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1. Limit models in strictly stable abstract elementary classes

   https://doi.org/10.1002/malq.202200075  

   Boney, Will; VanDieren, Monica_M (December 2024, Mathematical Logic Quarterly)

   Abstract In this paper, we examine the locality condition for non‐splitting and determine the level of uniqueness of limit models that can be recovered in some stable, but not superstable, abstract elementary classes. In particular we prove the following. Suppose that is an abstract elementary class satisfyingthe joint embedding and amalgamation properties with no maximal model of cardinality ,stability in ,,continuity for (i.e., if and is a limit model witnessed by for some limit ordinal and there exists so that does not ‐split over for all , then does not ‐split over ). Then for and limit ordinals both with cofinality , if satisfies symmetry for (or just ‐symmetry), then, for any and that are and ‐limit models over , respectively, we have that and are isomorphic over . Note that no tameness is assumed. 

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2. Local decoding and testing of polynomials over grids

   https://doi.org/10.1002/rsa.20933  

   Bafna, Mitali; Srinivasan, Srikanth; Sudan, Madhu (June 2020, Random Structures & Algorithms)

   We study the local decodability and (tolerant) local testability of low‐degreen‐variate polynomials over arbitrary fields, evaluated over the domain {0,1}ⁿ. We show that for every field there is a tolerant local test whose query complexity depends only on the degree. In contrast we show that decodability is possible over fields of positive characteristic, but not over the reals. 

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3. Base Change and Iwasawa Main Conjectures for GL2

   https://doi.org/10.1093/imrn/rnaf082  

   Burungale, Ashay; Castella, Francesc; Skinner, Christopher (April 2025, International Mathematics Research Notices)

   Abstract Let $$E$$ be an elliptic curve defined over $${\mathbb{Q}}$$ of conductor $$N$$, $$p$$ an odd prime of good ordinary reduction such that $E[p]$ is an irreducible Galois module, and $$K$$ an imaginary quadratic field with all primes dividing $Np$ split. We prove Iwasawa main conjectures for the $${\mathbb{Z}}_{p}$$-cyclotomic and $${\mathbb{Z}}_{p}$$-anticyclotomic deformations of $$E$$ over $${\mathbb{Q}}$$ and $K,$ respectively, dispensing with any of the ramification hypotheses on $E[p]$ in previous works. The strategy employs base change and the two-variable zeta element associated to $$E$$ over $$K$$, via which the sought after main conjectures are deduced from Wan’s divisibility towards a three-variable main conjecture for $$E$$ over a quartic CM field containing $$K$$ and certain Euler system divisibilities. As an application, we prove cases of the two-variable main conjecture for $$E$$ over $$K$$. The aforementioned one-variable main conjectures imply the $$p$$-part of the conjectural Birch and Swinnerton-Dyer formula for $$E$$ if $$\operatorname{ord}_{s=1}L(E,s)\leq 1$$. They are also an ingredient in the proof of Kolyvagin’s conjecture and its cyclotomic variant in our joint work with Grossi [1]. 

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4. Notes on solutions of KZ equations modulo $p^s$ and $$p$$-adic limit $$s\to\infty$$

   Alexander Varchenko (January 2022, Contemporary mathematics)

   We consider the KZ equations over C in the case, when the hypergeometric solutions are hyperelliptic integrals of genus g. Then the space of solutions is a 2g-dimensional complex vector space. We also consider the same equations modulo ps, where p is an odd prime and s is a positive integer, and over the field Q_p of p-adic numbers. We construct polynomial solutions of the KZ equations modulo ps and study the space Mps of all constructed solutions. We show that the p-adic limit of Mps as s→∞ gives us a g-dimensional vector space of solutions of the KZ equations over Qp. The solutions over Qp are power series at a certain asymptotic zone of the KZ equations. In the appendix written jointly with Steven Sperber we consider all asymptotic zones of the KZ equations in the case g=1 of elliptic integrals. The p-adic limit of Mps as s→∞ gives us a one-dimensional space of solutions over Qp at every asymptotic zone. We apply Dwork's theory and show that our germs of solutions over Qp defined at different asymptotic zones analytically continue into a single global invariant line subbundle of the associated KZ connection. Notice that the corresponding KZ connection over C does not have proper nontrivial invariant subbundles, and therefore our invariant line subbundle is a new feature of the KZ equations over Qp. We describe the Frobenius transformations of solutions of the KZ equations for g=1 and then recover the unit roots of the zeta functions of the elliptic curves defined by the equations y2=βx(x−1)(x−α) over the finite field Fp. Here α,β∈F×p,α≠1 

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5. 2-Adic point counting on K3 surfaces

   https://doi.org/10.1007/s40993-022-00378-x  

   Elsenhans, Andreas-Stephan; Jahnel, Jörg (December 2022, Research in Number Theory)

   Abstract This article reports on an approach to point counting on algebraic varieties over finite fields that is based on a detailed investigation of the 2-adic orthogonal group. Combining the new approach with a p -adic method, we count the number of points on some K 3 surfaces over the field $$\mathbb {F}_{\!p}$$ F p , for all primes $$p < 10^8$$ p < 10 8 . 

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