ± (a/pn, ) = b (Z/p n+1 Z) µ ± ({x Z p : x = b mod p n+1 }) b (Z/p n+1 Z) I ± (b/pn+1, ). ( ) We ve used quotation marks above because both equalities are alse, though only mildly so. [PS, p.18] explains this lucidly, and we consolidate their explanation here or convenience. The right side o ( ) looks an awul lot like the action o the T p Hecke operator on the modular symbol I ± (a/pn, ), and indeed, the last equation above almost says that I ± (a/pn, ) is an eigensymbol o T p. But there are two problems with this. (1) The right side is not literally the action o T p on I ± (a/pn, ). We irst specialize the above discussion to the case where is an ordinary eigenorm. This means that T p I ± = a pi ± with a p a p-adic unit. Then a p I ± ( a p, ) = T pi ± n ( a p, ) ( ( n = I ± p 0 )) ( a p n, ) + = I ± ( a, ) + I ± pn 1 + jpn (a, ). p n+1 ( ( I ± 1 j 0 p )) ( a p n, ) The irst summand in this last expression appears to get in the way o the distribution property or µ ±. But it wouldn t be there i had been a unction o the U p operator, which ( ) p 0 is a sum over all o the above cosets except or. Unortunately, when we view as an old orm or Γ 0 (Np), it is not a U p -eigenunction. But span{(z), (pz)} is invariant under the U p operator, and it happens to have characteristic polynomial the Hecke polynomial x 2 a p x + p. Since a p is a p-adic unit, one o the roots α is a p-adic unit as well, and the second β is not. By an explicit computation, we ind that α (z) = (z) β(pz), β (z) = (z) α(pz) 6

7 are eigenvectors o U p with respective eigenvalues α and β. It thus ollows that αi ± α ( a p n, ) = I ± α ( a + jpn, ). p n+1 (2) The above equation would be the additivity we need, i not or the annoying α. We deal with this by setting µ ± ({x Z p : x = a mod p n }) = 1 α n I± α ( a p n, ). Finally, this satisies the additivity property we need. It is clear that integrating Dirichlet characters o p-power conductor against this modiied character µ χ( 1) will interpolate something related to special values o L(, χ, 1)/Ω χ( 1). We just need to igure out what. We can compute that I ± α = I ± 1 α I± ( p 0 This implies that or χ a primitive Dirichlet character modulo p n, that Z p χdµ χ( 1) = 1 α n = 1 α n = 1 α n a Z/p n Z χ(a)i χ( 1) a Z/p n Z χ(a)i χ( 1) a Z/p n Z χ(a)i χ( 1) But since χ is primitive, the inner sum ( a p n, ) 1 α n ). a Z/p n Z 1 ( a p n, ) 1 α n ( a p n, ) ( α χ(a)iχ( 1) p 0 1 α χ(a)iχ( 1) ( a Z/p n Z b (Z/p n 1 Z) a b mod p n 1 χ(a) = 0 ) ( a p n, ) a, ) pn 1 χ(a) 1 α Iχ( 1) a b mod p n 1 ( b, ). pn 1 or each b (Z/p n 1 Z). Thus, the inal line in the above equation simpliies tremendously to give 1 α n χ(a)i χ( 1) a Z/p n Z ( a p, ) = 1 τ(χ, 1)L(, χ, n 1)/Ωχ( 1) αn by our earlier comptuations using Birch s Lemma. Note the importance o having α be a p-adic unit. In order or dµ ± to be a measure, it is necessary and suicient or its values to be p-adically bounded as we integrate inite order characters. But this is the case because α is a p-adic unit, and the Z-module o (period-normalized) modular symbols is initely generated. Thus, the p-adic absolute value o any modular symbol is no greater than the maximum over a inite generating set. 7

8 Appendix: Hecke Operators Later on, we will encounter various expressions that look like Hecke actions on X 0 (N), which can be somewhat tricky when p divides N. The goal o this section is to understand this properly, especially the distinction between T p and U p. Suppose that p does not divide N. For any positive integer M, X 0 (M) is a coarse moduli space or the unctor S (E/S, C/S)/ = o isomorphism classes generalized elliptic curves together with a intie lat cyclic subgroup o order M E sm. Remark. There are subtleties with the notion o cyclicity. But we ll really only need Hecke operators in the case o S = C, where cyclic subgroups are just vanilla cyclic groups. Since C is cyclic o order Np, it has a unique subgroup, C 1, and a unique quotient C/C 2 which are order N. Then we can deine the Hecke correspondence T p through two degeneracy maps The Hecke operator T p is deined to be X 0 (N) π 1 X 0 (Np) π 2 X 0 (N) (E, C 1 ) π 1 (E, C) π 2 (E/C 2, C/C 2 ). T p = (π 2 ) π 1. When p divides the level N, we rename this Hecke operator to be U p. Note that T p always has degree p + 1, whereas U p has degree p. This comes down to some group theory. Suppose that (N, p) = 1. Then i (E, C) is an elliptic curve with a cyclic subgroup o order N, how many elements lie in its preimage by π 1? We must ind all subgroups o (Z/NpZ) 2 = (Z/NZ) 2 (Z/pZ) 2 which contain a ixed cyclic subgroup o order N. The cyclic subgroup must be contained entirely within the (Z/NZ) 2 actor, and we need to count the number o cyclic subgroups o (Z/pZ) 2 o order p. This is exactly p + 1 (the number o lines in a 2 dimensional F p vector space). Suppose that N = p r M or some r > 0, (M, p) = 1. Then we must ind all cyclic subgroups o order Np = p r+1 M contained in (Z/NpZ) 2 = (Z/MZ) 2 (Z/p r+1 Z) 2 which contain a ixed cyclic group o order Mp r. Equivalently, we must ind the number o cyclic subgroups D o order p r+1 contained in the cyclic subgroup (Z/p r Z) 2 which contain a ixed cyclic subgroup C o order p r. GL 2 (Z/p r+1 Z) acts transitively on the set o such lags C D. For the standard lag, C D consisting o appropriate multiples o the irst standard basis vector, the elements o GL 2 (Z/p r+1 Z) which stabilize C are ( ) unit ap r unit 8

9 and the elements which stabilize D are ( ) unit 0 unit The ratio Stab(C)/Stab(D) = p, and so U p has degree p. Explicitly, with respect to the isomorphism we see that and Γ 0 (N)\H X 0 (N)(C) : Γ 0 (N)z ( z, 1, 1/N ) T p (z) = ( p 0 U p (z) = ) z + ( 1 j 0 p ( 1 j 0 p The Hecke action on a weight 2 cusp orm (z) is then computed through its induced pullback ollowed by pushorward action on the dierential (z)dz: (( ) ) (( ) ) p 0 p 0 T p ((z)dz) = z d z + (( ) ) (( ) ) 1 j 1 j z d z 0 p 0 p and similarly or U p (just excluding the irst summand rom the above expression). The action at the level o the q-expansion o is thereby readily computed: ) z. U p ( a n q n ) = a np q n ) z T p ( a n q n ) = a np q n + p a n q pn. Earlier in our exposition o p-stabilization, we made use o the ollowing: Lemma 1. Suppose that is an eigenorm o T p acting on S 2 (Γ 0 (N)) with (N, p) = 1. Then span{(z), (pz)} is preserved by the U p operator acting on S 2 (Γ 0 (Np)). Proo. Note that or S 2 (Γ 0 (N)), we can express ( ) p 0 T p (z) = U p (z) + = U p (z) + (pz). But we can directly express the U p action on the q-expansion o. I (z) = a n q n, then (pz) = a n q pn and So i T p = a p, then U p ((pz)) = U p ( a n q pn ) = a n q n = (z). so the span is clearly stable. U p (z) = a p (z) (pz), U p (pz) = (z). 9

10 I α and β are the roots o the above characteristic polynomial x 2 a p x + p, then α = (z) β(pz), β = (z) α(pz) are eigenvectors o U p with respective eigenvalues α and β. I is an ordinary eigenorm or T p, then a p is a p-adic unit and so one root α is a p-adic unit too. We call α the ordinary p-stabilization o (and β the non-ordinary p-stabilization o ). Reerences Mazur, Swinnerton-Dyer. The Arithmetic o Weil Curves. Inventiones (1974). Pollack, Stevens. Overcovergent Modular Symbols. Arizona Winter School Notes (2011). 10