Geometry-of-numbers methods are extended to count orbits in coregular spaces over arbitrary global fields, yielding bounds on average ranks and Selmer sizes for elliptic curves and hyperelliptic Jacobians.
Siad.Monogenic fields with odd class number Part II: even degree
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Establishes that averages of distinct n-periodic points for maps φ_{p^ℓ,c} and φ_{(p-1)^ℓ,c} over Z_p and F_p[t] are unbounded/zero or 1/2/0 as c varies, then derives counting results for irreducibles, zeta functions, and L-functions.
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Geometry-of-numbers methods over global fields II: Coregular representations
Geometry-of-numbers methods are extended to count orbits in coregular spaces over arbitrary global fields, yielding bounds on average ranks and Selmer sizes for elliptic curves and hyperelliptic Jacobians.
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Counting the number of $n$-periodic $\mathbb{Z}_{p}$-and $\mathbb{F}_{p}[t]$-points of a discrete dynamical system with applications from arithmetic statistics, VI
Establishes that averages of distinct n-periodic points for maps φ_{p^ℓ,c} and φ_{(p-1)^ℓ,c} over Z_p and F_p[t] are unbounded/zero or 1/2/0 as c varies, then derives counting results for irreducibles, zeta functions, and L-functions.