Let $F$ be a real quadratic field in which a fixed prime $p$ is inert, and $E_0$ be an imaginary quadratic field in which $p$ splits; put $E=E_{0}F$. Let $X$ be the fiber over ${\mathbb{F}}_{p^2}$ of the Shimura variety for $G\left(U\left(1,n-1\right)\times U\left(n-1,1\right)\right)$ with hyperspecial level structure at $p$ for some integer $n\ge 2$. We show that under some genericity conditions the middle-dimensional Tate classes of $X$ are generated by the irreducible components of its supersingular locus. We also discuss a general conjecture regarding special cycles on the special fibers of unitary Shimura varieties, and on their relation to Newton stratification.

In this paper we study the action of complex conjugation on Shimura varieties and the problem of descending Shimura varieties to the maximal totally real field of the reflex field. We prove the existence of such a descent for many Shimura varieties whose associated adjoint group has certain factors of type $A$ or $D$. This includes a large family of Shimura varieties of abelian type. Our considerations and constructions are carried out purely at the level of Shimura data and group theory.

We establish a height inequality, in terms of an (ample) line bundle, for a sum of subschemes located in $\ell$-subgeneral position in an algebraic variety, which extends a result of McKinnon and Roth (2015). The inequality obtained in this paper connects the result of McKinnon and Roth (the case when the subschemes are points) and the results of Corvaja and Zannier (2004), Evertse and Ferretti (2008), Ru (2017), and Ru and Vojta (2016) (the case when the subschemes are divisors). Furthermore, our approach gives an alternative short and simpler proof of McKinnon and Roth’s result.

Building on the construction of big Heegner points in the quaternionic setting by Longo and Vigni, and their relation to special values of Rankin–Selberg $L$-functions established by Castella and Longo, we obtain anticyclotomic analogues of the results of Emerton, Pollack and Weston on the variation of Iwasawa invariants in Hida families. In particular, combined with the known cases of the anticyclotomic Iwasawa main conjecture in weight $2$, our results yield a proof of the main conjecture for $p$-ordinary newforms of higher weights and trivial nebentypus.

We show that the Ambro–Kawamata nonvanishing conjecture holds true for a quasismooth WCI $X$ which is Fano or Calabi–Yau, i.e., we prove that, if $H$ is an ample Cartier divisor on $X$, then $\left|H\right|$ is not empty. If $X$ is smooth, we further show that the general element of $\left|H\right|$ is smooth. We then verify the Ambro–Kawamata conjecture for any quasismooth weighted hypersurface. We also verify Fujita’s freeness conjecture for a Gorenstein quasismooth weighted hypersurface.

For the proofs, we introduce the arithmetic notion of regular pairs and highlight some interesting connections with the Frobenius coin problem.

In previous work, we described conditions under which a single geometric representation ${\mathrm{\Gamma }}_F\to H\left({\overline{\mathbb{Q}}}_{\ell }\right)$ of the Galois group of a number field $F$ lifts through a central torus quotient $\tilde{H}\to H$ to a geometric representation. In this paper, we prove a much sharper result for systems of $\ell$-adic representations, such as the $\ell$-adic realizations of a motive over $F$, having common “good reduction” properties. Namely, such systems admit geometric lifts with good reduction outside a common finite set of primes. The method yields new proofs of theorems of Tate (the original result on lifting projective representations over number fields) and Wintenberger (an analogue of our main result in the case of a central isogeny $\tilde{H}\to H$).

W. Zhang’s arithmetic fundamental lemma (AFL) is a conjectural identity between the derivative of an orbital integral on a symmetric space with an arithmetic intersection number on a unitary Rapoport–Zink space. In the minuscule case, Rapoport, Terstiege and Zhang have verified the AFL conjecture via explicit evaluation of both sides of the identity. We present a simpler way for evaluating the arithmetic intersection number, thereby providing a new proof of the AFL conjecture in the minuscule case.