Abstract
This paper introduces the twin concepts of sampling spaces and arithmetic dimension, which together address the question of how to count the number, or measure the size of, families of objects over a number field or global field. It can be seen as an alternative to coarse moduli schemes, with more attention to the arithmetic properties of the ambient base field, and which leads to concrete algorithmic applications and natural height functions. It is compared to the definition of essential dimension.
To Serge. It was an honor to be your favorite sophomore.
Mathematics Subject Classification (2010): 11G35
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Notes
- 1.
Thanks to René Schoof for these models.
- 2.
This idea of compression in the case of a faithful representation comes from the papers of Reichstein.
- 3.
What if we can’t solve for λ? This happens when one and more of X, Y, and Z is zero. Two being zero would mean the point is rational, which gives us the trivial cocycle. Acting by T above if necessary (which only changes the cocycle mod coboundaries) we can assume our point looks like (0; 1; α), and so the corresponding cocycle ξ sends σ to (0; 1; σ(α)), but on the other hand sends it to itself acted on by ξ(σ) ∈ E[3]. We see then that ξ(σ) = i ⋅S for some i, so ξ pulls back to \({H}^{1}({G}_{K}, \mathbb{Z}/3\mathbb{Z})\cong{K}^{{_\ast}}/{K}^{{_\ast}3}.\) However these cocycles are also covered by the points (1; α; α2), which have well-defined λ.
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O’Neil, C. (2012). Sampling spaces and arithmetic dimension. In: Goldfeld, D., Jorgenson, J., Jones, P., Ramakrishnan, D., Ribet, K., Tate, J. (eds) Number Theory, Analysis and Geometry. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-1260-1_23
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