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Let be a curve over a complete discretely valued field . We give tropical descriptions of the weight function attached to a pluricanonical form on and the essential skeleton of . We show that the Laplacian of the weight function equals the pluricanonical divisor on Berkovich skeleta, and we describe the essential skeleton of as a combinatorial skeleton of the Berkovich skeleton of the minimal snc-model. In particular, if has semistable reduction, then the essential skeleton coincides with the minimal skeleton. As an intermediate step, we describe the base loci of logarithmic pluricanonical line bundles on minimal snc-models.
A long-standing conjecture states that every positive integer other than
is a sum of at most seven positive cubes. This was first observed by Jacobi in 1851 on the basis of extensive calculations performed by the famous computationalist Zacharias Dase. We complete the proof of this conjecture, building on previous work of Linnik, Watson, McCurley, Ramaré, Boklan, Elkies, and many others.
We introduce a new category of coefficients for -adic cohomology called constructible isocrystals. Conjecturally, the category of constructible isocrystals endowed with a Frobenius structure is equivalent to the category of perverse holonomic arithmetic -modules. We prove here that a constructible isocrystal is completely determined by any of its geometric realizations.
Let be a number field and let be a curve of genus with Jacobian variety . We study the canonical height . More specifically, we consider the following two problems, which are important in applications:
for a given , compute efficiently;
for a given bound , find all with .
We develop an algorithm running in polynomial time (and fast in practice) to deal with the first problem. Regarding the second problem, we show how one can tweak the naive height that is usually used to obtain significantly improved bounds for the difference , which allows a much faster enumeration of the desired set of points.
Our approach is to use the standard decomposition of as a sum of local “height correction functions”. We study these functions carefully, which leads to efficient ways of computing them and to essentially optimal bounds. To get our polynomial-time algorithm, we have to avoid the factorization step needed to find the finite set of places where the correction might be nonzero. The main innovation is to replace factorization into primes by factorization into coprimes.
Most of our results are valid for more general fields with a set of absolute values satisfying the product formula.
In this article we study combinatorial degenerations of minimal surfaces of Kodaira dimension 0 over local fields, and in particular show that the “type” of the degeneration can be read off from the monodromy operator acting on a suitable cohomology group. This can be viewed as an arithmetic analogue of results of Persson and Kulikov on degenerations of complex surfaces, and extends various particular cases studied by Matsumoto, Liedtke and Matsumoto, and Hernández Mada. We also study “maximally unipotent” degenerations of Calabi–Yau threefolds, following Kollár and Xu, showing in this case that the dual intersection graph is a 3-sphere.
We prove a Voronoi formula for coefficients of a large class of -functions including Maass cusp forms, Rankin–Selberg convolutions, and certain noncuspidal forms. Our proof is based on the functional equations of -functions twisted by Dirichlet characters and does not directly depend on automorphy. Hence it has wider application than previous proofs. The key ingredient is the construction of a double Dirichlet series.
Let be an algebraically closed field of characteristic zero. In joint work with J. Cuadra, we showed that a semisimple Hopf action on a Weyl algebra over a polynomial algebra factors through a group action, and this in fact holds for any finite dimensional Hopf action if . We also generalized these results to finite dimensional Hopf actions on algebras of differential operators. In this work we establish similar results for Hopf actions on other algebraic quantizations of commutative domains. This includes universal enveloping algebras of finite dimensional Lie algebras, spherical symplectic reflection algebras, quantum Hamiltonian reductions of Weyl algebras (in particular, quantized quiver varieties), finite -algebras and their central reductions, quantum polynomial algebras, twisted homogeneous coordinate rings of abelian varieties, and Sklyanin algebras. The generalization in the last three cases uses a result from algebraic number theory due to A. Perucca.