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1. Every finite abelian group is the group of rational points of an ordinary abelian variety over 𝔽₂, 𝔽₃ and 𝔽₅

   https://doi.org/10.1090/proc/16127  

   Marseglia, Stefano; Springer, Caleb (February 2023, Proceedings of the American Mathematical Society)

   We show that every finite abelian group occurs as the group of rational points of an ordinary abelian variety over ${\mathbb{F}}_2$, ${\mathbb{F}}_3$and ${\mathbb{F}}_5$. We produce partial results for abelian varieties over a general finite field  ${\mathbb{F}}_q$. In particular, we show that certain abelian groups cannot occur as groups of rational points of abelian varieties over ${\mathbb{F}}_q$when $q$is large. Finally, we show that every finite cyclic group arises as the group of rational points of infinitely many simple abelian varieties over  ${\mathbb{F}}_2$. 

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2. The distribution of negative eigenvalues of Schrödinger operators on asymptotically hyperbolic manifolds

   https://doi.org/10.4171/JST/561  

   Sá_Barreto, Antônio; Wang, Yiran (May 2025, Journal of Spectral Theory)

   We study the asymptotic behavior of the counting function of negative eigenvalues of Schrödinger operators with real valued potentials which decay at infinity on asymptotically hyperbolic manifolds. We establish conditions on the rate of decay of the potential that determine if there are finitely or infinitely many negative eigenvalues. In the latter case, they may only accumulate at zero and we obtain the asymptotic behavior of the counting function of eigenvalues in an interval(-\infty, -E)asE\rightarrow 0. 

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3. Multiplicative dependence of rational values modulo approximate finitely generated groups

   https://doi.org/10.1017/S0305004124000173  

   BÉRCZES, ATTILA; BUGEAUD, YANN; GYŐRY, KÁLMÁN; MELLO, JORGE; OSTAFE, ALINA; SHA, MIN (July 2024, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract In this paper, we establish some finiteness results about the multiplicative dependence of rational values modulo sets which are ‘close’ (with respect to the Weil height) to division groups of finitely generated multiplicative groups of a number fieldK. For example, we show that under some conditions on rational functions$$f_1, \ldots, f_n\in K(X)$$, there are only finitely many elements$$\alpha \in K$$such that$$f_1(\alpha),\ldots,f_n(\alpha)$$are multiplicatively dependent modulo such sets. 

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4. An almost linear time algorithm testing whether the Markoff graph modulo p is connected

   https://doi.org/10.1007/s40993-024-00592-9  

   Brown, Colby Austin (March 2025, Research in Number Theory)

   Abstract The Markoff graphs modulopwere proven by Chen (Ann Math 199(1), 2024) to be connected for all but finitely many primes, and Baragar (The Markoff equation and equations of Hurwitz. Brown University, 1991) conjectured that they are connected for all primes, equivalently that every solution to the Markoff equation moduloplifts to a solution over$$\mathbb {Z}$$ $Z$. In this paper, we provide an algorithmic realization of the process introduced by Bourgain et al. [arXiv:1607.01530] to test whether the Markoff graph modulopis connected for arbitrary primes. Our algorithm runs in$$o(p^{1 + \epsilon })$$ $o\left(p^{1+ϵ}\right)$time for every$$\epsilon > 0$$ $ϵ>0$. We demonstrate this algorithm by confirming that the Markoff graph modulopis connected for all primes less than one million. 

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5. A new proof of Chen’s theorem for Markoff graphs

   https://doi.org/10.1007/s00222-025-01346-9  

   Martin, Daniel E (August 2025, Inventiones mathematicae)

   Abstract In 2021, Chen proved that the size of any connected component of the Markoff mod$$p$$ $p$graph is divisible by$$p$$ $p$. In combination with the work of Bourgain, Gamburd, and Sarnak, Chen’s result resolves a conjecture of Baragar for all but finitely many primes: the Markoff mod$$p$$ $p$graph is connected. In particular, strong approximation for Markoff triples holds for all but finitely many primes. We provide an alternative proof of Chen’s theorem. 

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