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论文题目：A basis theorem for the affine Kauffman category and its cyclotomic quotients论文作者：Mengmeng Gao, Hebing Rui, Linliang Song发表刊物：Journal of Algebra成果介绍：The affine Kauffman category is a strict monoidal category and can be considered as a $q$-analogue of the affine Brauer category in (Rui and Song in Math.Zeit.293,503-550,2019).In this paper,we prove a basis theorem for the...

成果介绍：We develop the theory of fundamental classes in the setting of motivic homotopy theory. Using this we construct, for any motivic spectrum, an associated twisted bivariant theory, extending the formalism of Fulton and MacPherson. We import the tools of Fulton’s intersection theory into this setting: (refined) Gysin maps, specialization maps, and formulas for excess of intersection, self-intersections, and blow-ups. We also develop a theory of Euler classes of vector bundles in this setting. For the Milnor–Witt spectrum recently constructed by Déglise–Fasel, we get a bivariant theory extending the Chow–Witt groups of Barge–Morel, in the same way the higher Chow groups extend the classical Chow groups. As another application we prove a motivic Gauss–Bonnet formula, computing Euler characteristics in the motivic homotopy category.

成果介绍：In the present paper, we generalize the celebrated classical lemma of Birch and Heegner on quadratic twists of elliptic curves over We prove the existence of explicit infinite families of quadratic twists with analytic ranks 0 and 1 for a large class of elliptic curves, and use Heegner points to explicitly construct rational points of infinite order on the twists of rank 1. In addition, we show that these families of quadratic twists satisfy the 2-part of the Birch and Swinnerton-Dyer conjecture when the original curve does. We also prove a new result in the direction of the Goldfeld conjecture. All these results apply to a large class of elliptic curves, especially for elliptic curves without CM. As applications, we present examples of elliptic curves of small conductors, of Newmann-Setzer curves and also examples of elliptic curves without CM for which the full Birch and Swinnerton-Dyer conjecture holds.