Equidistribution of Hecke points on the supersingular module

For a fixed prime p, we consider the (finite) set of supersingular elliptic curves over $\bar{\mathbb{F}}$. Hecke operators act on this set. We compute the asymptotic frequence with which a given supersingular elliptic curve visits another under this action.


Introduction
Let p be a prime number. We denote by E = {E 1 , . . . , E n } the set of isomorphism classes of supersingular elliptic curves over F p . We denote by S := ⊕ n i=1 ZE i the supersingular module in characteristic p (i.e. S is the free abelian group spanned by the elements of E). Hecke operators act on S by where C runs through the subgroup schemes of E i of rank m. This definition is extended by linearity to S and to S R := S ⊗ R. For an integer m ≥ 1 we put We have that T m E i = n j=1 B i,j (m)E j . The the matrix B i,j (m) n i,j=1 is known as the Brandt matrix of order m.
For a given D = Let M be the set of probability measures on E. For every i = 1, . . . , n, we denote by δ E i ∈ M the Dirac measure supported on E i . Let We have that Θ D is a probability measure on E and every element of M has this form. Hence, there is a natural action of the Hecke operators on M, given by T m Θ D := Θ TmD .
Each E i has a finite number of automorphisms. We define We denote by Θ := Θ e . Equation (1.1) implies that T m Θ = Θ for all m ≥ 1. Let C(E) ∼ = C n be the space of complex valued functions on E. For f ∈ C(E), we denote by f = max i |f (E i )| and For a positive integer m, we write m = p k m p with p ∤ m p . In this note, we will prove the following result: where m runs through a set of positive integers such that m p grows to infinity, is equidistributed with respect to Θ. More precisely, for all ε > 0, there exists C ε > 0 such that, for every f ∈ C(E), and for every sequence of integers m such that m p → ∞, we have that We study the asymptotic frequence of the multiplicity of E j inside T m E i . That is, we investigate the behavoir of the ratio B i,j (m)/ deg(T m ) when m varies. We will prove Theorem 1.1 in the equivalent formulation: In particular, .
The proof of this assertion is found in section 1.2.
. We will also consider the action induced on M by h * Θ D := Θ h(D) . Corollary 1.4. Let q = p be a prime number. Let h : E → E be a function such that h • T q = T q • h. Then h * Θ = Θ. In other words, h can be identified with a permutation τ ∈ S n by h(E i ) = E τ (i) and we have that w i = w τ (i) for all i = 1, . . . , n.
Proof: since T q k is a polynomial in T q , we also have that h We have that The statement Theorem 1.1, using the Hecke invariant measure Θ, has been included to emphasize the analogy with the fact that Hecke orbits are equidistributed on the modular curve SL 2 (Z)\H with respect to the hyperbolic measure, which is Hecke invariant (e.g. see [1], Section 2).
1.1. Weight 2 Eisenstein series for Γ 0 (p). The modular curve X 0 (p) has two cusps, represented by 0 and ∞. We denote by Γ ∞ (resp. Γ 0 ) the stabilizer of ∞ (resp. 0). The associated weight 2 Eisenstein series are given by The functions E ∞ and E 0 are weight 2 modular forms for Γ 0 (p) and they are Hecke eigenforms. The Fourier expansions at i∞ are ( [5], Theorem 7.2.12, p. 288) with the sequences a n and b n given by: • if p ∤ n, then a n = b n = σ 1 (n) = d|n d • if k ≥ 1, then b p k = p + 1 − p k+1 and a p k = p k • if p ∤ m and k ≥ 1, then b p k m = −b p k b m and a p k m = a p k a m . By taking an appropriate linear combination, we obtain a non cuspidal, holomorphic at i∞ modular form Since we have that this shows that f is holomorphic at Γ 0 (p)0 as well. Since dim C M 2 (Γ 0 (p)) = 1 + dim C S 2 Γ 0 (p) and since f is holomorphic, non zero and non cuspidal, we have the decomposition (1.6) M 2 (Γ 0 (p)) = S 2 (Γ 0 (p)) ⊕ Cf 0 .
Hence, to prove Theorem 1.2 we may assume p ∤ m, which is what we will do in what follows.
Our method is based on the interpretation of the multiplicities B i,j (m) as Fourier coefficients of a modular form. Theorem 1.5. For every 0 ≤ i, j ≤ n, there exists a weight 2 modular form f i,j for Γ 0 (p) such that its q-expansion at ∞ is Proof: this fact is stated in [4], p.118. It is a particular case of [3], Chapter II, Theorem 1 (D = p, H = 1, l = 0 in Eichler's notation). We remark that the theorem in loc. cit. states modularity of a theta series constructed from an order in a quaternion algebra. The fact that this theta series is the same as our f i,j is a consequence of [4], Proposition 2.3 Using (1.6), we can decompose f i,j = g i,j + c i,j f 0 , g i,j ∈ S 2 (Γ 0 (p)), c i,j ∈ C. Comparing the q-expansions, we get c i,j = 1 2w j . We have that The coefficient c m depends on (i, j), but we don't include this dependence in the notation in order to simplify it. Since p ∤ m, we have that deg(T m ) = σ 1 (m) and Hence, Using Deligne's theorem ( [2], théorème 8.2, previously Ramanujan's conjecture), we have that c m = O ε (m 1/2+ε ), concluding the proof.