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For an algebraic number $alpha$ and $gammain mathbb{R}$, $h(alpha)$ be the (logarithmic) Weil height, and $h_gamma(alpha)=(mathrm{deg}alpha)^gamma h(alpha)$ be the $gamma$-weighted (logarithmic) Weil height of $alpha$. Let $f:overline{mathbb{Q}}to [0,infty)$ be a function on the algebraic numbers $overline{mathbb{Q}}$, and let $Ssubset overline{mathbb{Q}}$. The Northcott number $mathcal{N}_f(S)$ of $S$, with respect to $f$, is the infimum of all $Xgeq 0$ such that ${alpha in S; f(alpha)< X}$ is infinite. This paper studies the set of Northcott numbers $mathcal{N}_f(mathcal{O})$ for subrings of $overline{mathbb{Q}}$ for the house, the Weil height, and the $gamma$-weighted Weil height. We show: (1) Every $tgeq 1$ is the Northcott number of a ring of integers of a field w.r.t. the house. (2) For each $tgeq 0$ there exists a field with Northcott number in $ [t,2t]$ w.r.t. the Weil height $h(cdot)$. (3) For all $0leq gammaleq 1$ and $gamma<gamma$ there exists a field $K$ with $mathcal{N}_{h_{gamma}}(K)=0$ and $mathcal{N}_{h_gamma}(K)=infty$. For $(1)$ we provide examples that satisfy an analogue of Julia Robinons property (JR), examples that satisfy an analogue of Vidaux and Videlas isolation property, and examples that satisfy neither of those. Item $(2)$ concerns a question raised by Vidaux and Videla due to its direct link with decidability theory via the Julia Robinson number. Item (3) is a strong generalisation of the known fact that there are fields that satisfy the Lehmer conjecture but which are not Bogomolov in the sense of Bombieri and Zannier.
We prove a new Bertini-type Theorem with explicit control of the genus, degree, height, and the field of definition of the constructed curve. As a consequence we provide a general strategy to reduce certain height and rank estimates on abelian variet
Let $q$ be an odd power of a prime $pin mathbb{N}$, and $mathrm{PPSP}(sqrt{q})$ be the finite set of isomorphism classes of principally polarized superspecial abelian surfaces in the simple isogeny class over $mathbb{F}_q$ corresponding to the real W
This paper explores analogies between the Weil proof of the Riemann hypothesis for function fields and the geometry of the adeles class space, which is the noncommutative space underlying Connes spectral realization of the zeros of the Riemann zeta f
We construct special cycles on the moduli stack of unitary shtukas. We prove an identity between (1) the r-th central derivative of non-singular Fourier coefficients of a normalized Siegel--Eisenstein series, and (2) the degree of special cycles of v
We investigate the $p$-adic valuation of Weil sums of the form $W_{F,d}(a)=sum_{x in F} psi(x^d -a x)$, where $F$ is a finite field of characteristic $p$, $psi$ is the canonical additive character of $F$, the exponent $d$ is relatively prime to $|F^t