 Award ID(s):
 2134315
 Publication Date:
 NSFPAR ID:
 10325819
 Journal Name:
 Journal of the American Mathematical Society
 ISSN:
 08940347
 Sponsoring Org:
 National Science Foundation
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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 nonabelian generalisation of Coleman’s effective Chabauty theorem.

Abstract We investigate the properties of a special class of singular solutions for a selfgravitating perfect fluid in general relativity: the singular isothermal sphere. For arbitrary constant equationofstate parameter
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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 “boundederror 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 »

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 GoldreichLevin (1989), Grigorescu et al. (2006), Dinur et al. (2008), and Guo and Sudan (2014), we further expand the range of groups for which local listdecoding 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 (mindistepsilon) 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 listdecoding for this case. We also obtain efficient local listdecoding 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^{n1}/sqrt{n}. The limitationsmore »

Abstract Consider a oneparameter 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 3rdnamed authors. In addition we obtain uniform bounds for the number of torsion points in the Jacobian that lie in each fiber of the family.