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Emmanuel Lecouturier

Publications and source records attributed to Emmanuel Lecouturier.

13 recordsLinked to original sources
arXiv2026

Indivisibility of ray class groups of real quadratic fields

Let ${\ell}$, p $\ge$ 5 be primes such that p | (${\ell}$ -1). Let $\Delta$ > 0 be the fundamental discriminant of a real quadratic field in which ${\ell}$ splits. We denote by h - ${\ell}$ ($\Delta$) the order of the minus part (for the Galois action) of the ray class group of Q( $\sqrt$ $\Delta$) of modulus ${\ell}$. In this paper, we study the indivisibility of h - ${\ell}$ ($\Delta$) by p, and prove that under the assumption that this set is non-empty. This lower bound is made unconditional if ${\ell}$ = 2p + 1, i.e. if p is a Sophie Germain prime. Our result can be viewed as being in the continuity of the results of Kohnen-Ono, Ono, Byeon, Beckwith etc. regarding the class numbers of quadratic fields, in the sense that we rely on techniques from the theory of half-integral weight modular forms. Significant difficulties however arise in our study, as we have to study Eisenstein congruences for cuspforms of weight 3 2 , and use a generalized Shimura correspondence of Baruch-Mao. Combined with the results of Lecouturier-Wang, our result has implications eg. for the 5-part of BSD for even quadratic twists of X 0 (11).

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arXiv2025

Eisenstein cocycles for imaginary quadratic fields

We construct Eisenstein cocycles for arithmetic subgroups of GL_2 of imaginary quadratic fields valued in second K-groups of products of two CM elliptic curves. We use these to construct maps from the first homology groups of Bianchi spaces to corresponding second K-groups of ray class fields and to verify the Eisenstein property of these maps for prime-to-level Hecke operators.

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arXiv2023

On the arithmetic of special values of $L$-functions for certain abelian varieties with a rational isogeny

Let $N$ and $p$ be primes $\geq 5$ such that $p \mid \mid N-1$. In this situation, Mazur defined and studied the $p$-Eisenstein quotient $\tilde{J}^{(p)}$ of $J_0(N)$. We prove a kind of modulo $p$ version of the Birch and Swinnerton-Dyer conjecture for the ``$p$-Eisenstein part'' of even quadratic twists of $\tilde{J}^{(p)}$. Our result is the analogue for even quadratic twists of a result of Mazur concerning odd quadratic twists.

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arXiv2022

Level compatibility in Sharifi's conjecture

Sharifi has constructed a map from the first homology of the modular curve $X_1(M)$ to the $K$-group $K_2(\mathbf{Z}[ζ_M, \frac{1}{M}])$, where $ζ_M$ is a primitive $M$th root of unity. We study how these maps relate when $M$ varies. Our method relies on the techniques developed by Sharifi and Venkatesh.

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arXiv2022

On triple product L-functions and a conjecture of Harris--Venkatesh

Harris and Venkatesh made a conjecture relating the derived Hecke operators and the adjoint motivic cohomology in the setting of weight one modular forms. This conjecture was proved under some conditions in the dihedral case by Darmon--Harris--Rotger--Venkatesh. We use a new approach to prove more general cases of the conjecture (up to sign). Our approach relies on Waldspurger's formula for the central value of Rankin L-series and Ichino's formula for the triple product L-function.

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arXiv2019

On the Mazur--Tate conjecture for prime conductor and Mazur's Eisenstein ideal

In 1995, Ehud de Shalit proved an analogue of a conjecture of Mazur--Tate for the modular Jacobian $J_0(p)$. His main result was valid away from the Eisenstein primes. We complete the work of de Shalit by including the Eisenstein primes, and give some applications such as an elementary combinatorial identity involving discrete logarithms of difference of supersingular $j$-invariants. An important tool is our recent work on the so called "generalized cuspidal $1$-motive".

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arXiv2019

On a conjecture of Sharifi and Mazur's Eisenstein ideal

Let $N$ and $p$ be prime numbers $\geq 5$ such that $p$ divides $N-1$. Let $I$ be Mazur's Eisenstein ideal of level $N$ and $H_+$ be the plus part of $H_1(X_0(N), \mathbf{Z}_p)$ for the complex conjugation. We give a conjectural explicit description of the group $I\cdot H_+/I^2\cdot H_+$ in terms of the second $K$-group of the cyclotomic field $\mathbf{Q}(ζ_N)$. We prove that this conjecture follows from a conjecture of Sharifi about some Eisenstein ideal of level $Γ_1(N)$. Following the work of Fukaya--Kato, we prove partial results on Sharifi's conjecture. This allows us to prove partial results on our conjecture.

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arXiv2019

Mixed modular symbols and the generalized cuspidal 1-motive

We define and study the space of mixed modular symbols for a given finite index subgroup $Γ$ of $SL_2(\mathbf{Z})$. 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 $X_0(p)$.

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arXiv2018

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

Let $N$ and $p$ be primes such that $p$ divides the numerator of $\frac{N-1}{12}$. In this paper, we study the rank $g_p$ of the completion of the Hecke algebra acting on cuspidal modular forms of weight $2$ and level $Γ_0(N)$ at the $p$-maximal Eisenstein ideal. We give in particular an explicit criterion to know if $g_p \geq 3$, thus answering partially a question of Mazur. In order to study $g_p$, we develop the theory of \textit{higher Eisenstein elements}, and compute the first few such elements in four different Hecke modules. This has applications such as generalizations of the Eichler mass formula in characteristic $p$.

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arXiv2017

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 $λ$-invariants. Let $F_p(X,Y) \in \mathbf{Z}[X,Y]$ be the $p$-th modular polynomial for $λ$-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 $R(λ)$ if $λ$ is a supersingular. This formula is related to the Manin--Drinfeld pairing used in the $p$-adic uniformization of the modular curve $X(Γ_0(p)\cap Γ(2))$. This pairing was computed explicitly modulo principal units in a previous work of both authors. Furthermore, if $λ$ is supersingular and lives in $\mathbf{F}_p$, then we also express $R(λ)$ in terms of a CM lift (which are showed to exist) of the Legendre elliptic curve associated to $λ$.

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arXiv2016

Sur le p-rang du groupe des classes de Q(N^1/p)

Let N and p be two prime numbers > 3 such that p divides N-1. We estimate the p-rank of the class group of Q(N^(1/p)) in terms of the discrete logarithm, with values un F_p, of certain units. Using the Gross--Koblitz formula and identities on the N-adic Gamma function, we explicitly compute these logarithms. A special case (for which we don't have an elementary proof) of our formula is the following: assume there are some integers $a$, $b$ such that N = (a^p+b^p)/(a+b). Then (a+b)*\prod_{k=1}^{(N-1)/2} k^{8k} is a p-th power modulo N. Furthermore we give a new proof which doesn't use modular forms of a result of Calegari and Emerton.

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arXiv2010

On a conjecture of H. Gupta

Denote by r(n) the length of a shortest integer sequence on a circle containing all permutations of the set {1,2,...,n} as subsequences. Hansraj Gupta conjectured in 1981 that r(n) <= n^2/2. In this paper we confirm the conjecture for the case where n is even, and show that r(n) < n^2/2 + n/4 -1 if n is odd.

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