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Fix a prime $p > 2$. Let $rho : text{Gal}(overline{mathbb{Q}}/mathbb{Q}) to text{GL}_2(mathbb{I})$ be the Galois representation coming from a non-CM irreducible component $mathbb{I}$ of Hidas $p$-ordinary Hecke algebra. Assume the residual representation $bar{rho}$ is absolutely irreducible. Under a minor technical condition we identify a subring $mathbb{I}_0$ of $mathbb{I}$ containing $mathbb{Z}_p[[T]]$ such that the image of $rho$ is large with respect to $mathbb{I}_0$. That is, $text{Im} rho$ contains $text{ker}(text{SL}_2(mathbb{I}_0) to text{SL}_2(mathbb{I}_0/mathfrak{a}))$ for some non-zero $mathbb{I}_0$-ideal $mathfrak{a}$. This paper builds on recent work of Hida who showed that the image of such a Galois representation is large with respect to $mathbb{Z}_p[[T]]$. Our result is an $mathbb{I}$-adic analogue of the description of the image of the Galois representation attached to a non-CM classical modular form obtained by Ribet and Momose in the 1980s.
In this paper we explicitly compute mod-l Galois representations associated to modular forms. To be precise, we look at cases with l<=23 and the modular forms considered will be cusp forms of level 1 and weight up to 22. We present the result in term
For every prime number $pgeq 3$ and every integer $mgeq 1$, we prove the existence of a continuous Galois representation $rho: G_mathbb{Q} rightarrow Gl_m(mathbb{Z}_p)$ which has open image and is unramified outside ${p,infty}$ (resp. outside ${2,p,i
Let $p$ be a prime, let $r$ and $q$ be powers of $p$, and let $a$ and $b$ be relatively prime integers not divisible by $p$. Let $C/mathbb F_{r}(t)$ be the superelliptic curve with affine equation $y^b+x^a=t^q-t$. Let $J$ be the Jacobian of $C$. By w
For positive integers $n$, the truncated binomial expansions of $(1+x)^n$ which consist of all the terms of degree $le r$ where $1 le r le n-2$ appear always to be irreducible. For fixed $r$ and $n$ sufficiently large, this is known to be the case. W
We compute the E-polynomials of a family of twisted character varieties by proving they have polynomial count, and applying a result of N. Katz on the counting functions. To compute the number of GF(q)-points of these varieties as a function of q,