Pär Kurlberg

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The number of points on a hyperelliptic curve over a field of q elements may be expressed as q +1+S where S is a certain character sum. We study fluctuations of S as the curve varies over a large family of hyperelliptic curves of genus g. For fixed genus and growing q, Katz and Sarnak showed that S/ √ q is distributed as the trace of a random 2g×2g unitary(More)
Let K be a number field, let φ ∈ K (t) be a rational map of degree at least 2, and let α, β ∈ K . We show that if α is not in the forward orbit of β, then there is a positive proportion of primes p of K such that α mod p is not in the forward orbit of β mod p. Moreover, we show that a similar result holds for several maps and several points. We also present(More)
The phase space density f of a dilute gas evolves according to the Boltzmann equation. In the physically relevant case, the gas would be confined to a subset Ω ⊂ R, and then f(x, v, t) : Ω×R3×R+ → R, where x denotes a position in space, v ∈ R is a velocity, and t denotes the time. From a mathematical point of view, it is equally natural to consider the(More)
Let f1, . . . , fg ∈ C(z) be rational functions, let Φ = (f1, . . . , fg) denote their coordinatewise action on (P1)g, let V ⊂ (P1)g be a proper subvariety, and let P be a point in (P1)g(C). We show that if S = {n > 0 : Φn(P ) ∈ V (C)} does not contain any infinite arithmetic progressions, then S must be a very sparse set of integers. In particular, for any(More)