Abstract In this paper, we study the variation of Selmer groups in families of modular Galois representations that are congruent modulo a fixed prime $p \geq 5$ . Motivated by analogies with Goldfeld’s conjecture on ranks in quadratic twist families of elliptic curves, we investigate the stability of Selmer groups defined over $\mathbb{Q}$ via Greenberg’s local conditions under congruences of residual Galois representations. Let X be a positive real number. Fix a residual representation $\bar{\rho}$ and a corresponding modular form f of weight 2 and optimal level. We count the number of level-raising modular forms g of weight 2 that are congruent to f modulo p , with level $N_g\leq X$ , such that the p -rank of the Selmer groups of g equals that of f . Under some mild assumptions on $\bar{\rho}$ , we prove that this count grows at least as fast as $X (\log X)^{\alpha - 1}$ as $X \to \infty$ , for an explicit constant $\alpha \gt 0$ . The main result is a partial generalisation of theorems of Ono and Skinner on rank-zero quadratic twists to the setting of modular forms and Selmer groups.

Let f and g be two different newforms without complex multiplication having the same coefficient field. The main result of the present article proves that an isomorphism between the residual Galois representations attached to f and to g for a large prime p (depending only on g ) implies that the endomorphism algebra of the abelian variety A_{f} , attached to f by the Eichler–Shimura construction (after tensoring with \mathbb{Q} ), is a subalgebra of the endomorphism algebra of the abelian variety A_{g} attached to g . This implies important relations between their building blocks. A non-trivial application of our result is that for all prime numbers d congruent to 3 modulo 8 satisfying that the class number of \mathbb{Q}(\sqrt{-d}) is prime to 3 , the equation x^{4}+dy^{2} =z^{p} has no non-trivial primitive solutions when p is large enough. We prove a similar result for the equation x^{2}+dy^{6}=z^{p} .

Abstract Letpbe an odd prime. Letf1{f_{1}}andf2{f_{2}}be weight 2 cuspidal Hecke eigenforms with isomorphic residual Galois representations atp. Greenberg–Vatsal and Emerton–Pollack–Weston showed that ifpis a good ordinary prime for the two forms, the Iwasawa invariants of theirp-primary Selmer groups andp-adicL-functions over the cyclotomicℤp{\mathbb{Z}_{p}}-extension ofℚ{\mathbb{Q}}are closely related. The goal of this article is to generalize these results to the anticyclotomic setting. More precisely, letKbe an imaginary quadratic field wherepsplits. Suppose that the generalized Heegner hypothesis holds with respect to both(f1,K){(f_{1},K)}and(f2,K){(f_{2},K)}. We study relations between the Iwasawa invariants of the BDP Selmer groups and the BDPp-adicL-functions off1{f_{1}}andf2{f_{2}}.

Altres ajuts: We acknowledge the financial support of ANR-14-CE-25-0015 Gardio (N. Billerey), an NSERC Discovery Grant (I. Chen), and the grant Proyecto RSME-FBBVA 2015 José Luis Rubio de Francia (N. Freitas).

For every prime number p ≥ 3 p\geq 3 and every integer m ≥ 1 m\geq 1 , we prove the existence of a continuous Galois representation ρ : G Q → G l m ( Z p ) \rho : G_\mathbb {Q} \rightarrow Gl_m(\mathbb {Z}_p) which has open image and is unramified outside { p , ∞ } \{p,\infty \} if p ≡ 3 p\equiv 3 mod 4 4 and is unramified outside { 2 , p , ∞ } \{2,p,\infty \} if p ≡ 1 p \equiv 1 mod 4 4 . We also revisit the question of the lifting of residual Galois representations in terms of embedding problems; that allows us to produce Galois representations with open image in the group of upper triangular matrices with diagonal entries equal to 1 1 , unramified outside { p , ∞ } \{p,\infty \} , for m m “small” comparing to p p .

We show that crystalline points are Zariski dense in the deformation space of a representation of the absolute Galois group of a p-adic field. We also show that these points are dense in the subspace parameterizing deformations with the determinant equal to a fixed crystalline character. Our proof is purely local and works for all p-adic fields and all residual Galois representations.

The purpose of this paper is to show how the modular method together with different techniques can be used to prove non-existence of primitive non-trivial solutions of the equation [Formula: see text] for square-free values [Formula: see text]. The key ingredients are: the approach presented in [A. Pacetti and L. V. Torcomian, [Formula: see text]-curves, Hecke characters and some Diophantine equations, Math. Comp. 91(338) (2022) 2817–2865] (in particular its recipe for the space of modular forms to be computed) together with the use of the symplectic method (as developed in [E. Halberstadt and A. Kraus, Courbes de Fermat: Résultats et problèmes, J. Reine Angew. Math. 548 (2002) 167–234], although we give a variant over ramified extensions needed in our applications) to discard solutions and the use of a second Frey curve, aiming to prove large image of residual Galois representations.

Abstract Let f be a primitive Hilbert modular form over F of weight k with coefficient field $E_f$ , generated by the Fourier coefficients $C(\mathfrak {p}, f)$ for $\mathfrak {p} \in \mathrm {Spec}(\mathcal {O}_F)$ . Under certain assumptions on the image of the residual Galois representations attached to f, we calculate the Dirichlet density of $\{\mathfrak {p} \in \mathrm {Spec}(\mathcal {O}_F)| E_f = \mathbb {Q}(C(\mathfrak {p}, f))\}$ . For $k=2$ , we show that those assumptions are satisfied when $[E_f:\mathbb {Q}] = [F:\mathbb {Q}]$ is an odd prime. We also study analogous results for $F_f$ , the fixed field of $E_f$ by the set of all inner twists of f. Then, we provide some examples of f to support our results. Finally, we compute the density of $\{\mathfrak {p} \in \mathrm {Spec}(\mathcal {O}_F)| C(\mathfrak {p}, f) \in K\}$ for fields K with $F_f \subseteq K \subseteq E_f$ .

Abstract We prove new cases of the Inverse Galois Problem by considering the residual Galois representations arising from a fixed newform. Specific choices of weight $3$ newforms will show that there are Galois extensions of ${\mathbb Q}$ with Galois group $\operatorname {PSL}_2({\mathbb F}_p)$ for all primes p and $\operatorname {PSL}_2({\mathbb F}_{p^3})$ for all odd primes $p \equiv \pm 2, \pm 3, \pm 4, \pm 6 \ \pmod {13}$ .

Let [Formula: see text] be the residual Galois representation attached to a newform [Formula: see text] and a prime ideal [Formula: see text] in the integer ring of its coefficient field. In this paper, we prove explicit bounds for the residue characteristic of the prime ideals [Formula: see text] such that [Formula: see text] is exceptional, that is reducible, of projective dihedral image, or of projective image isomorphic to [Formula: see text], [Formula: see text] or [Formula: see text]. We also develop explicit criteria to check the reducibility of [Formula: see text], leading to an algorithm that computes the exact set of such [Formula: see text]’s. We have implemented this algorithm in PARI/GP. Along the way, we construct lifts of Katz’ [Formula: see text] operator in characteristic zero, and we prove a new Sturm bound theorem.

In the present article, we study the conjecture of Sharifi on the surjectivity of the map ϖ θ \varpi _{\theta } . Here θ \theta is a primitive even Dirichlet character of conductor N p Np , which is exceptional in the sense of Ohta. After localizing at the prime ideal p \mathfrak {p} of the Iwasawa algebra related to the trivial zero of the Kubota–Leopoldt p p -adic L L -function L p ( s , θ − 1 ω 2 ) L_p(s,\theta ^{-1}\omega ^2) , we compute the image of ϖ θ , p \varpi _{\theta ,\mathfrak {p}} in a local Galois cohomology group and prove that it is an isomorphism. Also, we prove that the residual Galois representations associated to the cohomology of modular curves are decomposable after taking the same localization.

It is known that if $p>37$ is a prime number and $E/\\mathbb{Q}$ is an\nelliptic curve without complex multiplication, then the image of the mod $p$\nGalois representation $$\n\\bar{\\rho}_{E,p}:\\operatorname{Gal}(\\overline{\\mathbb{Q}}/\\mathbb{Q})\\rightarrow\n\\operatorname{GL}(E[p]) $$\n of $E$ is either the whole of $\\operatorname{GL}(E[p])$, or is\n\\emph{contained} in the normaliser of a non-split Cartan subgroup of\n$\\operatorname{GL}(E[p])$. In this paper, we show that when $p>1.4\\times 10^7$,\nthe image of $\\bar{\\rho}_{E,p}$ is either $\\operatorname{GL}(E[p])$, or the\n\\emph{full} normaliser of a non-split Cartan subgroup. We use this to show the\nfollowing result, partially settling a question of Najman. For $d\\geq 1$, let\n$I(d)$ denote the set of primes $p$ for which there exists an elliptic curve\ndefined over $\\mathbb{Q}$ and without complex multiplication admitting a degree\n$p$ isogeny defined over a number field of degree $\\leq d$. We show that, for\n$d\\geq 1.4\\times 10^7$, we have\n $$\n I(d)=\\{p\\text{ prime}:p\\leq d-1\\}. $$\n

Let [Formula: see text] denote the mod [Formula: see text] local Hecke algebra attached to a normalized Hecke eigenform [Formula: see text], which is a commutative algebra over some finite field [Formula: see text] of characteristic [Formula: see text] and with residue field [Formula: see text]. By a result of Carayol we know that, if the residual Galois representation [Formula: see text] is absolutely irreducible, then one can attach to this algebra a Galois representation [Formula: see text] that is a lift of [Formula: see text]. We will show how one can determine the image of [Formula: see text] under the assumptions that (i) the image of the residual representation contains [Formula: see text], (ii) [Formula: see text] and (iii) the coefficient ring is generated by the traces. As an application we will see that the methods that we use allow to deduce the existence of certain [Formula: see text]-elementary abelian extensions of big non-solvable number fields.

Variations of Lehmer's Conjecture for Ramanujan's tau-function

We consider certain [Formula: see text]-ordinary non-CM Hida families with full residual Galois representation and give mild conditions under which every arithmetic point in these families is locally indecomposable when [Formula: see text]. The proof uses methods from deformation theory and mostly works for any odd prime [Formula: see text], but ultimately relies on the existence of a weight [Formula: see text] form in an auxiliary family which is available only for [Formula: see text]. We end by giving several non-trivial examples of [Formula: see text]-ordinary non-CM locally indecomposable modular forms of small level with full residual Galois representation.

Modular forms of arbitrary even weight with no exceptional primes

In this paper we study deformations of mod $p$ Galois representations $\\tau$\n(over an imaginary quadratic field $F$) of dimension $2$ whose\nsemi-simplification is the direct sum of two characters $\\tau_1$ and $\\tau_2$.\nAs opposed to our previous work we do not impose any restrictions on the\ndimension of the crystalline Selmer group $H^1_{\\Sigma}(F, {\\rm Hom}(\\tau_2,\n\\tau_1)) \\subset {\\rm Ext}^1(\\tau_2, \\tau_1)$. We establish that there exists a\nbasis $\\mathcal{B}$ of $H^1_{\\Sigma}(F, {\\rm Hom}(\\tau_2, \\tau_1))$ arising\nfrom automorphic representations over $F$ (Theorem 8.1). Assuming among other\nthings that the elements of $\\mathcal{B}$ admit only finitely many crystalline\ncharacteristic 0 deformations we prove a modularity lifting theorem asserting\nthat if $\\tau$ itself is modular then so is its every crystalline\ncharacteristic zero deformation (Theorems 8.2 and 8.5).\n

In this article we prove a congruence formula for the special values of certain dihedral twists of two primitive modular forms of weight two with isomorphic residual Galois representation at a prime p p .

Let F be a totally real field and p be an odd prime which splits completely in F. We show that a generic p-ordinary non-CM primitive Hilbert modular cuspidal eigenform over F of parallel weight two or more must have a locally non-split p-adic Galois representation, at at least one of the primes of F lying above p. This is proved under some technical assumptions on the global residual Galois representation. We also indicate how to extend our results to nearly ordinary families and forms of non-parallel weight.

We construct infinitely ramified Galois representations ρ such that the al(ρ)'s have distributions in contrast to the statements of Sato–Tate, Lang–Trotter and others. Using similar methods we deform a residual Galois representation for number fields and obtain an infinitely ramified representation with very large image, generalizing a result of Ramakrishna.