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Thread: What's the determinant of this lattice?

  1. #1
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    What's the determinant of this lattice?

    Say $\displaystyle a$ and $\displaystyle b$ are both coprime to $\displaystyle c$. Let $\displaystyle L$ be the lattice

    $\displaystyle L = \{ (i,j) \in \mathbb{Z}^2: a^i b^j = 1 \text{ in } (\frac{\mathbb{Z}}{c \mathbb{Z}})^{\times} \}$.

    What is the determinant of $\displaystyle L$? I think the answer may be that it's the cardinality of the subset of $\displaystyle (\frac{\mathbb{Z}}{c \mathbb{Z}})^{\times} $ generated by $\displaystyle a$ and $\displaystyle b$ but I can't prove it. Is this a standard result?
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  2. #2
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    Quote Originally Posted by amanda19 View Post
    Say $\displaystyle a$ and $\displaystyle b$ are both coprime to $\displaystyle c$. Let $\displaystyle L$ be the lattice

    $\displaystyle L = \{ (i,j) \in \mathbb{Z}^2: a^i b^j = 1 \text{ in } (\frac{\mathbb{Z}}{c \mathbb{Z}})^{\times} \}$.

    What is the determinant of $\displaystyle L$? I think the answer may be that it's the cardinality of the subset of $\displaystyle (\frac{\mathbb{Z}}{c \mathbb{Z}})^{\times} $ generated by $\displaystyle a$ and $\displaystyle b$ but I can't prove it. Is this a standard result?
    What does any of this mean? It seems like this is bound to be an infite set, so how does one define the determinant here? What is $\displaystyle \displaystyle \frac{\mathbb{Z}}{c\mathbb{Z}}$, a weird way of writing $\displaystyle \mathbb{Z}/c\mathbb{Z}$?
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  3. #3
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    Yes, $\displaystyle \frac{\mathbb{Z}}{c\mathbb{Z}}$ means $\displaystyle \mathbb{Z}/c\mathbb{Z}$, but I'll write $\displaystyle \mathbb{Z}/c\mathbb{Z}$ if it's confusing. And yes, the lattice is going to be an infinite set of points (and then the determinant is defined in the standard way - see eg. Lattice (group) - Wikipedia, the free encyclopedia )
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