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Case g = 2 and q = 5. As pointed out in remark 3.9, there is an issue when q + 1 − t < 0 (for example when t = 7). Indeed, H (q, 7) = 0 because q + 1 − t represents the number of Fq-rational points of a curve. Instead, both ν (q, 7) ≈ 0.0009 and ν∞(q, 7) ≈ 0.0011 are strictly positive.
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We discuss, in a non-Archimedean setting, the distribution of the coefficients of L-polynomials of curves of genus g over $\mathbb{F}_q$ . Among other results, this allows us to prove that the $\mathbb{Q}$ -vector space spanned by such characteristic polynomials has dimension g + 1. We also state a conjecture about the Archimedean distribution of t...
Context in source publication
Context 1
... of the distribution H (q, t) An advantage of working with PPAVs rather than curves is that the former always admit quadratic twists, which implies that the distribution of their traces is always symmetric around 0. This is further indication that perhaps conjecture 3.4 is more natural for the family of PPAVs. In fact, we remark that while ν (q, t) is symmetric (that is, ν (q, −t) = ν (q, t)), this is not necessarily the case for H (q, t) as soon as g ≥ 3, as one can see for example in [9, figure 4], or below in our own figure 3. See also [9, § 5] for a more extensive discussion of the asymmetry of H (q, t). In particular, we note again that one cannot have an exact equality H (q, t) = ν (q, t) for general g, because the right-hand side is easily seen to be symmetric. ...
