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Let $n=\prod_{i=1}^np_i^{e_i}$ with $p_i$ different prime numbers and $e_i$ positive integers. Given a Dirichlet character $\chi$ modulo $n$ we can define the characters $\chi_i$ (modulo $p_i^{e_i}$) by $\chi_i(a):=\chi(a_i)$ where $a_i$ is such that $a_i\equiv a \mod p_i^{e_i}$ and $a_i\equiv 1\mod p_j^{e_j}$ for $j\neq i$ (by using CRT). Using these characters one can factor $\chi$ uniquely as $\chi=\chi_1\ldots\chi_r$. Let $c_{\phi}$ be the conductor of the Dirichlet character $\phi$ modulo $k$, i.e. the smallest number $d$ such that $\phi(a)=1$ for all $a\in\mathbb{Z}$ satisfying $\gcd(a,k)=1$ and $a\equiv1\mod d$.

I want to show that $c_{\chi}=c_{\chi_1}\ldots c_{\chi_r}$ and I can prove that $c_{\chi}\mid c_{\chi_1}\ldots c_{\chi_r}$. So it only remains to show that we cannot have $c_{\chi}<c_{\chi_1}\ldots c_{\chi_r}$. In the book "Modular Forms, a Computational Approach'' by William A. Stein there is a solution (see page 72 of https://books.google.nl/books?id=blaZAwAAQBAJ&printsec=frontcover&hl=nl#v=onepage&q&f=false) which I don't fully understand. The argument starts by showing that $c_{\chi}\mid c_{\chi_1}\ldots c_{\chi_r}$ and then proceeds by assuming that there is a prime $p$ such that $\text{ord}_{p}(c_{\chi})<\text{ord}_p(c_{\chi_j})$ for some $j$ to arrive at a contradiction. The proof says that if this is the case then we can factor $\chi$ as a product of (local) prime power characters differently (see first paragraph). Which gives the desired contradiction.

My question is why we can factor $\chi$ differently if we assume $\text{ord}_{p}(c_{\chi})<\text{ord}_p(c_{\chi_j})$?

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