Does there exist anywhere a comprehensive list of small genus modular curves $X_G$, for G a subgroup of GL(2,Z/(n))$? (say genus <= 2), together with equations? I'm particularly …

Let $X/K$ be a variety over a global field $K$. When (and why) is the Galois representation $H^i_{et}(X \times_K \bar{K}, \mathbf{Q}_\ell)$ unramified at a place $v$ of $K$? I gue …

This is the converse question to an earlier question. More precisely, Let $X/K$ be a curve(or variety) over a global field $K$. We consider the Galois representation obtained by t …

I have been looking at Serre's conjecture and noticed that there are two conventions in the literature for a p-adic representation $\rho:\mbox{Gal}(\bar{\mathbb Q}/\mathbb Q)\to \m …

Galois deformations are an important tool in Wiles' arsenal for proving FLT. Are there any more elementary aspects (I'm thinking of 1-dimensional Galois representations attached t …

I would like to better understand the simplest case of the correspondence between Galois representations and (phi,Gamma)-modules. Namely, consider 1-dimensional Galois representat …

Let $\mathbf{T}$ be the reduced nearly ordinary Hecke algebra of level $N$ of Hida theory for $\operatorname{GL}_{2}$ over $\mathbb{Q}$ (or more generally over a totally real field …

I've just finished reading Ash and Gross's Fearless Symmetry, a wonderful little pop mathematics book on, among other things, Galois representations. The book made clear a very in …

Let me stress that I am only interested in $p$-adic fields in this question, for reasons that will become clear later. Let me also stress that in some sense I am basically assuming …

Suppose that $f$ is a weight $k$ newform for $\Gamma_1(N)$ with attached $p$-adic Galois representation $\rho_f$. Denote by $\rho_{f,p}$ the restriction of $\rho_f$ to a decomposi …

I often use the following theorem (that one can state more generally) in my research. Let E/Q be an elliptic curve of conductor N corresponding to a modular form f(E), l a prime o …

Let p be a prime and let K be a field containing the p'th roots of unity. Let E be an elliptic curve over K. We consider the the moduli problem $Y_E(p)$, which sends L to set of el …

First of all, let me apologize in advance for the terseness of this question. It seems that by now there are well-developed techniques (the "Taylor-Wiles-Kisin" method) for provin …

Let $X$ be a smooth projective variety over $F$, and $E/F$ - finite Galois extension. There is an extension of scalars map $CH^*(X) \to CH^*(X_E)$. The image lands in the Galois in …

I am in search of a concrete example [a concrete elliptic curve in Weierstrass form] of how Galois theory helps to find rational points on an elliptic curve. Chapter VI of Silverma …