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In order to add my bit to the already rich content on this site, here is a nice family of galoisian extensions $K_l|{\bf Q}$ with group ${\rm GL}_2({\bf F}_l)$ (indexed by primes $l\neq5$) for which a reciprocity law'' can be written down explicitly. I've come across this family recently while writing an expository article.

Let $E$ be the elliptic curve (over $\bf Q$) of conductor $11$ defined by $y^2+y=x^3-x^2$, with associated modular form $$\eta_{1^2,11^2}=q\prod_{k>0}(1-q^k)^2(1-q^{11k})^2=\sum_{n>0}c_nq^n.$$ Let $K_l={\bf Q}(E[l])$, which is thus galoisian over $\bf Q$ and unramified at every prime $p\neq11,l$.

One can deduce from cor.1 on p.308 of Serre (Inventiones 1972) that for every prime $l\neq5$, the representation $$\rho_{E,l}:{\rm Gal}(K_l|{\bf Q})\rightarrow{\rm GL}_2({\bf F}_l)$$ we get upon choosing an ${\bf F}_l$-basis of $E[l]$ is an isomorphism; cf. the online notes on Serre's conjecture by Ribet and Stein. Shimura did this for $l\in[9,97]$ (Crelle 1966).

Suppose henceforth that $l$ is a prime $\neq5$ and that $p$ is a prime $\neq11,l$. The characteristic polynomial of $\rho_{E,l}({\rm Frob}_p)\in{\rm GL}_2({\bf F}_l)$ is $$T^2-\bar c_pT+\bar p\in{\bf F}_l[X].$$ The prime $p$ splits completely in $K_l$ if and only if ${\rm Frob}_p=1$ in ${\rm Gal}(K_l|{\bf Q})$, which happens if and only if $$\rho_{E,l}({\rm Frob}_p)=\pmatrix{1&0\cr0&1}.$$ If so, then $p,c_p\equiv1,2\pmod l$ but not conversely, for the matrix $\displaystyle\pmatrix{1&1\cr0&1}$ also has the characteristic polynomial $T^2-\bar2T+\bar1$. But these congruences on $p,c_p$ do rule out an awful lot of primes as not splitting completely in $K_l$.

In summary, we have the following reciprocity law" for $K_l$ : $$\hbox{(p splits completely in K_l)} \quad\Leftrightarrow\quad E_p[l]\subset E_p({\bf F}_p),$$ where $E_p$ is the reduction of $E$ modulo $p$. Indeed, reduction modulo $p$ identifies $E[l]$ with $E_p[l]$ and the action of ${\rm Frob}_p$ on the former space with the action of the canonical generator $\varphi_p\in{\rm Gal}(\bar{\bf F}_p|{\bf F}_p)$ on the latter space. To say that $\varphi_p$ acts trivially on $E_p[l]$ is the same as saying that $E_p[l]$ is contained in the ${\bf F}_p$-rational points of $E_p$. The analogy with the multiplicative group $\mu$ is perfect: $$\hbox{(p\neq l splits completely in {\bf Q}(\mu[l]))} \quad\Leftrightarrow\quad \mu_p[l]\subset \mu_p({\bf F}_p)$$ ($\Leftrightarrow l|p-1\Leftrightarrow p\equiv1\pmod l$), where $\mu_p$ is not the $p$-torsion of $\mu$ but the reduction of $\mu$ modulo $p$.

I requested Tim Dokchitser to compute the first ten $p$ which split completely in $K_7$, and his instantaneous response was 4831, 22051, 78583, 125441, 129641, 147617, 153287, 173573, 195581, and 199501.

It is true that all this (except the list of these ten primes) was known before Serre's conjecture was proved (2006--9) or even formulated (1973--87), but I find this example a very good illustration of the kind of reciprocity laws it provides.

I hope you enjoyed it as much as I did.

4 added 337 characters in body

In order to add my bit to the already rich content on this site, here is a nice family of galoisian extensions $K_l|{\bf Q}$ with group ${\rm GL}_2({\bf F}_l)$ (indexed by primes $l\neq5$) for which a reciprocity law'' can be written down explicitly. I've come across this family recently while writing an expository article.

Let $E$ be the elliptic curve (over $\bf Q$) of conductor $11$ defined by $y^2+y=x^3-x^2$, with associated modular form $$\eta_{1^2,11^2}=q\prod_{k>0}(1-q^k)^2(1-q^{11k})^2=\sum_{n>0}c_nq^n.$$ Let $K_l={\bf Q}(E[l])$, which is thus galoisian over $\bf Q$ and unramified at every prime $p\neq11,l$.

One can deduce from cor.1 on p.308 of Serre (Inventiones 1972) that for every prime $l\neq5$, the representation $$\rho_{E,l}:{\rm Gal}(K_l|{\bf Q})\rightarrow{\rm GL}_2({\bf F}_l)$$ we get upon choosing an ${\bf F}_l$-basis of $E[l]$ is an isomorphism; cf. the online notes on Serre's conjecture by Ribet and Stein. Shimura did this for $l\in[9,97]$ (Crelle 1966).

Suppose henceforth that $l$ is a prime $\neq5$ and that $p$ is a prime $\neq11,l$. The characteristic polynomial of $\rho_{E,l}({\rm Frob}_p)\in{\rm GL}_2({\bf F}_l)$ is $$T^2-\bar c_pT+\bar p\in{\bf F}_l[X].$$ The prime $p$ splits completely in $K_l$ if and only if ${\rm Frob}_p=1$ in ${\rm Gal}(K_l|{\bf Q})$, which happens if and only if $$\rho_{E,l}({\rm Frob}_p)=\pmatrix{1&0\cr0&1}.$$ If so, then $p,c_p\equiv1,2\pmod l$ but not conversely, for the matrix $\displaystyle\pmatrix{1&1\cr0&1}$ also has the characteristic polynomial $T^2-\bar2T+\bar1$. But these congruences on $p,c_p$ do rule out an awful lot of primes as not splitting completely in $K_l$.

In summary, we have the following reciprocity law" for $K_l$ : $$\hbox{(p splits completely in K_l)} \quad\Leftrightarrow\quad E_p[l]\subset E_p({\bf F}_p),$$ where $E_p$ is the reduction of $E$ modulo $p$. Indeed, reduction modulo $p$ identifies $E[l]$ with $E_p[l]$ and the action of ${\rm Frob}_p$ on the former space with the action of the canonical generator $\varphi_p\in{\rm Gal}(\bar{\bf F}_p|{\bf F}_p)$ on the latter space. To say that $\varphi_p$ acts trivially on $E_p[l]$ is the same as saying that $E_p[l]$ is contained in the ${\bf F}_p$-rational points of $E_p$. The analogy with the multiplicative group $\mu$ is perfect: $$\hbox{(p\neq l splits completely in {\bf Q}(\mu[l]))} \quad\Leftrightarrow\quad \mu_p[l]\subset \mu_p({\bf F}_p)$$ ($\Leftrightarrow l|p-1\Leftrightarrow p\equiv1\pmod l$), where $\mu_p$ is not the $p$-torsion of $\mu$ but the reduction of $\mu$ modulo $p$.

I requested Tim Dokchitser to compute the first ten $p$ which split completely in $K_7$, and his instantaneous response was 4831, 22051, 78583, 125441, 129641, 147617, 153287, 173573, 195581, and 199501.

It is true that all this (except the list of these ten primes) was known before Serre's conjecture was proved (2006--9) or even formulated (1973--87), but I find this example a very good illustration of the kind of reciprocity laws it provides.

I hope you enjoyed it as much as I did.

3 replaced some l's with p's

In order to add my bit to the already rich content on this site, here is a nice family of galoisian extensions $K_l|{\bf Q}$ with group ${\rm GL}_2({\bf F}_l)$ (indexed by primes $l\neq5$) for which a reciprocity law'' can be written down explicitly. I've come across this family recently while writing an expository article.

Let $E$ be the elliptic curve (over $\bf Q$) of conductor $11$ defined by $y^2+y=x^3-x^2$, with associated modular form $$\eta_{1^2,11^2}=q\prod_{k>0}(1-q^k)^2(1-q^{11k})^2=\sum_{n>0}c_nq^n.$$ Let $K_l={\bf Q}(E[l])$, which is thus galoisian over $\bf Q$ and unramified at every prime $p\neq11,l$.

One can deduce from cor.1 on p.308 of Serre (Inventiones 1972) that for every prime $l\neq5$, the representation $$\rho_{E,l}:{\rm Gal}(K_l|{\bf Q})\rightarrow{\rm GL}_2({\bf F}_l)$$ we get upon choosing an ${\bf F}_l$-basis of $E[l]$ is an isomorphism; cf. the online notes on Serre's conjecture by Ribet and Stein. Shimura did this for $l\in[9,97]$ (Crelle 1966).

Suppose henceforth that $l$ is a prime $\neq5$ and that $p$ is a prime $\neq11,l$. The characteristic polynomial of $\rho_{E,l}({\rm Frob}_p)\in{\rm GL}_2({\bf F}_l)$ is $$T^2-\bar c_pT+\bar p\in{\bf F}_l[X].$$ The prime $p$ splits completely in $K_l$ if and only if ${\rm Frob}_p=1$ in ${\rm Gal}(K_l|{\bf Q})$, which happens if and only if $$\rho_{E,l}({\rm Frob}_p)=\pmatrix{1&0\cr0&1}.$$ If so, then $p,c_p\equiv1,2\pmod l$ but not conversely, for the matrix $\displaystyle\pmatrix{1&1\cr0&1}$ also has the characteristic polynomial $T^2-\bar2T+\bar1$. But these congruences on $p,c_p$ do rule out an awful lot of primes as not splitting completely in $K_l$.

In summary, we have the following reciprocity law" for $K_l$ : $$\hbox{(p splits completely in K_l)} \quad\Leftrightarrow\quad E_p[l]\subset E_p({\bf F}_l)F}_p),$$ where $E_p$ is the reduction of $E$ modulo $p$. Indeed, reduction modulo $p$ identifies $E[l]$ with $E_p[l]$ and the action of ${\rm Frob}_p$ on the former space with the action of the canonical generator $\varphi_p\in{\rm Gal}(\bar{\bf F}_l|{\bf F}_l)$ F}_p|{\bf F}_p)$on the latter space. To say that$\varphi_p$acts trivially on$E_p[l]$is the same as saying that$E_p[l]$is contained in the${\bf F}_l$-rational F}_p$-rational points of $E_p$.

I requested Tim Dokchitser to compute the first ten $p$ which split completely in $K_7$, and his instantaneous response was 4831, 22051, 78583, 125441, 129641, 147617, 153287, 173573, 195581, and 199501.

It is true that all this (except the list of these ten primes) was known before Serre's conjecture was proved (2006--9) or even formulated (1973--87), but I find this example a very good illustration of the kind of reciprocity laws it provides.

I hope you enjoyed it as much as I did.

2 added 606 characters in body
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