We prove a general formula for the p$p$-adic heights of Heegner points on modular abelian varieties with potentially ordinary (good or semistable) reduction at the primes above p$p$. The formula is in terms of the cyclotomic derivative of a Rankin–Selberg p$p$-adic L$L$-function, which we construct. It generalises previous work of Perrin-Riou, Howard, and the author to the context of the work of Yuan–Zhang–Zhang on the archimedean Gross–Zagier formula and of Waldspurger on toric periods. We further construct analytic functions interpolating Heegner points in the anticyclotomic variables, and obtain a version of our formula for them. It is complemented, when the relevant root number is +1$+1$ rather than -1$-1$, by an anticyclotomic version of the Waldspurger formula. When combined with work of Fouquet, the anticyclotomic Gross–Zagier formula implies one divisibility in a p$p$-adic Birch and Swinnerton-Dyer conjecture in anticyclotomic families. Other applications described in the text will appear separately.

Let A$A$ be a complete local ring with a coefficient field k$k$ of characteristic zero, and let Y$Y$ be its spectrum. The de Rham homology and cohomology of Y$Y$ have been defined by R. Hartshorne using a choice of surjection R\rightarrow A$R\rightarrow A$ where R$R$ is a complete regular local k$k$-algebra: the resulting objects are independent of the chosen surjection. We prove that the Hodge–de Rham spectral sequences abutting to the de Rham homology and cohomology of Y$Y$, beginning with their E_{2}$E_{2}$-terms, are independent of the chosen surjection (up to a degree shift in the homology case) and consist of finite-dimensional k$k$-spaces. These E_{2}$E_{2}$-terms therefore provide invariants of A$A$ analogous to the Lyubeznik numbers. As part of our proofs we develop a theory of Matlis duality in relation to {\mathcal{D}}${\mathcal{D}}$-modules that is of independent interest. Some of the highlights of this theory are that if R$R$ is a complete regular local ring containing k$k$ and {\mathcal{D}}={\mathcal{D}}(R,k)${\mathcal{D}}={\mathcal{D}}(R,k)$ is the ring of k$k$-linear differential operators on R$R$, then the Matlis dual D(M)$D(M)$ of any left {\mathcal{D}}${\mathcal{D}}$-module M$M$ can again be given a structure of left {\mathcal{D}}${\mathcal{D}}$-module, and if M$M$ is a holonomic {\mathcal{D}}${\mathcal{D}}$-module, then the de Rham cohomology spaces of D(M)$D(M)$ are k$k$-dual to those of M$M$.

In this paper we study the local cohomology modules of Du Bois singularities. Let (R,\mathfrak{m})$(R,\mathfrak{m})$ be a local ring; we prove that if R_{\text{red}}$R_{\text{red}}$ is Du Bois, then H_{\mathfrak{m}}^{i}(R)\rightarrow H_{\mathfrak{m}}^{i}(R_{\text{red}})$H_{\mathfrak{m}}^{i}(R)\rightarrow H_{\mathfrak{m}}^{i}(R_{\text{red}})$ is surjective for every i$i$. We find many applications of this result. For example, we answer a question of Kovács and Schwede [Inversion of adjunction for rational and Du Bois pairs, Algebra Number Theory 10 (2016), 969–1000; MR 3531359] on the Cohen–Macaulay property of Du Bois singularities. We obtain results on the injectivity of \operatorname{Ext}$\operatorname{Ext}$ that provide substantial partial answers to questions in Eisenbud et al. [Cohomology on toric varieties and local cohomology with monomial supports, J. Symbolic Comput. 29 (2000), 583–600] in characteristic 0$0$. These results can also be viewed as generalizations of the Kodaira vanishing theorem for Cohen–Macaulay Du Bois varieties. We prove results on the set-theoretic Cohen–Macaulayness of the defining ideal of Du Bois singularities, which are characteristic-0$0$ analogs and generalizations of results of Singh–Walther and answer some of their questions in Singh and Walther [On the arithmetic rank of certain Segre products, in Commutative algebra and algebraic geometry, Contemporary Mathematics, vol. 390 (American Mathematical Society, Providence, RI, 2005), 147–155]. We extend results on the relation between Koszul cohomology and local cohomology for F$F$-injective and Du Bois singularities first shown in Hochster and Roberts [The purity of the Frobenius and local cohomology, Adv. Math. 21 (1976), 117–172; MR 0417172 (54 #5230)]. We also prove that singularities of dense F$F$-injective type deform.

We present a complex analytic proof of the Pila–Wilkie theorem for subanalytic sets. In particular, we replace the use of C^{r}$C^{r}$-smooth parametrizations by a variant of Weierstrass division. As a consequence we are able to apply the Bombieri–Pila determinant method directly to analytic families without limiting the order of smoothness by a C^{r}$C^{r}$ parametrization. This technique provides the key inductive step for our recent proof (in a closely related preprint) of the Wilkie conjecture for sets definable using restricted elementary functions. As an illustration of our approach we prove that the rational points of height H$H$ in a compact piece of a complex-analytic set of dimension k$k$ in \mathbb{C}^{m}$\mathbb{C}^{m}$ are contained in O(1)$O(1)$ complex-algebraic hypersurfaces of degree (\log H)^{k/(m-k)}$(\log H)^{k/(m-k)}$. This is a complex-analytic analog of a recent result of Cluckers, Pila, and Wilkie for real subanalytic sets.

We prove that the standard motives of a semisimple algebraic group G$G$ with coefficients in a field of order p$p$ are determined by the upper motives of the group G$G$. As a consequence of this result, we obtain a partial version of the motivic rigidity conjecture of special linear groups. The result is then used to construct the higher indexes which characterize the motivic equivalence of semisimple algebraic groups. The criteria of motivic equivalence derived from the expressions of these indexes produce a dictionary between motives, algebraic structures and the birational geometry of twisted flag varieties. This correspondence is then described for special linear groups and orthogonal groups (the criteria associated with other groups being obtained in De Clercq and Garibaldi [Titsp$p$-indexes of semisimple algebraic groups, J. Lond. Math. Soc. (2) 95 (2017) 567–585]). The proofs rely on the Levi-type motivic decompositions of isotropic twisted flag varieties due to Chernousov, Gille and Merkurjev, and on the notion of pondered field extensions.