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Olivier
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Why is $n_{n^2-1}$ the smallest graph that clearly shows the structure of multiplication by $n$?
@HansStricker What I'd like to know is how you drew the pictures (and if that is not too much to ask, if you could send me a couple of the large ones as large png?).
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Why is $n_{n^2-1}$ the smallest graph that clearly shows the structure of multiplication by $n$?
Let me add that this is very easy to see for the smallish number you displayed (for instance for $5\leq m\leq 8$): one sees clearly the involutive aspect and the fact that each element of the smallest group of $n+1$ terms is sent to a different group, hence the nice display with $n-1$ cusps.
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Why is $n_{n^2-1}$ the smallest graph that clearly shows the structure of multiplication by $n$?
Because $n^2-1=(n-1)(n+1)$, multiplication by $n$ modulo $n^2-1$ is an involution that sends the $n+1$ terms $0,2,\cdots,n$ each to one of the $n-1$ another $n$-ade (decade for $n=10$). This it seems to me, entirely explains the nice (but trivially simple) geometric design observed.
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Why is $n_{n^2-1}$ the smallest graph that clearly shows the structure of multiplication by $n$?
So I don't quite understand what counts at undeniably displaying $n-1$ cusps but, as (hopefully) intelligible to every school kid, $n^2-1=(n-1)(n+1)$, so you have $n-1$ group linked by two-way edges (reflecting the fact that $n^2$ is $1$), which produces a nice figure as in your pictures. For smaller values, there is a perturbation related to the fact that either $n^2$ is not $1$ which destroys somewhat the symmetry or there are not enough groups (or both). To be more mathematically precise, I would need to be able to give a mathematical content to "undeniably displaying cusps".
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How does the Bernstein-Zelevinsky construction of irreducibles from supercuspidals parallel the representations of the Weil-Deligne group?
Oh, I see that I gave an identical answer to yours a couple of minutes after you. Sorry about that.
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Explicit elements of the first cohomology of modular curves
Do you want classes in the étale cohomology of $M$ over $\mathbb Q$? or over the algebraic closure? (If the latter a weight $k+2$ cuspform $f$ yields such an element via the holomorphic one-form $f(q)dq(q,1)^k$ attached to $f$).
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Some questions on the $p$-adic properties of special $L$-values
@DavidLoeffler Well, thanks to you David and all the other people that implemented built in functions in SAGE that pedestrians like me can then use to produce such examples.
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Generalizing a problem to make it easier
and requesting that nobody gives away the solution any more than it is already given away by my mentioning it here and then I die poor, alone and without a proof that the square of a repetitive number is repetitive.
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