We show how our p-adic method to compute Galois representations occurring in the torsion of Jacobians of algebraic curves can be adapted to modular curves. The main ingredient is the use of "moduli-friendly" Eisenstein series introduced by Makdisi, which allow us to evaluate modular forms at p-adic of modular curves points and dispenses us of the need for equations of modular curves and for q-expansion computations. The resulting algorithm compares very favourably to the complex-analytic method.
For a totally real field $F$, a finite extension $\mathbf{F}$ of $\mathbf{F}_p$ and a Galois character $χ: G_F \to \mathbf{F}^{\times}$ unramified away from a finite set of places $Σ\supset \{\mathfrak{p} \mid p\}$ consider the Bloch-Kato Selmer group $H:=H^1_Σ(F, χ^{-1})$. In an earlier paper of the authors it was proved that the number $d$ of isomorphism classes of (non-semisimple, reducible) residual representations $\overlineρ$ giving rise to lines in $H$ which are modular by some $ρ_f$ (also unramified outside $Σ$) satisfies $d \geq n:= \dim_{\mathbf{F}} H$. This was proved under the assu…
For certain algebraic Hecke characters chi of an imaginary quadratic field F we define an Eisenstein ideal in a p-adic Hecke algebra acting on cuspidal automorphic forms of GL_2/F. By finding congruences between Eisenstein cohomology classes (in the sense of G. Harder) and cuspidal classes we prove a lower bound for the index of the Eisenstein ideal in the Hecke algebra in terms of the special L-value L(0,chi). We further prove that its index is bounded from above by the order of the Selmer group of the p-adic Galois character associated to chi^{-1}. This uses the work of R. Taylor et al. on a…
Let $p\neq2$ be a prime. We show a technique based on local class field theory and on the expansions of certain resultants which allows to recover very easily Lbekkouri's characterization of Eisenstein polynomials generating cyclic wild extensions of degree $p^2$ over $\Q_p$, and to extend it to the case of the base field $K$ being an unramified extension of $\Q_p$. Furthermore, when a polynomial satisfies only some of the stated conditions, we show that the first unsatisfied condition gives information about the Galois group of the normal closure. This permits to give a complete classificatio…
In this article, we set up a strategy to prove one divisibility towards the main Iwasawa conjecture for the Selmer groups attached to the twisted adjoint modular Galois representations associated to Hida families. This conjecture asserts the equality of the p-adic L-function interpoling the critical values of the symmetric square of the modular forms in these families and the characteristic ideal of the associated Selmer group. The idea is to introduce a third characteristic ideal containing informations on the congruences between cuspidal Siegel modular forms of genus 2 and the Klingen type E…
We construct a norm compatible system of Galois cohomology classes in the cyclotomic extension of the field of rationnals giving rise (conjecturally) to the degree four p-adic L-function of the symplectic group GSp(4). These classes are defined as cup products of torsion sections of the elliptic polylogarithm pro-sheaf. We rely on the norm compatibility of the elliptic polylogarithm and on some weight computations in the cohomology of Siegel threefolds.
In this paper we prove some divisibility properties of the cardinality of elliptic curves modulo primes. These proofs explain the good behavior of certain parameters when using Montgomery or Edwards curves in the setting of the elliptic curve method (ECM) for integer factorization. The ideas of the proofs help us to find new families of elliptic curves with good division properties which increase the success probability of ECM.
We study the Eisenstein ideal for modular forms of even weight $k>2$ and prime level $N$. We pay special attention to the phenomenon of $\mathit{extra \ reducibility}$: the Eisenstein ideal is strictly larger than the ideal cutting out reducible Galois representations. We prove a modularity theorem for these extra reducible representations. As consequences, we relate the derivative of a Mazur-Tate $L$-function to the rank of the Hecke algebra, generalizing a theorem of Merel, and give a new proof of a special case of an equivariant main conjecture of Kato. In the second half of the paper, we r…
In this paper, we consider normalized newforms $f\in S_k(Γ_0(N),\varepsilon_f)$ whose non-constant term Fourier coefficients are congruent to those of an Eisenstein series modulo some prime ideal above a rational prime $p$. In this situation, we establish a congruence between the anticyclotomic $p$-adic $L$-function of Bertolini-Darmon-Prasanna and the Katz two-variable $p$-adic $L$-function. From this, we derive congruences between images under the $p$-adic Abel-Jacobi map of certain generalized Heegner cycles attached to $f$ and special values of the Katz $p$-adic $L$-function. In particular…
Let $k > 3$ be an integer and $p$ a prime with $p > 2k-2$. Let $f$ be a newform of weight $2k-2$ and level 1 so that $f$ is ordinary at $p$ and $\barρ_{f}$ is irreducible. Under some additional hypotheses we prove that $ord_{p}(L_{alg}(k,f)) \leq ord_{p}(# S)$ where $S$ is the Pontryagin dual of the Selmer group associated to $ρ_{f} \otimes ε^{1-k}$ with $ε$ the $p$-adic cyclotomic character. We accomplish this by first constructing a congruence between the Saito-Kurokawa lift of $f$ and a non-CAP Siegel cusp form. Once this congruence is established, we use Galois representations to obtain th…
In the present paper, we study the $p$-adic $L$-functions and the (strict) Selmer groups over $\mathbb{Q}_{\infty}$, the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$, of the $p$-adic weight one cusp forms $f$, obtained via the $p$-stabilization of weight one Eisenstein series, under the assumption that a certain Eisenstein component of the $p$-ordinary universal cuspidal Hecke algebra is Gorenstein. As an application, we compute the Iwasawa invariants of ordinary modular forms of weight $k\geq 2$ with the same residual Galois representations as the one of $f$, which in our setting, is r…
Let $F$ be a cuspidal eigenform of even weight and trivial nebentypus, let $p$ be a prime not dividing the level of $F$, and let $ρ_F$ be the $p$-adic Galois representation attached to $F$. Assume that the $L$-function attached to the symmetric cube of $ρ_F$ vanishes to odd order at its central point. Then under some mild hypotheses, and conditional on certain consequences of Arthur's conjectures, we construct a nontrivial element in the Bloch--Kato Selmer group of an appropriate twist of the symmetric cube of $ρ_F$, in accordance with the Bloch--Kato conjectures. Our technique is based on the…
We show that the periods of the holomorphic Eisenstein series of level 1, parallel weight 2, on a Hilbert modular surface are not rational, even up to scaling. This is deduced from a study of the mixed Hodge structure on the cohomology of the Hilbert modular surface, where we find extension classes related to the units of the real quadratic field. We prove similar results for Hilbert modular varieties of all dimensions, and produce extensions of Galois representations in étale cohomology. The key point is to study the restriction of the canonical extension of the Hodge bundle to the boundary o…
We compute the non-Eisenstein systems of Hecke eigenvalues contributing to the $p$-arithmetic homology of irreducible smooth mod $p$ representations $π$ of $\mathrm{GL}_2(\mathbb{Q}_p)$ and to the cohomology of their duals. We show that in most cases they are associated to odd irreducible 2-dimensional Galois representations whose local component at $p$ corresponds under the mod $p$ local Langlands correspondence to a smooth representation that contains $π$ as a subrepresentation.
Let $p$ be an odd prime. We prove the cyclotomic Iwasawa Main Conjecture of K.Kato for the motive attached to an eigencuspform $f\in S_{k}(Γ_{0}(N))$ with arbitrary reduction type at $p$ under mild assumptions on the residual Galois representation $\barρ_{f}$. Under the same hypotheses, we also prove the generalized Iwasawa Main Conjecture for $p$-adic families of modular forms. The Iwasawa Main Conjecture for $f$ is deduced by a limit argument involving fundamental lines from a universal Iwasawa Main Conjecture over the universal deformation space of $\barρ_{f}$, which itself follows from the…
Let $V$ be a two-dimensional vector space over a field $\mathbb F$ of characteristic not $2$ or $3$. We show there is a canonical surjection $ν$ from the set of suitably generic commutative algebra structures on $V$ modulo the action of $GL(V)$ onto the plane $\mathbb F^2$. In these coordinates, which are quotients of invariant quartic polynomials, properties such as associativity and the existence of zero divisors are described by simple algebraic conditions. The map $ν$ is a bijection over the complement of a degenerate elliptic curve $Γ$ and over $Γ$ we give an explicit parametrisation of t…
