Explicit Computation of a Galois Representation Attached to an Eigenform Over $${\text {SL}}_3$$ from the $${{\text {H}}}_{\acute{\mathrm{e}}\mathrm{t}}^2$$ of a Surface

We describe a method to compute mod $$\ell $$ ℓ Galois representations contained in the $${{\text {H}}}_{\acute{\mathrm{e}}\mathrm{t}}^2$$ H e ´ t 2 of surfaces. We apply this method to the case of a representation with values in $${\text {GL}}_3(\mathbb {F}_9)$$ GL 3 ( F 9 ) attached to an eigenform over a congruence subgroup of $${\text {SL}}_3$$ SL 3 . We obtain, in particular, a polynomial with Galois group isomorphic to the simple group $${\text {PSU}}_3(\mathbb {F}_9)$$ PSU 3 ( F 9 ) and ramified at 2 and 3 only.

Wiles defect for Hecke algebras that are not complete intersections

In his work on modularity theorems, Wiles proved a numerical criterion for a map of rings $R\to T$ to be an isomorphism of complete intersections. He used this to show that certain deformation rings and Hecke algebras associated to a mod $p$ Galois representation at non-minimal level are isomorphic and complete intersections, provided the same is true at minimal level. In this paper we study Hecke algebras acting on cohomology of Shimura curves arising from maximal orders in indefinite quaternion algebras over the rationals localized at a semistable irreducible mod $p$ Galois representation $\bar {\rho }$ . If $\bar {\rho }$ is scalar at some primes dividing the discriminant of the quaternion algebra, then the Hecke algebra is still isomorphic to the deformation ring, but is not a complete intersection, or even Gorenstein, so the Wiles numerical criterion cannot apply. We consider a weight-2 newform $f$ which contributes to the cohomology of the Shimura curve and gives rise to an augmentation $\lambda _f$ of the Hecke algebra. We quantify the failure of the Wiles numerical criterion at $\lambda _f$ by computing the associated Wiles defect purely in terms of the local behavior at primes dividing the discriminant of the global Galois representation $\rho _f$ which $f$ gives rise to by the Eichler–Shimura construction. One of the main tools used in the proof is Taylor–Wiles–Kisin patching.

ON NONCRITICAL GALOIS REPRESENTATIONS

We propose a conjecture that the Galois representation attached to every Hilbert modular form is noncritical and prove it under certain conditions. Under the same condition we prove Chida, Mok and Park’s conjecture that Fontaine-Mazur L-invariant and Teitelbaum-type L-invariant coincide with each other.

Taylor coefficients of Anderson generating functions and Drinfeld torsion extensions

We generalize our work on Carlitz prime power torsion extension to torsion extensions of Drinfeld modules of arbitrary rank. As in the Carlitz case, we give a description of these extensions in terms of evaluations of Anderson generating functions and their hyperderivatives at roots of unity. We also give a direct proof that the image of the Galois representation attached to the [Formula: see text]-adic Tate module lies in the [Formula: see text]-adic points of the motivic Galois group. This is a generalization of the corresponding result of Chang and Papanikolas for the [Formula: see text]-adic case.

A short note on inadmissible coefficients of weight 2 and $$2k+1$$ newforms

Let $$f(z)=q+\sum _{n\ge 2}a(n)q^n$$ f ( z ) = q + ∑ n ≥ 2 a ( n ) q n be a weight k normalized newform with integer coefficients and trivial residual mod 2 Galois representation. We extend the results of Amir and Hong in Amir and Hong (On L-functions of modular elliptic curves and certain K3 surfaces, Ramanujan J, 2021) for $$k=2$$ k = 2 by ruling out or locating all odd prime values $$|\ell |<100$$ | ℓ | < 100 of their Fourier coefficients a(n) when n satisfies some congruences. We also study the case of odd weights $$k\ge 1$$ k ≥ 1 newforms where the nebentypus is given by a quadratic Dirichlet character.

Irreducibility of mod p Galois representations of elliptic curves with multiplicative reduction over number fields

In this paper we prove that for every integer [Formula: see text], there exists an explicit constant [Formula: see text] such that the following holds. Let [Formula: see text] be a number field of degree [Formula: see text], let [Formula: see text] be any rational prime that is totally inert in [Formula: see text] and [Formula: see text] any elliptic curve defined over [Formula: see text] such that [Formula: see text] has potentially multiplicative reduction at the prime [Formula: see text] above [Formula: see text]. Then for every rational prime [Formula: see text], [Formula: see text] has an irreducible mod [Formula: see text] Galois representation. This result has Diophantine applications within the “modular method”. We present one such application in the form of an Asymptotic version of Fermat’s Last Theorem that has not been covered in the existing literature.

Images of Galois representations in mod p Hecke algebras

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.

Non-openness of v-adic Galois representation for A-motives

Drinfeld modules and [Formula: see text]-motives are the function field analogs of elliptic curves and abelian varieties. For both Drinfeld modules and [Formula: see text]-motives, one can construct their [Formula: see text]-adic Galois representations and ask whether the images are open. For Drinfeld modules, this question has been answered by Richard Pink and his co-authors; however, this question has not been addressed for [Formula: see text]-motives. Here, we clarify the rank-one case for [Formula: see text]-motives and show that the image of Galois is open if and only if the virtual dimension is prime to the characteristic of the ground field.