Let $\ell \geq 5$ be a prime and let $N$ be a square-free integer prime to $\ell$. For each prime $p$ dividing $N$, let $a_p$ be either $1$ or $-1$. We give sufficient criteria for the existence of a newform $f$ of weight 2 for $Γ_0(N)$ such that the mod $\ell$ Galois representation attached to $f$ is reducible and $U_p f = a_p f$ for primes $p$ dividing $N$. The main techniques used are level raising methods based on an exact sequence due to Ribet.
In this paper, we call strongly modular those reducible semi-simple odd mod $l$ Galois representations for which the conclusion of the strongest form of Serre's original modularity conjecture holds. Under the assumption that the Serre weight $k$ satisfies $l\textgreater{}k+1$, we give a precise characterization of strongly modular representations, hence generalizing a classical theorem of Ribet pertaining to the case of conductor $1$.When the representation $ρ$ is not strongly modular, we give a necessary and sufficient condition on the primes $p$ not dividing $Nl$ for which it arises in level…
The ghost conjecture, formulated by this article's authors, predicts the list of p-adic valuations of the non-zero p-th eigenvalues ("slopes") for overconvergent p-adic modular eigenforms in terms of the Newton polygon of an easy-to-describe power series (the "ghost series"). The prediction is restricted to eigenforms whose Galois representation modulo p is reducible on a decomposition group at p. It has been discovered, however, that the conjecture is not formulated correctly. Here we explain the issue and propose a salvage.
We extend the lifting methods of our previous paper to lift reducible odd representations $\barρ:\mathrm{Gal}(\overline{F}/F) \to G(k)$ of Galois groups of global fields $F$ valued in Chevalley groups $G(k)$. Lifting results, when combined with automorphy lifting results pioneered by Wiles in the number field case and the results on the global Langlands correspondence proved by Drinfeld and L. Lafforgue in the function field case, give the only known method to access modularity of mod $p$ Galois representations in both reducible and irreducible cases. In the reducible case this allows one to s…
In this paper we study deformations of mod $p$ Galois representations $τ$ (over an imaginary quadratic field $F$) of dimension $2$ whose semi-simplification is the direct sum of two characters $τ_1$ and $τ_2$. As opposed to our previous work we do not impose any restrictions on the dimension of the crystalline Selmer group $H^1_Σ(F, {\rm Hom}(τ_2, τ_1)) \subset {\rm Ext}^1(τ_2, τ_1)$. We establish that there exists a basis $\mathcal{B}$ of $H^1_Σ(F, {\rm Hom}(τ_2, τ_1))$ arising from automorphic representations over $F$ (Theorem 8.1). Assuming among other things that the elements of $\mathcal{…
We prove the following uniformity principle: if one of the Galois representations in the family attached to a genus two Siegel cusp form of weight $k>3$, "semistable" and with multiplicity one, is reducible (for an odd prime $p$),then all the representations in the family are reducible. This, combined with Serre's conjecture (which is now a theorem) gives a proof of the Endoscopy Conjecture.
We prove that every odd semisimple reducible (2-dimensional) mod l Galois representation arises from a cuspidal eigenform. In addition, we investigate the possible different types (level, weight, character) of such a modular form. When the representation is the direct sum of the trivial character and a power of the mod l cyclotomic character, we are able to characterize the primes that can arise as levels of the associated newforms. As an application, we determine a new explicit lower bound for the highest degree among the fields of coefficients of newforms of trivial Nebentypus and prime leve…
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 totally real field, and v a place of F dividing an odd prime p. We study the weight part of Serre's conjecture for continuous, totally odd, two-dimensional mod p representations rhobar of the absolute Galois group of F that are reducible locally at v. Let W be the set of predicted Serre weights for the semisimplification of rhobar restricted to the decomposition group at v. We prove that when the local representation is generic, the Serre weights in W for which rhobar is modular are exactly the ones that are predicted (assuming that rhobar is modular). We also determine precisely wh…
We classify the three-dimensional representations of the modular group that are reducible but indecomposable, and their associated spaces of holomorphic vector-valued modular forms. We then demonstrate how such representations may be employed to compute periods of modular curves. This technique obviates the use of Hecke operators, and therefore provides a method for studying noncongruence modular curves as well as congruence.
Let $a,k\in\mathbb{N}$. For the $k-1$-th iterate of the exponential function $x\mapsto a^x$, also known as tetration, we write \[ ^k a:=a^{a^{.^{.^{.^{a}}}}}. \] In this paper, we show how an efficient algorithm for tetration modulo natural numbers $N$ may be used to compute the prime factorization of $N$. In particular, we prove that the problem of computing the squarefree part of integers is deterministically polynomial-time reducible to modular tetration.
The theory of elliptic modular forms has gained significant momentum from the discovery of relaxed yet well-behaved notions of modularity, such as mock modular forms, higher order modular forms, and iterated Eichler-Shimura integrals. Applications beyond number theory range from combinatorics, geometry, and representation theory to string theory and conformal field theory. We unify these relaxed notions in the framework of vector-valued modular forms by introducing a new class of $\mathrm{SL}_{2}(\mathbb{Z})$-representations: virtually real-arithmetic types. The key point of the paper is that …
We prove modularity of certain residually reducible ordinary 2-dimensional $p$-adic Galois representations with determinant a finite order odd character $χ$. For certain non-quadratic $χ$ we prove an $R=T$ result for $T$ the weight 1 specialisation of the Hida Hecke algebra acting on non-classical weight 1 forms, under the assumption that no two Hida families congruent to an Eisenstein series cross in weight 1. For quadratic $χ$ we prove that the quotient of $R$ corresponding to deformations split at $p$ is isomorphic to the Hecke algebra acting on classical CM weight 1 modular forms.
Given a continuous, odd, reducible and semi-simple $2$-dimensional representation $\barρ_0$ of $G_{\mathbb{Q},Np}$ over a finite field of odd characteristic $p$, we study the relation between the universal deformation ring of the pseudo-representation corresponding to $\barρ_0$ (pseudo-deformation ring) and the big $p$-adic Hecke algebra to prove that the maximal reduced quotient of the pseudo-deformation ring is isomorphic to the local component of the big $p$-adic Hecke algebra corresponding to $\barρ_0$ if a certain global Galois cohomology group has dimension $1$. This partially extends th…
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 apply results from the theory of congruences of modular forms (control of reducible primes, level-lowering), the modularity of elliptic curves and Q-curves, and a couple of Frey curves of Fermat-Goldbach type, to show the existence of newforms of weight 2 and trivial nebentypus with coefficient fields of arbitrarily large degree and square-free or almost square-free level. More precisely, we prove that for any given numbers t and B, there exists a newform f of weight 2 and trivial nebentypus whose level N is square-free (almost square-free), N has exactly t prime divisors (t o…
Let $ρ$ f,$λ$ be the residual Galois representation attached to a newform f and a prime ideal $λ$ in the integer ring of its coefficient field. In this paper, we prove explicit bounds for the residue characteristic of the prime ideals $λ$ such that $ρ$ f,$λ$ is exceptional, that is reducible, of projective dihedral image, or of projective image isomorphic to A4, S4 or A5. We also develop explicit criteria to check the reducibility of $ρ$ f,$λ$ , leading to an algorithm that compute the exact set of such $λ$. We have implemented this algorithm in PARI/GP. Along the way, we construct lifts of Ka…
We study the crystalline universal deformation ring R (and its ideal of reducibility I) of a mod p Galois representation rho_0 of dimension n whose semisimplification is the direct sum of two absolutely irreducible mutually non-isomorphic constituents rho_1 and rho_2. Under some assumptions on Selmer groups associated with rho_1 and rho_2 we show that R/I is cyclic and often finite. Using ideas and results of (but somewhat different assumptions from) Bellaiche and Chenevier we prove that I is principal for essentially self-dual representations and deduce statements about the structure of R. Us…
We prove the modularity of minimally ramified ordinary residually reducible p-adic Galois representations of an imaginary quadratic field F under certain assumptions. We first exhibit conditions under which the residual representation is unique up to isomorphism. Then we prove the existence of deformations arising from cuspforms on GL_2(A_F) via the Galois representations constructed by Taylor et al. We establish a sufficient condition (in terms of the non-existence of certain field extensions which in many cases can be reduced to a condition on an L-value) for the universal deformation ring t…
We study algebras of meromorphic modular forms whose poles lie on Heegner divisors for orthogonal and unitary groups associated to root lattices. We give a uniform construction of $147$ hyperplane arrangements on type IV symmetric domains for which the algebras of modular forms with constrained poles are free and therefore the Looijenga compactifications of the arrangement complements are weighted projective spaces. We also construct $8$ free algebras of modular forms on complex balls with poles on hyperplane arrangements. The most striking example is the discriminant kernel of the $2U\oplus D…
We study the modular symmetry in $T^2$ and orbifold comfactifications with magnetic fluxes. There are $|M|$ zero-modes on $T^2$ with the magnetic flux $M$. Their wavefunctions as well as massive modes behave as modular forms of weight $1/2$ and represent the double covering group of $Γ\equiv SL(2,\mathbb{Z})$, $\widetildeΓ \equiv \widetilde{SL}(2,\mathbb{Z})$. Each wavefunction on $T^2$ with the magnetic flux $M$ transforms under $\widetildeΓ(2|M|)$, which is the normal subgroup of $\widetilde{SL}(2,\mathbb{Z})$. Then, $|M|$ zero-modes are representations of the quotient group $\widetildeΓ'_{2…
We compute explicit rational models for some Hilbert modular surfaces corresponding to square discriminants, by connecting them to moduli spaces of elliptic K3 surfaces. Since they parametrize decomposable principally polarized abelian surfaces, they are also moduli spaces for genus-2 curves covering elliptic curves via a map of fixed degree. We thereby extend classical work of Jacobi, Hermite, Bolza etc., and more recent work of Kuhn, Frey, Kani, Shaska, Völklein, Magaard and others, producing explicit families of reducible Jacobians. In particular, we produce a birational model for the modul…
Let F be a totally real field, v an unramified place of F dividing p and rho a continuous irreducible two-dimensional mod p representation of G_F such that the restriction of rho to G_{F_v} is reducible and sufficiently generic. If rho is modular (and satisfies some weak technical assumptions), we show how to recover the corresponding extension between the two characters of G_{F_v} in terms of the action of GL_2(F_v) on the cohomology mod p.
Given a reducible Galois representation $\overlineρ: G_{\mathbb{Q}} \rightarrow GL_2( \mathbb{F}_q)$ we show there exists an irreducible deformation $ρ: G_{\mathbb{Q}} \rightarrow GL_2 (\mathbb{W} [[T_1, T_2,.., T_r,....,]])$ of $\overlineρ$ ramified at infinitely many primes, where $\mathbb{W}$ denotes the ring of Witt vectors of $\mathbb{F}_q$. This is a modification of Ramakrishna's result for the irreducible case.
Crystabelline representations are representations of the absolute Galois group $G_{\mathbb{Q}_p}$ over $\mathbb{Q}_p$ that become crystalline on $G_{F}$ for some abelian extension $F/\mathbb{Q}_p$. Their relation to modular forms is that the representation associated with a finite slope newform of level divisible by $p^2$ is crystabelline. In this article we study the connection between the slopes of two-dimensional crystabelline representations and the reducibility of their modulo $p$ reductions. This question is inspired by a theorem by Buzzard and Kilford which implies that the slopes on th…
The global deformation theory of residually reducible Galois representations with fixed auxiliary conditions is studied. We show that $\barρ:\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow \operatorname{GL}_2(\bar{\mathbb{F}}_p)$ lifts to a Hida line for which the weights range over a congruence class modulo-$p^2$. The advantage of the purely Galois theoretic approach is that it allows us to construct $p$-adic families of Galois representations lifting the actual representation $\barρ$, and not just the semisimplification.
We give a method of constructing polynomials of arbitrarily large degree irreducible over a global field F but reducible modulo every prime of F. The method consists of finding quadratic f in F[x] whose iterates have the desired property, and it depends on new criteria ensuring all iterates of f are irreducible. In particular when F is a number field in which the ideal (2) is not a square, we construct infinitely many families of quadratic f such that every iterate f^n is irreducible over F, but f^n is reducible modulo all primes of F for n at least 2. We also give an example for each n of a q…
We develop methods to study $2$-dimensional $2$-adic Galois representations $ρ$ of the absolute Galois group of a number field $K$, unramified outside a known finite set of primes $S$ of $K$, which are presented as Black Box representations, where we only have access to the characteristic polynomials of Frobenius automorphisms at a finite set of primes. Using suitable finite test sets of primes, depending only on $K$ and $S$, we show how to determine the determinant $\detρ$, whether or not $ρ$ is residually reducible, and further information about the size of the isogeny graph of $ρ$ whose ver…
We prove the following theorem: Let $\bar\F_p$ be an algebraic closure of a finite field of characteristic $p$. Let $ρ$ be a continuous homomorphism from the absolute Galois group of $\Q$ to $\GL(3,\bar\F_p)$ which is isomorphic to a direct sum of a character and a two-dimensional odd irreducible representation. Under the condition that the conductor of $ρ$ is squarefree, we prove that $ρ$ is attached to a Hecke eigenclass in the homology of an arithmetic subgroup $Γ$ of $\GL(3,\Z)$. In addition, we prove that the coefficient module needed is, in fact, predicted by a conjecture of Ash, Doud, P…
We prove the modularity for an algebraic space curve $Y$ of genus $50$ in $\mathbb{P}^5$, which consists of $21$ quartic polynomials in six variables, by means of an explicit modular parametrization by theta constants of order $13$. This provides an example of modularity, explicit uniformization and hyperbolic uniformization of arithmetic type for a higher genus algebraic space curve. In particular, it gives a new example for Hilbert's 22nd problem. This gives $21$ modular equations of order $13$, which greatly improve the result of Ramanujan and Evans on the construction of modular equations …
Let K be a finite unramified extension of Q_p. We parametrize the (phi, Gamma)-modules corresponding to reducible two-dimensional mod p representations of G_K and characterize those which have reducible crystalline lifts with certain Hodge-Tate weights.
We give upper bounds on the numbers of various classes of polynomials reducible over the integers and over integers modulo a prime and on the number of matrices in SL(n), GL(n) and Sp(2n) with reducible characteristic polynomials, and on polynomials with non-generic Galois groups. We use our result to show that a random (in the appropriate sense) element of the mapping class group of a closed surface is pseudo-Anosov, and that a random automorphism of a free group is strongly irreducible (aka irreducible with irreducible powers). We also give a necessary condition for all powers of an algebrai…
