# Tagged Questions

69 views

### Expression and growth bound for $r_{p^m,k}(n)$

Let's define , $$R_{p^m,k}(n)=\#\{(a_1,\dots,a_k)\in\mathbb{Z}^k:\sum_{i=1}^ka_i^2\le n \ \text{and} \ p^m|\sum_{i=1}^ka_i^2\}$$ what will be growth bound of $R_{p^m,k}(n)$? This can be thought as a ...
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### What does the Jacquet-Langlands correspondence say about quaternion algebras of class number one?

If F is a totally real number field of degree n, and A is a definite quaternion algebra over F, I understand (not really) the Jacquet Langlands correspondence to construct a modular form in n ...
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### Which quaternion algebras have class number one?

Over Q, the definite quaternion algebras with a unique conjugacy class of maximal orders, i.e. "with class number one", are those with discriminant 2,3,5,7, and 13. Three questions: What is a ...
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### How do you find maximal orders in quaternion algebras?

Let A be the 4-dimensional algebra over Q with basis 1,i,j,k, and multiplication table $$i^2 = -1 \quad j^2 = -11 \quad k^2 = -11 \quad ij = k \quad jk = 11i \quad ki = j$$ So, A is the unique ...
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### Question about the definition of the genus 0 curves in Gross' paper “Heights and the Special values of L-series”

Let $N \in \mathbb{Z}$ be a prime number, and let $B = \left( \dfrac{a, b}{\mathbb{Q}} \right)$ be the unique quaternion algebra over $\mathbb{Q}$ ramified at $N$ and at $\infty$. Then, in section 3 ...
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### Octonions and the dance of the seven veils

Let us take the octonions as having all integer coefficients and the multiplication table at BAEZ We have a standard conjugation operator with $\bar{1} = 1$ and $\bar{e_i}= - e_i,$ extend by ...