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Joël Bellaïche

Publications and source records attributed to Joël Bellaïche.

8 recordsLinked to original sources
arXiv2020

On self-correspondences on curves

We study the algebraic dynamics of self-correspondences on a curve. A self-correspondence on a (proper and smooth) curve $C$ over an algebraically closed field is the data of another curve $D$ and two non-constant separable morphisms $π_1$ and $π_2$ from $D$ to $C$. A subset $S$ of $C$ is complete if $π_1^{-1}(S)=π_2^{-1}(S)$. We show that self-correspondences are divided into two classes: those that have only finitely many finite complete sets, and those for which $C$ is a union of finite complete sets. The latter ones are called finitary and have a trivial dynamics. For a non-finitary self-correspondence in characteristic zero, we give a sharp bound for the number of étale finite complete sets.

math.AG↗
arXiv2018

Congruences with Eisenstein series and mu-invariants

We study the variation of mu-invariants in Hida families with residually reducible Galois representations. We prove a lower bound for these invariants which is often expressible in terms of the p-adic zeta function. This lower bound forces these mu-invariants to be unbounded along the family, and moreover, we conjecture that this lower bound is an equality. When U_p-1 generates the cuspidal Eisenstein ideal, we establish this conjecture and further prove that the p-adic L-function is simply a power of p up to a unit (i.e. lambda=0). On the algebraic side, we prove analogous statements for the associated Selmer groups which, in particular, establishes the main conjecture for such forms.

math.NT↗
arXiv2017

Non-zero coefficients of half-integral weight modular forms mod $\ell$

We obtain new lower bounds for the number of Fourier coefficients of a weakly holomorphic modular form of half-integral weight not divisible by some prime $\ell$. Among the applications of this we show that there are $\gg \sqrt{X}/\log \log X$ integers $n \leq X$ for which the partition function $p(n)$ is not divisible by $\ell$, and that there are $\gg \sqrt{X}/\log \log X$ values of $n \leq X$ for which $c(n)$, the $n$th Fourier coefficient of the $j$-invariant, is odd.

math.NT↗
arXiv2016

Images of Pseudo-Representations and Coefficients of Modular Forms modulo p

We describe the image of general families of two-dimensional representations over compact semi-local rings. Applying this description to the family carried by the universal Hecke algebra acting on the space of modular forms of level $N$ modulo a prime $p$, we prove new results about the coefficients of modular forms mod $p$. If $f=\sum_{n=0}^\infty a_n q^n$ is such a form, for which we can assume without loss of generality that $a_n=0$ if $(n,Np)>1$, calling $δ(f)$ the density of the set of primes $\ell$ such that $a_\ell \neq 0$, we prove that $δ(f)>0$ provided that $f$ is not zero (and if $p=2$, not a multiple of $Δ$). More importantly, we prove, when $p>2$, a {\it uniform} version of this result, namely that there exists a constant $c>0$ depending only on $N$ and $p$ such that $δ(f)>c$ for all forms $f$ except for those in an explicit subspace of infinite codimension of the space of all modular forms mod $p$ of level $N$. Forms in this subspace, called {\it special} modular forms mod $p$, are proved to be closely related to certain classes of modular forms mod $p$ previously studied by the author, Nicolas and Serre, called cyclotomic and CM modular forms mod $p$.

math.NT↗
arXiv2014

On the Eigencurve at classical weight one points

We show that the p-adic Eigencurve is smooth at classical weight one points which are regular at p and give a precise criterion for etaleness over the weight space at those points. Our approach uses deformations of Galois representations.

math.NT↗
arXiv2013

Théorème de Chebotarev et complexité de Littlewood

The effective version of Chebotarev's density theorem under the Generalized Riemann Hypothesis and the Artin conjecture (cf. Iwaniec and Kowalski's Analytic Number Theory, 5.13) involves a numerical invariant of a subset $D$ of a finite group $G$ that we call the Littlewood Complexity of $D$. We study this invariant in detail. Using this study, and a new application of the large sieve, we give improved versions of two standard questions related to Chebotarev: the bound on the first prime in a Frobenian set, and the asymptotic of the set of primes with given Frobenius in an infinite Galois extension. We then give concrete applications to various problems in number theory, such as the first primitive root modulo a prime $\ell$, the factorization of an integral polynomial modulo primes, the Lang-Trotter conjecture and its generalizations, and Serre's uniformity conjecture.

math.NT↗
arXiv2012

The p-adic L-functions of Evil Eisenstein Series

We compute the $p$-adic $L$-functions of evil Eisenstein series, showing that they factor as products of two Kubota--Leopoldt $p$-adic $L$-functions times a logarithmic term. This proves in particular a conjecture of Glenn Stevens.

math.NT↗