For a given -adic sheaf on a commutative algebraic group over a finite field and an integer , we define the th local norm -function of at a point and prove its rationality. This function gives information on the sum of the local Frobenius traces of over the points of (where is the extension of degree of ) with norm . For the -dimensional affine line or the torus, these sums can in turn be used to estimate the number of rational points on curves or the absolute value of exponential sums which are invariant under a large group of translations or homotheties.
"Rationality of trace and norm -functions." Duke Math. J. 161 (9) 1751 - 1795, 15 June 2012. https://doi.org/10.1215/00127094-1593371