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For each integer , we prove an unconditional upper bound on the size of the -torsion subgroup of the class group, which holds for all but a zero-density set of field extensions of of degree , for any fixed (with the additional restriction in the case that the field be non-). For sufficiently large (specified explicitly), these results are as strong as a previously known bound that is conditional on GRH. As part of our argument, we develop a probabilistic “Chebyshev sieve,” and give uniform, power-saving error terms for the asymptotics of quartic (non-) and quintic fields with chosen splitting types at a finite set of primes.
By a classical method due to Roitman, a complete intersection of sufficiently small degree admits a rational decomposition of the diagonal. This means that some multiple of the diagonal by a positive integer , when viewed as a cycle in the Chow group, has support in , for some divisor and a finite set of closed points . The minimal such is called the torsion order. We study lower bounds for the torsion order following the specialization method of Voisin, Colliot-Thélène, and Pirutka. We give a lower bound for the generic complete intersection with and without point. Moreover, we use methods of Kollár and Totaro to exhibit lower bounds for the very general complete intersection.
We define and construct integral canonical models for automorphic vector bundles over Shimura varieties of abelian type.
More precisely, we first build on Kisin’s work to construct integral canonical models over for Shimura varieties of abelian type with hyperspecial level at all primes not dividing compatible with Kisin’s construction. We then define a notion of an integral canonical model for the standard principal bundles lying over Shimura varieties and proceed to construct them in the abelian type case. With these in hand, one immediately also gets integral models for automorphic vector bundles.
Given a ring object in a symmetric monoidal category, we investigate what it means for the extension to be (quasi-)Galois. In particular, we define splitting ring extensions and examine how they occur. Specializing to tensor-triangulated categories, we study how extension-of-scalars along a quasi-Galois ring object affects the Balmer spectrum. We define what it means for a separable ring to have constant degree, which is a necessary and sufficient condition for the existence of a quasi-Galois closure. Finally, we illustrate the above for separable rings occurring in modular representation theory.
Let be a totally real field with ring of integers and be an odd prime unramified in . Let be a prime above . We prove that a mod Hilbert modular form associated to is determined by its restriction to the partial Serre–Tate deformation space (-rigidity). Let be an imaginary quadratic CM extension such that each prime of above splits in and a Hecke character of . Partly based on -rigidity, we prove that the -invariant of the anticyclotomic Katz -adic L-function of equals the -invariant of the full anticyclotomic Katz -adic L-function of . An analogue holds for a class of Rankin–Selberg -adic L-functions. When is self-dual with the root number , we prove that the -invariant of the cyclotomic derivatives of the Katz -adic L-function of equals the -invariant of the cyclotomic derivatives of the Katz -adic L-function of . Based on previous works of the authors and Hsieh, we consequently obtain a formula for the -invariant of these -adic L-functions and derivatives. We also prove a -version of a conjecture of Gillard, namely the vanishing of the -invariant of the Katz -adic L-function of .
Let be a smooth cubic hypersurface of dimension . It is well-known that new rational points may be obtained from old ones by secant and tangent constructions. In view of the Mordell–Weil theorem for , Manin (1968) asked if there exists a finite set from which all other rational points can be thus obtained. We give an affirmative answer for , showing in fact that we can take the generating set to consist of just one point. Our proof makes use of a weak approximation theorem due to Skinner, a theorem of Browning, Dietmann and Heath-Brown on the existence of rational points on the intersection of a quadric and cubic in large dimension, and some elementary ideas from differential geometry, algebraic geometry and numerical analysis.
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