algebraic-number-theoryclass-field-theorygn.general-topologytopological-groups
I was reading this question The connected component of the idele class group but I am very confused about the structure of the solenoids $(\widehat{\mathbb{Z}}\times\mathbb{R})/\mathbb{Z}$, (where $\mathbb{Z}$ acts diagonally), more specifically why it is connected. More generally, if we replace $\widehat{\mathbb{Z}}$ by $\mathbb{Z}_p$, is the resulting quotient group $(\mathbb{Z}_p\times \mathbb{R})/\mathbb{Z}$ still connected?
Hoping if someone can give me any hints or answers!!
What you are looking for is the correspondence between algebraic Hecke characters over a number field $F$ and compatible families of $l$-adic characters of the absolute Galois group of $F$. This is laid out beautifully in the first section of Laurent Fargues's notes here.
EDIT: In more detail, as Kevin notes in the comments above, an automorphic representation of $GL(1)$ over $F$ is nothing but a Hecke character; that is, a continuous character $$\chi:F^\times\setminus\mathbb{A}_F^\times\to\mathbb{C}^\times$$ of the idele class group of $F$. You can associate $L$-functions to these things: they admit analytic continuation and satisfy a functional equation. This is the automorphic side of global Langlands for $GL(1)$.
How to go from here to the Galois side? Well, let's start with the local story. Fix some prime $v$ of $F$; then the automorphic side is concerned with characters $$\chi_v:F_v^\times\to\mathbb{C}^\times$$ Local class field theory gives you the reciprocity isomorphism $$rec_v:W_{F_v}\to F_v^\times,$$ where $W_{F_v}$ is the Weil group of $F_v$. Then $\chi_v\circ rec_v$ gives you a character of $W_{F_v}$. This is local Langlands for $GL(1)$. The matching up local $L$-functions and $\epsilon$-factors is basically tautological.
We return to our global Hecke character $\chi$. Recall that global class field theory can be interpreted as giving a map (the Artin reciprocity map) $$Art_F:F^\times\setminus\mathbb{A}_F^\times\to Gal(F^{ab}/F),$$ where $F^{ab}$ is the maximal abelian extension of $F$. Local-global compatibility here means that, for each prime $v$ of $F$, the restriction $Art_F\vert_{F_v^\times}$ agrees with the inverse of the local reciprocity map $rec_v$.
Since $Art_F$ is not an isomorphism, we do not expect every Hecke character to be associated with a Galois representation. What is true is that $Art_F$ induces an isomorphism from the group of connected components of the idele class group to $Gal(F^{ab}/F)$. In particular, any Hecke character with finite image will factor through the reciprocity map, and so will give rise to a character of $Gal(F^{ab}/F)$. This is global Langlands for Dirichlet characters (or abelian Artin motives).
But we can say more, supposing that we have a certain algebraicity (or arithmeticity) condition on our Hecke character $\chi$ at infinity. The notes of Fargues referenced above have a precise definition of this condition; I believe the original idea is due to Weil. The basic idea is that the obstruction to $\chi$ factoring through the group of connected components of the idele class group (and hence through the abelianized Galois group) lies entirely at infinity. The algebraicity condition lets us "move" this persnickety infinite part over to the $l$-primary ideles (for some prime $l$), at the cost of replacing our field of coefficients $\mathbb{C}$ by some finite extension $E_\lambda$ of $\mathbb{Q}_l$. This produces a character
$$\chi_l:F^\times\setminus\mathbb{A}_F^\times\to E_\lambda^\times$$
that shares its local factors away from $l$ and $\infty$ with $\chi$, but now factors through $Art_F$. Varying over $l$ gives us a compatible family of $l$-adic characters associated with our automorphic representation $\chi$ of $GL(1)$. The $L$-functions match up since their local factors do.
This question deserves an expert answer such as this one by Emerton, but allow me to offer an outsider's perspective. The following remarks are taken from my expository article arXiv:1007.4426.
First recall that the proportion of primes $p$ for which $T^2+1$ has no roots (resp. two distinct roots) in $\mathbf{F}_p$ is $1/2$ (resp. $1/2$), and that the proportion of $p$ for which $T^3-T-1$ has no roots (resp. exactly one root, resp. three distinct roots) in $\mathbf{F}_p$ is $1/3$ (resp. $1/2$, resp. $1/6$).
What is the analogue of the foregoing for the number of roots $N_p(f)$ of $f=S^2+S-T^3+T^2$ in $\mathbf{F}_p$? A theorem of Hasse implies that $a_p=p-N_p(f)$ lies in the interval $[-2\sqrt p,+2\sqrt p]$, so $a_p/2\sqrt p$ lies in $[-1,+1]$. What is the proportion of primes $p$ for which $a_p/2\sqrt p$ lies in a given interval $I\subset[-1,+1]$? It was predicted by Sato (on numerical grounds) and Tate (on theoretical grounds), not just for this $f$ but for all $f\in\mathbf{Z}[S,T]$ defining an "elliptic curve without complex multiplications", that the proportion of such $p$ is equal to the area $$ {2\over\pi}\int_{I}\sqrt{1-x^2}\;dx. $$ of the portion of the unit semicircle projecting onto $I$. The Sato-Tate conjecture for elliptic curves over $\mathbf{Q}$ was settled in 2008 by Clozel, Harris, Shepherd-Barron and Taylor.
There is an analogue for "higher weights". Let $c_n$ (for $n>0$) be the coefficient of $q^n$ in the formal product $$ \eta_{1^{24}}= q\prod_{k=1}^{+\infty}(1-q^{k})^{24}=0+1.q^1+\sum_{n>1}c_nq^n. $$ In 1916, Ramanujan had made some deep conjectures about these $c_n$; some of them, such as $c_{mm'}=c_mc_{m'}$ if $\gcd(m,m')=1$ and $$ c_{p^r}=c_{p^{r-1}}c_p-p^{11}c_{p^{r-2}} $$ for $r>1$ and primes $p$, which can be more succintly expressed as the identity $$ \sum_{n>0}c_nn^{-s}=\prod_p{1\over 1-c_p.p^{-s}+p^{11}.p^{-2s}} $$ when the real part of $s$ is $>(12+1)/2$, were proved by Mordell in 1917. The last of Ramanujan's conjectures was proved by Deligne only in the 1970s: for every prime $p$, the number $t_p=c_p/2p^{11/2}$ lies in the interval $[-1,+1]$.
All these properties of the $c_n$ follow from the fact that the corresponding function $F(\tau)=\sum_{n>0}c_ne^{2i\pi\tau.n}$ of a complex variable $\tau=x+iy$ ($y>0$) in $\mathfrak{H}$ is a "primitive eigenform of weight $12$ and level $1$" (which basically amounts to the identity $F(-1/\tau)=\tau^{12}F(\tau)$).
(Incidentally, Ramanujan had also conjectured some congruences satisfied by the $c_p$ modulo $2^{11}$, $3^7$, $5^3$, $7$, $23$ and $691$, such as $c_p\equiv1+p^{11}\pmod{691}$ for every prime $p$; they were at the origin of Serre's modularity conjecture recently proved by Khare-Wintenberger and Kisin.)
We may therefore ask how these $t_p=c_p/2p^{11/2}$ are distributed: for example are there as many primes $p$ with $t_p\in[-1,0]$ as with $t_p\in[0,+1]$? Sato and Tate predicted in the 1960s that the precise proportion of primes $p$ for which $t_p\in I$, for given interval $I\subset[-1,+1]$, is $$ {2\over\pi}\int_{I}\sqrt{1-x^2}\;dx. $$ This is expressed by saying that the $t_p=c_p/2p^{11/2}$ are equidistributed in the interval $[-1,+1]$ with respect to the measure $(2/\pi)\sqrt{1-x^2}\;dx$. Recently Barnet-Lamb, Geraghty, Harris and Taylor have proved that such is indeed the case.
Their main theorem implies many such equidistribution results, including the one recalled above for the elliptic curve $S^2+S-T^3+T^2=0$; for an introduction to such density theorems, see Taylor's review article Reciprocity laws and density theorems.
Best Answer
No originality here, but I would tell the story as follows. Consider the subgroup $(\mathbb{Z}\times\mathbb{R})/\mathbb{Z}$ of $(\mathbb{Z}_p\times\mathbb{R})/\mathbb{Z}$. It is dense, because $\mathbb{Z}$ is dense in $\mathbb{Z}_p$. It is also connected, because it is isomorphic to $\mathbb{R}$ (with a coarser topology than the standard one). Therefore, in the group $(\mathbb{Z}_p\times\mathbb{R})/\mathbb{Z}$, the connected component of the identity is dense, whence it equals $(\mathbb{Z}_p\times\mathbb{R})/\mathbb{Z}$.
The same proof works for $(\widehat{\mathbb{Z}}\times\mathbb{R})/\mathbb{Z}$.