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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. A note on rank one quadratic twists of elliptic curves and the non-degeneracy of 𝑝-adic regulators at Eisenstein primes

   https://doi.org/10.1090/bproc/144  

   Burungale, Ashay; Skinner, Christopher (February 2023, Proceedings of the American Mathematical Society, Series B)

   We show that for certain non-CM elliptic curves E / Q E_{/\mathbb {Q}} such that 3 3 is an Eisenstein prime of good reduction, a positive proportion of the quadratic twists E ψ E_{\psi } of E E have Mordell–Weil rank one and the 3 3 -adic height pairing on E ψ ( Q ) E_{\psi }(\mathbb {Q}) is non-degenerate. We also show similar but weaker results for other Eisenstein primes. The method of proof also yields examples of middle codimensional algebraic cycles over number fields of arbitrarily large dimension (generalized Heegner cycles) that have non-zero p p -adic height. It is not known – though expected – that the archimedian height of these higher-codimensional cycles is non-zero. 

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3. Quadratic Chabauty for modular curves: algorithms and examples

   https://doi.org/10.1112/S0010437X23007170  

   Balakrishnan, Jennifer S.; Dogra, Netan; Müller, J. Steffen; Tuitman, Jan; Vonk, Jan (June 2023, Compositio Mathematica)

   We describe how the quadratic Chabauty method may be applied to determine the set of rational points on modular curves of genus$$g>1$$whose Jacobians have Mordell–Weil rank$$g$$. This extends our previous work on the split Cartan curve of level 13 and allows us to consider modular curves that may have few known rational points or non-trivial local height contributions at primes of bad reduction. We illustrate our algorithms with a number of examples where we determine the set of rational points on several modular curves of genus 2 and 3: this includes Atkin–Lehner quotients$$X_0^+(N)$$of prime level$$N$$, the curve$$X_{S_4}(13)$$, as well as a few other curves relevant to Mazur's Program B. We also compute the set of rational points on the genus 6 non-split Cartan modular curve$$X_{\scriptstyle \mathrm { ns}} ^+ (17)$$. 

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4. Sporadic Cubic Torsion

   https://doi.org/10.2140/ant.2021.15.1837  

   Derickx, Maarten; Etropolski, Anastassia; van Hoeij, Mark; Morrow, Jackson S.; Zureick-Brown, David. (November 2021, Algebra and number theory)

   Let K be a number field, and let E/K be an elliptic curve over K. The Mordell–Weil theorem asserts that the K-rational points E(K) of E form a finitely generated abelian group. In this work, we complete the classification of the finite groups which appear as the torsion subgroup of E ( K ) for K a cubic number field. To do so, we determine the cubic points on the modular curves X1(N) for N = 21,22,24,25,26,28,30,32,33,35,36,39,45,65,121. As part of our analysis, we determine the complete lists of N for which J0(N), J1(N), and J1(2,2N) have rank 0. We also provide evidence to a generalized version of a conjecture of Conrad, Edixhoven, and Stein by proving that the torsion on J1(N)(Q) is generated by Galois-orbits of cusps of X1(N) for N ≤55, N ̸=54. 

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5. THE DYNAMICAL MORDELL–LANG CONJECTURE FOR ENDOMORPHISMS OF SEMIABELIAN VARIETIES DEFINED OVER FIELDS OF POSITIVE CHARACTERISTIC

   https://doi.org/10.1017/S1474748019000318  

   Corvaja, Pietro; Ghioca, Dragos; Scanlon, Thomas; Zannier, Umberto (March 2021, Journal of the Institute of Mathematics of Jussieu)

   null (Ed.)

   Abstract Let $$K$$ be an algebraically closed field of prime characteristic $$p$$ , let $$X$$ be a semiabelian variety defined over a finite subfield of $$K$$ , let $$\unicode[STIX]{x1D6F7}:X\longrightarrow X$$ be a regular self-map defined over $$K$$ , let $$V\subset X$$ be a subvariety defined over $$K$$ , and let $$\unicode[STIX]{x1D6FC}\in X(K)$$ . The dynamical Mordell–Lang conjecture in characteristic $$p$$ predicts that the set $$S=\{n\in \mathbb{N}:\unicode[STIX]{x1D6F7}^{n}(\unicode[STIX]{x1D6FC})\in V\}$$ is a union of finitely many arithmetic progressions, along with finitely many $$p$$ -sets, which are sets of the form $$\{\sum _{i=1}^{m}c_{i}p^{k_{i}n_{i}}:n_{i}\in \mathbb{N}\}$$ for some $$m\in \mathbb{N}$$ , some rational numbers $$c_{i}$$ and some non-negative integers $$k_{i}$$ . We prove that this conjecture is equivalent with some difficult diophantine problem in characteristic 0. In the case $$X$$ is an algebraic torus, we can prove the conjecture in two cases: either when $$\dim (V)\leqslant 2$$ , or when no iterate of $$\unicode[STIX]{x1D6F7}$$ is a group endomorphism which induces the action of a power of the Frobenius on a positive dimensional algebraic subgroup of $$X$$ . We end by proving that Vojta’s conjecture implies the dynamical Mordell–Lang conjecture for tori with no restriction. 

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