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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.
2. Exploration of a singular fluid spacetime

   https://doi.org/10.1007/s10714-021-02873-5  

   Remmen, Grant N. ( November 2021 , General Relativity and Gravitation)

   We investigate the properties of a special class of singular solutions for a self-gravitating perfect fluid in general relativity: the singular isothermal sphere. For arbitrary constant equation-of-state parameter$$w=p/\rho $$$w=p/\rho$, there exist static, spherically-symmetric solutions with density profile$$\propto 1/r^2$$$\propto 1/r^2$, with the constant of proportionality fixed to be a special function ofw. Like black holes, singular isothermal spheres possess a fixed mass-to-radius ratio independent of size, but no horizon cloaking the curvature singularity at$$r=0$$$r=0$. For$$w=1$$$w=1$, these solutions can be constructed from a homogeneous dilaton background, where the metric spontaneously breaks spatial homogeneity. We study the perturbative structure of these solutions, finding the radial modes and tidal Love numbers, and also find interesting properties in the geodesic structure of this geometry. Finally, connections are discussed between these geometries and dark matter profiles, the double copy, and holographic entropy, as well as how the swampland distance conjecture can obscure the naked singularity.
3. Interpolation for Curves in Projective Space with Bounded Error

   https://doi.org/10.1093/imrn/rnz136  

   Larson, Eric ( July 2019 , International Mathematics Research Notices)

   Abstract Given $n$ general points $p_1, p_2, \ldots , p_n \in{\mathbb{P}}^r$ it is natural to ask whether there is a curve of given degree $d$ and genus $g$ passing through them; by counting dimensions a natural conjecture is that such a curve exists if and only if $$\begin{equation*}n \leq \left\lfloor \frac{(r + 1)d - (r - 3)(g - 1)}{r - 1}\right\rfloor.\end{equation*}$$The case of curves with nonspecial hyperplane section was recently studied in [2], where the above conjecture was shown to hold with exactly three exceptions. In this paper, we prove a “bounded-error analog” for special linear series on general curves; more precisely we show that existence of such a curve subject to the stronger inequality $$\begin{equation*}n \leq \left\lfloor \frac{(r + 1)d - (r - 3)(g - 1)}{r - 1}\right\rfloor - 3.\end{equation*}$$Note that the $-3$ cannot be replaced with $-2$ without introducing exceptions (as a canonical curve in ${\mathbb{P}}^3$ can only pass through nine general points, while a naive dimension count predicts twelve). We also use the same technique to prove that the twist of the normal bundle $N_C(-1)$ satisfies interpolation for curves whose degree is sufficiently large relative to their genus, and deduce from this that the number of generalmore »points contained in the hyperplane section of a general curve is at least $$\begin{equation*}\min\left(d, \frac{(r - 1)^2 d - (r - 2)^2 g - (2r^2 - 5r + 12)}{(r - 2)^2}\right).\end{equation*}$$ As explained in [7], these results play a key role in the author’s proof of the maximal rank conjecture [9].« less
4. List-Decoding Homomorphism Codes with Arbitrary Codomains

   Babai, Laszlo ; Black, Timothy ; Wuu, Angela ( August 2018 , Leibniz international proceedings in informatics)

   The codewords of the homomorphism code aHom(G,H) are the affine homomorphisms between two finite groups, G and H, generalizing Hadamard codes. Following the work of Goldreich-Levin (1989), Grigorescu et al. (2006), Dinur et al. (2008), and Guo and Sudan (2014), we further expand the range of groups for which local list-decoding is possible up to mindist, the minimum distance of the code. In particular, for the first time, we do not require either G or H to be solvable. Specifically, we demonstrate a poly(1/epsilon) bound on the list size, i. e., on the number of codewords within distance (mindist-epsilon) from any received word, when G is either abelian or an alternating group, and H is an arbitrary (finite or infinite) group. We conjecture that a similar bound holds for all finite simple groups as domains; the alternating groups serve as the first test case. The abelian vs. arbitrary result permits us to adapt previous techniques to obtain efficient local list-decoding for this case. We also obtain efficient local list-decoding for the permutation representations of alternating groups (the codomain is a symmetric group) under the restriction that the domain G=A_n is paired with codomain H=S_m satisfying m < 2^{n-1}/sqrt{n}. The limitationsmore »on the codomain in the latter case arise from severe technical difficulties stemming from the need to solve the homomorphism extension (HomExt) problem in certain cases; these are addressed in a separate paper (Wuu 2018). We introduce an intermediate "semi-algorithmic" model we call Certificate List-Decoding that bypasses the HomExt bottleneck and works in the alternating vs. arbitrary setting. A certificate list-decoder produces partial homomorphisms that uniquely extend to the homomorphisms in the list. A homomorphism extender applied to a list of certificates yields the desired list.« less
5. Uniform Bound for the Number of Rational Points on a Pencil of Curves

   https://doi.org/10.1093/imrn/rnz248  

   Dimitrov, Vesselin ; Gao, Ziyang ; Habegger, Philipp ( November 2019 , International Mathematics Research Notices)

   Consider a one-parameter family of smooth, irreducible, projective curves of genus $g\ge 2$ defined over a number field. Each fiber contains at most finitely many rational points by the Mordell conjecture, a theorem of Faltings. We show that the number of rational points is bounded only in terms of the family and the Mordell–Weil rank of the fiber’s Jacobian. Our proof uses Vojta’s approach to the Mordell Conjecture furnished with a height inequality due to the 2nd- and 3rd-named authors. In addition we obtain uniform bounds for the number of torsion points in the Jacobian that lie in each fiber of the family.