On the $p$-adic periods of the modular curve $X(\Gamma _0(p) \cap \Gamma (2))$

Betina, Adel; Lecouturier, Emmanuel

doi:10.1090/tran/7236

Cited by 4 publications

(10 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Proof. The first assertion follows by comparing Theorem [1,1] and the Theorem above. Let λ ∈ F p 2 which is not supersingular.…”

Section: Introductionmentioning

confidence: 89%

“…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…”

Section: P-adic Uniformization and The Reduction Mapmentioning

confidence: 99%

“…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 .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Congruence formulas for Legendre modular polynomials

Betina

Lecouturier

2018

Journal of Number Theory

Self Cite

View full text Add to dashboard Cite

Let p ≥ 5 be a prime number. We generalize the results of E. de Shalit [13] about supersingular j-invariants in characteristic p.We consider supersingular elliptic curves with a basis of 2-torsion over F p , or equivalently supersingular Legendre λ-invariants. Let F p (X, Y ) ∈ Z[X, Y ] be the p-th modular polynomial for λ-invariants. A simple generalization of Kronecker's classical congruence shows that R(X) :=We give a formula for R(λ) if λ is a supersingular. This formula is related to the Manin-Drinfeld pairing used in the p-adic uniformization of the modular curve X(Γ 0 (p) ∩ Γ(2)). This pairing was computed explicitly modulo principal units in a previous work of both authors. Furthermore, if λ is supersingular and lives in F p , then we also express R(λ) in terms of a CM lift (which are showed to exist) of the Legendre elliptic curve associated to λ.1 2 ADEL BETINA AND EMMANUEL LECOUTURIER By combining our previous work (cf.[1]) on the p-adic uniformization of our modular curve and the present results, we obtain an elementary formula for values taken by F p (X, X) modulo p 2 (which does not however give us a formula for the polynomial itself). We now give more precise details about ours results.Let M Γ 0 (p)∩Γ(2) be the stack over Z[1/2] whose S-points are the isomorphism classes of generalized elliptic curves E/S, endowed with a locally free subgroup A of rank p such that A + E[2] meets each irreducible component of any geometric fiber of E (E[2] is the subgroup of 2-torsion points of E) and a basis of the 2-torsion (i.e. an isomorphism α 2 : E[2] ≃ (Z/2Z) 2 ). Deligne and Rapoport proved in [2] that M Γ 0 (p)∩Γ(2) is a regular algebraic stack, proper, of pure dimension 2 and flat over Z[1/2].Let M Γ 0 (p)∩Γ(2) be the coarse space of the algebraic stack M Γ 0 (p)∩Γ(2) over Z[1/2]. Deligne-Rapoport proved that M Γ 0 (p)∩Γ(2) is a normal scheme and proper flat of relative dimension one over Z[1/2]. Moreover, Deligne-Rapoport proved that M Γ 0 (p)∩Γ (2) is smooth over Z[1/2] outside the points associated to supersingular elliptic curves in characteristic p and that M Γ 0 (p)∩Γ(2) is a semi-stable regular scheme (cf. [2, V.1.14, Variante] and [1, Proposition 2.1] for more details).Let K be the unique quadratic unramified extension of Q p , O K be the ring of integers of K and k be the residual field. Let X be the base change M Γ 0 (p)∩Γ(2) ⊗ O K ; it is the coarse moduli space of the base change M Γ 0 (p)∩Γ(2) ⊗O K (because the formation of coarse moduli space commutes with flat base change).Let M Γ(2) be the model over Z[1/2] of the modular curve X(2) introduced by Igusa [9]. The special fiber of the scheme X is the union of two copies of M Γ(2) ⊗k meeting transversally at the supersingular points, and such that a supersingular point x of the first copy is identified with the point x p = Frob p (x) of the second copy (the supersingular points of the special fiber of X are k-rational). Moreover, we have M Γ(2) ⊗k ≃ P 1 k (cf. [1, Proposition 2.1]).The cusps of M Γ 0 (p)∩Γ(2) correspond to Néron 2-gons or 2p-...

show abstract

“…Proof. The first assertion follows by comparing Theorem [1,1] and the Theorem above. Let λ ∈ F p 2 which is not supersingular.…”

Section: Introductionmentioning

confidence: 89%

Section: P-adic Uniformization and The Reduction Mapmentioning

confidence: 99%

See 1 more Smart Citation

Congruence formulas for Legendre modular polynomials

Betina

Lecouturier

2018

Journal of Number Theory

Self Cite

View full text Add to dashboard Cite

show abstract

“…In particular, MΓ has the required rank. There is a boundary map ∂ Γ : MΓ → Z[C Γ ] 0 (the upper 0 meaning the augmentation subgroup), whose kernel contains H 1 (Y Γ , Z) ֒→ MΓ with finite index equal to 1 dΓ c∈CΓ e c . 1.2.…”

Section: Introductionmentioning

confidence: 99%

“…Remark 1.1. There exists a perfect pairing of Galois modules V ℓ × V ℓ → Z ℓ (1), where as usual Z ℓ (1) is Z ℓ with the Galois action given by χ ℓ (cf. Proposition 3.10).…”

Section: Introductionmentioning

confidence: 99%

Mixed modular symbols and the generalized cuspidal 1-motive

Lecouturier¹

2019

Preprint

Self Cite

View full text Add to dashboard Cite

We define and study the space of mixed modular symbols for a given finite index subgroup Γ of SL 2 (Z). This is an extension of the usual space of modular symbols, which in some cases carries more information about Eisenstein series. We make use of mixed modular symbols to construct some 1-motives related to the generalized Jacobian of modular curves. In the case Γ = Γ 0 (p) for some prime p, we relate our construction to the work of Ehud de Shalit on p-adic periods of X 0 (p).

show abstract

Higher Eisenstein elements, higher Eichler formulas and rank of Hecke algebras

Lecouturier

2020

Invent. math.

View full text Add to dashboard Cite

On the $p$-adic periods of the modular curve $X(\Gamma _0(p) \cap \Gamma (2))$

Abstract: Abstract. We prove a variant of Oesterlé's conjecture describing p-adic periods of the modular curve X 0 (p), with an additional Γ(2)-structure (and also Γ(3) ∩ Γ 0 (p) if p ≡ 1 (mod 3)). We use de Shalit's techniques and p-adic uniformization of curves with semi-stable reduction.

Cited by 4 publications

References 10 publications

Congruence formulas for Legendre modular polynomials

Congruence formulas for Legendre modular polynomials

Mixed modular symbols and the generalized cuspidal 1-motive

Higher Eisenstein elements, higher Eichler formulas and rank of Hecke algebras

Contact Info

Product

Resources

About