All Questions

Let $L/\mathbb{Q}$ be a finite Galois extension, and let $\mathcal{O}_L$ be the ring of integers of $L$. We have $tr_{L/\mathbb{Q}}(\mathcal{O}_L)=d\mathbb{Z}$ for some $d\geq 1.$ Fact. $d=1$ if ...

Let $\mathbb{F}_{q^\ell}/\mathbb{F}_{q}$ be the extension of finite filed $\mathbb{F}_{q}$, where $\ell$ be a odd prime and $(\neq q)$. Take $\zeta\in\mathbb{F}_{q^\ell}$. Does there exist different $...

Let $F$ be a finite extension of $\mathbb{Q}_p$ (p is prime) and $K/F$ be a unramified extension of prime degree $\ell (\neq p)$. Denote $\mu_K$ be the group of roots of unity in $K.$ Does there exist ...

I am currently working on level issues in the division central simple algebra case, say $D$ over a local non-archimedean field $F$ (e.g. $\mathbf{Q}_p$). Let say that $\mathcal{O}_D$ and $\mathcal{O}...

Let $n\in\mathbb N$ and $p$ be any prime. Denote by $\mathbb Q_p$ the $p$-adic numbers. For a field extension $L/K$ denote by $Tr_{L/K}$ the corresponding trace function. Let $\zeta_n$ be a primitve $...

If RH is not true, we have that Weil's explicit formula still holds: $$ \sum_{\gamma} h(\gamma) = h(i/2)+h(-i/2)-2 \sum_{n=1}^{\infty} \frac{ \Lambda(n)}{ \sqrt n}g(logn)+\frac{1}{2\pi} \int_{-\infty}...