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1. An effective Chabauty–Kim theorem

   https://doi.org/10.1112/S0010437X19007243  

   Balakrishnan, Jennifer S.; Dogra, Netan (June 2019, Compositio Mathematica)

   The Chabauty–Kim method allows one to find rational points on curves under certain technical conditions, generalising Chabauty’s proof of the Mordell conjecture for curves with Mordell–Weil rank less than their genus. We show how the Chabauty–Kim method, when these technical conditions are satisfied in depth 2, may be applied to bound the number of rational points on a curve of higher rank. This provides a non-abelian generalisation of Coleman’s effective Chabauty theorem. 

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2. Rational points on hyperelliptic Atkin-Lehner quotients of modular curves and their coverings

   https://doi.org/10.1007/s40993-022-00388-9  

   Adžaga, Nikola; Chidambaram, Shiva; Keller, Timo; Padurariu, Oana (October 2022, Research in Number Theory)

   Abstract We complete the computation of all$$\mathbb {Q}$$ $Q$-rational points on all the 64 maximal Atkin-Lehner quotients$$X_0(N)^*$$ $X_0{\left(N\right)}^{\ast }$such that the quotient is hyperelliptic. To achieve this, we use a combination of various methods, namely the classical Chabauty–Coleman, elliptic curve Chabauty, quadratic Chabauty, and the bielliptic quadratic Chabauty method (from a forthcoming preprint of the fourth-named author) combined with the Mordell-Weil sieve. Additionally, for square-free levelsN, we classify all$$\mathbb {Q}$$ $Q$-rational points as cusps, CM points (including their CM field andj-invariants) and exceptional ones. We further indicate how to use this to compute the$$\mathbb {Q}$$ $Q$-rational points on all of their modular coverings. 

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3. Bounding conjugacy depth functions for wreath products of finitely generated abelian groups

   https://doi.org/10.46298/jgcc.2023.15.1.11728  

   Ferov, Michal; Pengitore, Mark (September 2023, journal of Groups, complexity, cryptology)

   In this article, we study the asymptotic behaviour of conjugacy separabilityfor wreath products of abelian groups. We fully characterise the asymptoticclass in the case of lamplighter groups and give exponential upper and lowerbounds for generalised lamplighter groups. In the case where the base group isinfinite, we give superexponential lower and upper bounds. We apply our resultsto obtain lower bounds for conjugacy depth functions of various wreath productsof groups where the acting group is not abelian. 

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4. Normal measures on large cardinals

   https://doi.org/10.1090/btran/132  

   Apter, Arthur; Cummings, James (February 2023, Transactions of the American Mathematical Society, Series B)

   The space of normal measures on a measurable cardinal is naturally ordered by the Mitchell ordering. In the first part of this paper we show that the Mitchell ordering can be linear on a strong cardinal where the Generalised Continuum Hypothesis fails. In the second part we show that a supercompact cardinal at which the Generalised Continuum Hypothesis fails may carry a very large number of normal measures of Mitchell order zero. 

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5. Refined Chabauty–Kim computations for the thrice-punctured line over $$\mathbb {Z}[1/6]$$

   https://doi.org/10.1007/s40993-024-00597-4  

   Lüdtke, Martin (March 2025, Research in Number Theory)

   Abstract The Chabauty–Kim method and its refined variant by Betts and Dogra aim to cut out theS-integral points$$X(\mathbb {Z}_S)$$ $X\left(Z_S\right)$on a curve inside thep-adic points$$X(\mathbb {Z}_p)$$ $X\left(Z_p\right)$by producing enough Coleman functions vanishing on them. We derive new functions in the case of the thrice-punctured line whenScontains two primes. We describe an algorithm for computing refined Chabauty–Kim loci and verify Kim’s Conjecture over$$\mathbb {Z}[1/6]$$ $Z[1/6]$for all choices of auxiliary prime $$p < 10{,}000$$ $p<10,000$. 

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