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We compute the Brauer group of , the moduli stack of elliptic curves, over , its localizations, finite fields of odd characteristic, and algebraically closed fields of characteristic not . The methods involved include the use of the parameter space of Legendre curves and the moduli stack of curves with full (naive) level structure, the study of the Leray–Serre spectral sequence in étale cohomology and the Leray spectral sequence in fppf cohomology, the computation of the group cohomology of in a certain integral representation, the classification of cubic Galois extensions of , the computation of Hilbert symbols in the ramified case for the primes and , and finding -adic elliptic curves with specified properties.
We enumerate smooth rational curves on very general Weierstrass fibrations over hypersurfaces in projective space. The generating functions for these numbers lie in the ring of classical modular forms. The method of proof uses topological intersection products on a period stack and the cohomological theta correspondence of Kudla and Millson for special cycles on a locally symmetric space of orthogonal type. The results here apply only in base degree , but heuristics for higher base degree match predictions from the topological string partition function.
We explore a number of problems related to the quadratic Chabauty method for determining integral points on hyperbolic curves. We remove the assumption of semistability in the description of the quadratic Chabauty sets containing the integral points of an elliptic curve of rank at most . Motivated by a conjecture of Kim, we then investigate theoretically and computationally the set-theoretic difference . We also consider some algorithmic questions arising from Balakrishnan and Dogra’s explicit quadratic Chabauty for the rational points of a genus-two bielliptic curve. As an example, we provide a new solution to a problem of Diophantus which was first solved by Wetherell.
Computationally, the main difference from the previous approach to quadratic Chabauty is the use of the -adic sigma function in place of a double Coleman integral.
We use the theta correspondence between and to study the -distinction problems over a quadratic extension of nonarchimedean local fields of characteristic . With a similar strategy, we investigate the distinction problem for the pair , where is the unique inner form of defined over . Then we verify the Prasad conjecture for a discrete series representation of .
Let be a nowhere vanishing holomorphic function on the Drinfeld space of dimension , where . The logarithm of its absolute value may be regarded as an affine function on the attached Bruhat–Tits building . Generalizing a construction of van der Put in case , we relate the group of such with the group of integer-valued harmonic 1-cochains on . This also gives rise to a natural -structure on the first (-adic or de Rham) cohomology of .
The Shafarevich conjecture for K3 surfaces asserts the finiteness of isomorphism classes of K3 surfaces over a fixed number field admitting good reduction away from a fixed finite set of finite places. André proved this conjecture for polarized K3 surfaces of fixed degree, and recently She proved it for polarized K3 surfaces of unspecified degree. We prove a certain generalization of their results, which is stated by the unramifiedness of -adic étale cohomology groups for K3 surfaces over finitely generated fields of characteristic . As a corollary, we get the original Shafarevich conjecture for K3 surfaces without assuming the extendability of polarization, which is stronger than the results of André and She. Moreover, as an application, we get the finiteness of twists of K3 surfaces via a finite extension of characteristic fields.
We give an explicit recipe for determining iterated local cohomology groups with support in ideals of minors of a generic matrix in characteristic zero, expressing them as direct sums of indecomposable -modules. For nonsquare matrices these indecomposables are simple, but this is no longer true for square matrices where the relevant indecomposables arise from the pole order filtration associated with the determinant hypersurface. Specializing our results to a single iteration, we determine the Lyubeznik numbers for all generic determinantal rings, thus answering a question of Hochster.
Let be a prime and let denote the -th layer of the cyclotomic -extension of . We prove the effective asymptotic FLT over for all and all primes that are non-Wieferich, i.e., . The effectivity in our result builds on recent work of Thorne proving modularity of elliptic curves over .
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