Congruence formulae for Legendre modular polynomials

Let $p\geq 5$ be a prime number. We generalize the results of E. de Shalit about supersingular $j$-invariants in characteristic $p$. We consider supersingular elliptic curves with a basis of $2$-torsion over $\overline{\mathbf{F}}_p$, or equivalently supersingular Legendre $\lambda$-invariants. Let $F_p(X,Y) \in \mathbf{Z}[X,Y]$ be the $p$-th modular polynomial for $\lambda$-invariants. A simple generalization of Kronecker's classical congruence shows that $R(X):=\frac{F_p(X,X^{p})}{p}$ is in $\mathbf{Z}[X]$. We give a formula for $R(\lambda)$ if $\lambda$ is a supersingular. This formula is related to the Manin--Drinfeld pairing used in the $p$-adic uniformization of the modular curve $X(\Gamma_0(p)\cap \Gamma(2))$. This pairing was computed explicitly modulo principal units in a previous work of both authors. Furthermore, if $\lambda$ is supersingular and lives in $\mathbf{F}_p$, then we also express $R(\lambda)$ in terms of a CM lift (which are showed to exist) of the Legendre elliptic curve associated to $\lambda$.


Introduction
Let p ≥ 5 be a prime number. We are interested in this article in the modular curve X(Γ(2) ∩ Γ 0 (p)). A plane equation of this curve is given by the classical p-th modular polynomialà la Legendre, which we denote by F p (X, Y ). It is shown that it satisfies the same properties as the classical modular polynomials for the j-invariants, namely it is symmetric, has integer coefficients and we have the Kronecker congruence F p (X, Y ) ≡ (X p −Y )(X −Y p ) modulo p.
This last congruence can be roughly interpreted by saying that the reduction modulo p of X(Γ(2) ∩ Γ 0 (2)) is a union of two irreducible components isomorphic to P 1 . In this work, we show a congruence formula for F p (X, X) modulo p 2 , which intuitively gives us information about the reduction of our curve modulo p 2 . The tools that we use to study this reduction is the p-adic uniformization (due to Mumford and Manin-Drinfeld). This was already used by E. de Shalit in his paper [13], and we follow his method in our case. We do a deep study of some annuli in the supersingular residue disks of the rigid modular curve.
Let ∆ be the dual graph of the special fiber of X. Mumford's construction shows that Γ is isomorphic to the fundamental group π 1 (∆). The abelianization of Γ is isomorphic to to the augmentation subgroup of the free Z-module with basis the isomorphism classes of supersingular elliptic curves over F p . Let S := {e i } be the set of supersingular points of X k . We proved in [1], using the ideas of [12], that the pairing Φ can be expressed, modulo the principal units, in terms of the modular invariant λ as follow.
where the sign ± is + except possibly if p ≡ 3 (modulo 4) and λ(e i ) ∈ F p .

Remark 1.
i) We have also proved an analogue of the above result when N = 3 and p ≡ 1 (mod 3), for a suitable model X of the modular curve of level Γ 0 (p) ∩ Γ(3) over Z p .
ii) The above formula was first conjectured by Oesterlé using the modular invariant j instead of the modular invariant λ for the modular curve X 0 (p) instead of X, and E. de Shalit proved this conjecture in [12] (up to a sign if p ≡ 3 (mod 4)).
We recall that the Lambda modular invariant λ : M Γ(2) ⊗ Q → P 1 Q is an isomorphism of curves. Let F p (X, Y ) ∈ C[X, Y ] be the unique polynomial such that for all τ in the complex upper-half plane, we have: Note that this polynomial has much smaller coefficients than the corresponding polynomial for the j-invariants. For example, we have: while the corresponding polynomials for j-invariant are enormous.
This agrees with the philosophical principle that adding a Γ(2) structure simplifies a lot the computations. Another instance of this principle was applied in a paper of the second author about the Eisenstein ideal and the supersingular module. Also, in the case of λ-invariants, there are no complications due to the elliptic points, so the formula are smoother and there is no conjectural sign as in the j case of E. de Shalit. This principle is one of the motivation we had to generalize E. de Shalit's results to our case.
In this article, we prove that the affine scheme Spec Q[X, Y ]/(F p (X, Y )) is a plane model of M Γ 0 (p)∩Γ(2) over Q (i.e both curves are birational), and that the polynomial F p (X, Y ) satisfies the same basic properties as Kronecker's p-th modular polynomial for the modular curve X 0 (p). We derive another formula for the diagonal values ofΦ, related to the polynomial F p as follow.
and F p gives a plane model the modular curve Our approach is based on the technics of p-adic uniformization of [12] and [13], on a deep analysis of the supersingular annuli in X an and on the action of the Atkin-Lehner involution w p : X ≃ X. The key point is to relate the diagonal elements of the extended period matrixΦ to the polynomial F p (X, Y ).
, andR(X) ∈ F p be the reduction of R(X) mod p. Let λ(e i ) ∈ F p 2 be the λ-invariant of a supersingular elliptic curve e i . Then, we have:R (λ(e i )) = ±(−1) where the ± sign is + except possibly if p ≡ 3 (modulo 4) and λ(e i ) ∈ F p . On the other hand, if λ ∈ F p 2 is not a supersingular invariant, thenR(λ) = 0.
Proof. The first assertion follows by comparing Theorem [1,1] and the Theorem above. Let λ ∈ F p 2 which is not supersingular. LetX be the scheme over O K defined by But it is clear from the Kronecker's congruence that F X (β, β p ) and F X (β, β p ) are divisible by p. Thus, our regularity conditions is equivalent to the fact thatR(λ) = 0. Corollary 2.3 shows that (λ, λ p ) is a singular point of the special fiber ofX. But E is ordinary, so the corresponding point on X is smooth. Since, the minimal regular model of the normalization ofX is unique, it is X. Since any local regular ring is normal (since it is factorial), the point x is not regular inX andR(λ(E)) = 0.

Notation
(i) For any algebraic extension k of the field Z/pZ, we denote byk the separable closure of k. (ii) For any congruence subgroup Γ of SL 2 (Z), we denote by M Γ the stack over Z whose S-points classify generalized elliptic curves over S with a Γ-level structure. (iii) For any congruence subgroup Γ of SL 2 (Z) and any c ∈ P 1 (Q), we denote by [c] Γ the cusp of Γ\(H ∪ P 1 (Q)) corresponding to the class of c, where H is the (complex) upper-half plane. (iv) For two congruence subgroups Γ and L, we denote by m Γ∩L for the fiber product of algebraic stacks M Γ × M M L , where M is the stack over Z whose S-points classify generalized elliptic curves over the scheme S. (vi) For any proper and flat scheme X over O K , we denote by X an K the rigid analytic space given by the generic fiber of the completion of X along its special fiber (i.e. X K (K) ≃ X an K (K)). (vii) Let | . | p be a p-adic valuation onQ p , then we shall denote by x ∼ y if and only if | xy −1 − 1 | p < 1.
Acknowledgements. The first author has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 682152). The first named author (A.B.) would like to thank the Université Paris 7, where most of our discussions took place, for its hospitality. The second named author (E.L.) has received funding from Université Paris 7 for his Phd thesis and would like to thank this institution. By the universal property of the coarse moduli space attached to an algebraic stack over a noetherian scheme, we have a coarse moduli map such that for any field L of characteristic different from two, g induces a bijection For any elliptic curve E with a basis of its 2-torsion over a field L of characteristic different from 2, E is isomorphic to a unique Legendre curve E λ : with basis of 2-torsion the points (0, 0) and (0, 1). Hence, we have a bijection , associating to an elliptic curve E its lambda invariant λ.

Proof.
We use similar arguments to those given in the proof of [2, VI. Theorem 1.1]. Let c = [1] Γ(2) be the cusp such that the complex modular invariant λ has a pole. Since a cusp of M Γ(2) is given by a section Spec Z[1/2] → M Γ(2) , then by composing with the coarse moduli map g, a cusp of M Γ(2) is also given by a section Spec giving rise to a Cartier divisor D of M Γ(2) . By proposition [2, V. 5.5], the geometric fibers of M Γ(2) are absolutely irreducible. The genus is constant on the geometric fibers of M Γ(2) and equals the genus of the complex modular curves M Γ(2) (C), which is zero (see proposition [3, 7.9]). Hence, by applying Riemann-Roch to each geometric fiber of M Γ(2) and using the base change compatibility of cohomology on geometric fibers (see [8, III. Corollary 9.4]), we obtain: (which we normalize so that it coincides with the Legendre lambda-invariant on L-points as above). On each geometric fiber M Γ(2) ⊗k away from characteristic 2, we see that the degree of the divisor Dk corresponding to D on M Γ(2) ⊗k is 1, hence Dk is very ample (see [8, IV. corollary 3.2]). Thus, D is relatively very ample over Z[1/2] (see [4, 9.6.5]) and λ is an isomorphism. (2) be the inverse image of M ′ Γ(2) by c, which is an affine scheme since c is a finite morphism. Denote by w p the Atkin-Lehner involution on . Thus, we obtain finite maps Since the degree of the two projections c, c • w p : (2) is injective outside CM-points and the special fiber at p, the degree of F as a polynomial in X equals to the degree of F as a polynomial in Y , equals to p + 1. Thus, we have F (X, We must have α = 1 since else, we have F (X, X) = 0. This is impossible since this implies that every elliptic curve over C has CM by some quadratic order. Thus we can assume that We have thus proved the first part of the following result. (2) ) be the field of rational functions of M Γ 0 (p)∩Γ (2) and and by comparing degrees, we have If E is an elliptic curve over F p and E → E (p) be the Frobenius, then E is ordinary if and only if the kernel of Frobenius is isomorphic to the finite flat group scheme µ p . The Atkin-Lehner involution w p sends the multiplicative component of the special fiber of M Γ 0 (p)∩Γ (2) to theétale component via λ → λ p .
Proof. Let x be an element of X(k) corresponding to (E, α 2 , H) such that E is not supersingular. We have two cases: If H is a multiplicative subgroup of order p, then from the discussion above, it is clear that λ(x) p = λ(w p (x)).
Otherwise, H isétale and λ(x) = λ(w p (w p (x))) = λ(w p (x)) p . Moreover, since the open given by the complementary of supersingular elliptic curves is dense in the special fiber of X, the zeros of the polynomial ( Y )) is the scheme theoretic image of (c, c • w p )). Thus, in this ring we have (X p − Y )(Y p − X) = 0 and by comparing the degree, we have the equality.
Remark 2. This corollary could be proved in a more down-to-earth way, like in [19, Chapter 5 §2].

p-adic uniformization and the reduction map
Let X be the modular curve M Γ(p)∩Γ(2) ⊗O K . Since the singularities of the special fiber of X are k-points, Mumford's Theorem [11] shows the existence of a free discrete subgroup Γ ⊂ PGL 2 (K) (i.e. a Schottky group) and of a Gal(K/K)-equivariant morphism of rigid spaces: τ : H Γ → X an K inducing an isomorphism X an K ≃ H Γ /Γ, where H Γ = P 1 K − L and L is the set of limit points of Γ. Note that H Γ is an admissible open of the rigid projective line P 1 K . Let T Γ be the subtree of the Bruhat-Tits tree for PGL 2 (K) generated by the axes whose ends correspond to the limit points of Γ. Mumford constructed in [11] a continuous map ρ : H Γ → T Γ called the reduction map.
The special fiber of X has two components, and each component has 3 cusps. One of these components, which we call theétale component, classifies elliptic curves or 2p-sided Néron polygons overk with anétale subgroup of order p and a basis of the 2-torsion. The other component, which we call the multiplicative component, classifies elliptic curves or 2-sided Néron polygons overk with a multiplicative subgroup of order p and a basis of the 2-torsion. The involution w p sends a 2p-gon to a 2-gon. Let c and c ′ = w p (c) be two cusps of M Γ 0 (p)∩Γ(2) (C) such that c is above c Γ(2) and we know also by proposition [1, 7.4] that c ′ is also above c Γ(2) (ξ c corresponds to a 2p-gon and ξ c ′ = w p (ξ c ) corresponds to a 2-gon).
The dual graph ∆ of the special fiber of X has two vertices v c ′ and v c indexed respectively by the cusps ξ c ′ and ξ c . There are g + 1 edges e i (i ∈ {0, ..., g}) corresponding to supersingular elliptic curves with a Γ(2)-structure. We orient these edges so that they point out of v c ′ .
The Atkin-Lehner involution w p exchanges the two vertices v c ′ and v c and also acts on edges (reversing the orientation). More precisely, if E i is a supersingular elliptic curve corresponding to e i , then w p (e i ) = e j where e j is the elliptic curve associated to E (p) i = w p (E i ) (here w p is the Frobenius). Thanks to Lemma 3.1 below, one can identify the generators Letṽ c andṽ c ′ be two neighbour vertices of T Γ reducing to v c ′ and v c respectively, such that the edge linkingṽ c toṽ c ′ reduces to e 0 modulo Γ. For 0 ≤ i ≤ g, letẽ ′ i be an edge pointing out ofṽ c ′ and reducing to e i modulo Γ. Letẽ i be oriented edges of T Γ lifting e i and pointing toṽ c . Note thatẽ 0 =ẽ ′ 0 . (i) B i is the open residue disk in the closed unit disk of P 1 K which reduces to λ(e i ) p , The annulus c i is isomorphic, as a rigid analytic space, to {z, |p| < |z| < 1}.
The results of Mumford [11] imply that we can identify Γ ab with Z[S] 0 . and that T Γ is the universal covering of the graph ∆. Moreover, Manin and Drinfeld proved that v K • Φ is positive definite (v K is the p-adic valuation of K). According to lemma [1, 4.2], the pairing Φ takes values in Q × p . We recall that we defined in [1] an extension Φ : Z[S] × Z[S] → K × as follow: For all 0 ≤ i ≤ g, we had chosen ξ (i) c (resp. ξ (i) c ′ ) in H Γ which reduces modulo Γ to the cusp ξ c ⊗ Q p (resp. ξ c ′ ⊗ Q p ), and such that ξ Thus, we had chosen ξ We can assume also without losing in generality that z 0 = ∞.
Thus, for any a ∈ H Γ , we had defined an extension of Φ to a pairing on Z[S] 0 × Z[S] (and taking values in K) as follow : for all α ∈ Γ, The Atkin-Lehner involution acts on Γ\T Γ and lifts to an orientation reversing involution w p of T Γ (by the universal covering property). By [7] ch. VII Sect. 1, there is a unique class in N(Γ)/Γ (where N(Γ) is the normalizer of Γ in PGL 2 (K)) inducing w p on T Γ . We denote by w p the induced map of H Γ (it is only unique modulo Γ). Fix c ′ such that τ (z) = w p (τ (z ′ )). Recall that by hypothesis, ξ [1] as follow: where z and z ′ approach ξ c ′ respectively. Since at z = z 0 , λ ′ • τ has a simple pole and the numerator and denominator have a simple zero and simple pole respectively, Φ(e i , e j ) is finite, and is in K × since K is complete (we choose z and z ′ in K to compute the limit).

4.1.
Case where λ(e 0 ) ∈ F p . Assume that λ(e 0 ) ∈ F p . We can choose a lift of the involution w p tow p of N(Γ) ⊂ PGL 2 (K) preserving the edge e 0 and reversing the orientation of this edge. Thus, w p preserves the annulusẽ 0 ′ , so sends A to A ′ . Hence,w p is an involution (we havew 2 p ∈ Γ and the stabilizer of an edge in Γ is trivial, sow 2 p = 1).
for any lift β 0 of λ(e i ) in K.
The proof is really the same as [13, 3.1-3.3], using our analogous fundamental domain for Γ, and replacing j by λ ′ . Thus, we shall be really sketchy and refer the reader to de Shalit's paper for details.
We recall that have ord p (π) = 1. By slight abuse of notation we shall denote λ for λ • c and λ ′ for λ • c • w p .
Let y = z − ζ; it identifies the annulus a = ρ −1 (ẽ 0 ) with Consider the map Ψ : a → B 0 defined by This is a covering of B 0 by a since a is the intersection of our fundamental domain D with B 0 (although it might seems surprising compared to the classical complex situation, such a covering indeed exists).
For t close enough to 1 and y ∈ a(t, 1), we set We have, by definition: This gives us, using partial derivatives and Corollary 2.3: We work modulo the ideal I(t) generated by rigid analytic functions on a(t, 1) which are strictly smaller than | p | p in absolute value. The term −pR(λ) − p · h(u, λ) is congruent to −pR(λ) modulo I(t). A simple computation using Lemma 4.2 shows that we must have u ≡ π/y modulo I(t) and This shows what we needed to conclude the proof of point (ii) of Theorem 1.1: (π/y) · y ∼ pR(λ) .

Existence of CM lifts.
In this section, we prove part (iii) of Theorem 1.1 in the case λ(e 0 ) ∈ F p . ] and F be the fraction field of O. Let a an element of the ideal class group of O. We denote by j(a) the j-invariant of the isomorphism class of the elliptic curve C/a. It is classical (cf. for instance [16] Theorem 5.6) that if λ is any λ-invariant above j(a), then F (λ) is an extension of F (j) contained in the ray class field of F of conductor 2 · O. Proof. If p ≡ −1 (modulo 8), then 2 splits in F , so √ −p − 1 ∈ p 2 if p 2 is any prime ideal of O above (2). Thus, by class field theory, since ( √ −p) is principal, it splits in the ray class field of conductor 2 · O and we are done. If p ≡ 3 (modulo 8), then 2 is inert in F . The prime ideal above 2 in O is P 2 = (2, Thus we have √ −p − 1 ∈ P 2 . As above, class field theory shows that ( √ −p) splits in the ray class field of F of conductor P 2 , which concludes the proof of the lemma.
Lemma 4.5. Let λ ∈ Z p such that the Legendre curve E λ : y 2 = x(x − 1)(x − λ) has supersingular reduction. Then λ is a root of F p (X, X) if and only if E λ has CM by ]. Furthermore in this case λ is a simple root of F p (X, X).
Proof. It is clear that E λ has CM by a quadratic order O such that p either splits or ramifies in the fraction field. But p has to ramify since the reduction of E λ is supersingular (this comes from the standard description of the local galois representation attached to a supersingular elliptic curve). Furthermore, there is an endomorphism of E whose square is −p.  Thus, any supersingular λ-invariant in F p is a double root of F p (X, X). Using the previous lemma, we get: We now finish the proof of point (iii) of 1.1. This is done in a similar way as [13] p. 146. Let λ 1 and λ 2 be the two CM values of lambda invariants in Q p ( √ −p) which lift λ(e i ) (which exist by Proposition 4.3) . It is clear that λ 1 and λ 2 are not in Q p , so they must be conjugate. Write λ 1 = a + b √ −p and λ 2 = a − b √ −p for some a, b ∈ Z p . By point (ii) of 1.1, it suffices to prove: We know that F p (λ 1 , λ 1 ) = 0. Therefore, we have: 0 = F p (λ 1 , λ 1 ) = F p (λ 1 , λ p 1 +(λ 1 −λ p 1 )) ≡ F p (λ 1 , λ p 1 )+(λ 1 −λ p 1 )·(λ p 2 1 −λ 1 ) (modulo p √ −p) where the last congruence follows from Corollary 2.3 (which gives ∂ Y F p (X, Y ) ≡ −(X − Y p ) (modulo p)). Since λ p 2 1 ≡ λ p 1 ≡ a (modulo p) and λ 1 ≡ λ p 1 (modulo √ −p), we get: which concludes the proof of Theorem 1.1 if λ(e 0 ) ∈ F p .
We refer as before to [13, for details in the j-invariant case.