In this paper we study a kind of spherical function, which we call a Siegel–Whittaker function, related to Fourier expansions of automorphic forms on Hermitian symmetric domains of type IV. We obtain a multiplicity-one theorem and an integral representation of this spherical function.

In this paper we show how to predict relative trace identities from the computation of Jacquet modules of the Weil representations. Many previously considered special cases of relative trace identities fit the principle we develop here, including those with important applications on L-functions. We also show how to prove these identities using the Weil representation. We give a proof of the relative trace identities between the distributions on SO(n + 1, n) and $\widetilde{\textit{Sp}}(m)$$(n\geq m)$. The proof should serve as a model to the other cases conjectured in the paper.

We prove a relation between a generating series for the heights of Heegner cycles on the arithmetic surface associated with a Shimura curve and the second term in the Laurent expansion at s = ½ of an Eisenstein series of weight $\frac32$ for SL(2). On the geometric side, a typical coefficient of the generating series involves the Faltings heights of abelian surfaces isogenous to a product of CM elliptic curves, an archimedean contribution, and contributions from vertical components in the fibers of bad reduction. On the analytic side, these terms arise via the derivatives of local Whittaker functions. It should be noted that s = ½ is not the central point for the functional equation of the Eisenstein series in question. Moreover, the first term of the Laurent expansion at s = ½ coincides with the generating function for the degrees of the Heegner cycles on the generic fiber and, in particular, does not vanish.

Following Katz–Sarnak, Iwaniec–Luo–Sarnak and Rubinstein, we use the one- and two-level densities to study the distribution of low-lying zeros for one-parameter rational families of elliptic curves over $\mathbb{Q}(t)$. Modulo standard conjectures, for small support the densities agree with Katz and Sarnak's predictions. Further, the densities confirm that the curves' L-functions behave in a manner consistent with having r zeros at the critical point, as predicted by the Birch and Swinnerton-Dyer conjecture. By studying the two-level densities of some constant sign families, we find the first examples of families of elliptic curves where we can distinguish SO(even) from SO(odd) symmetry.

Let (X, 0) be a germ of an analytic space and f the germ of an analytic function on X. We show that the polar filtration of the local Milnor fiber, defined by a projection on a complex disc, is diffeomorphic to a valuative filtration of the Milnor fiber called Hironaka's filtration, a result which links invariants associated with the singularities of the projection to those associated with a desingularisation.

Using the weak factorization theorem, we give a simple presentation for the value group of the universal Euler characteristic with compact support for varieties of characteristic zero and describe the value group of the universal Euler characteristic of pairs. This gives a new proof for the existence of natural Euler characteristics with values in the Grothendieck group of Chow motives. A generalization of the presentation to the relative setting allows us to define duality and the six operations.

Which algebraic surfaces occur as covers of moduli spaces for curves? We show that surfaces of a special type over number fields do, with possible exception for non-elliptic K3 surfaces.

We prove a generalization to the context of real geometry of an intersection formula for the vanishing cycle functor, which in the complex context is due to Dubson, Lê, Ginsburg and Sabbah (after a conjecture of Deligne). It is also a generalization of similar results of Kashiwara and Schapira, where these authors work with a suitable assumption about the micro-support of the corresponding constructible complex of sheaves. We only use a similar assumption about the support of the corresponding characteristic cycle so that our result can be formulated in the language of constructible functions and Lagrangian cycles.

The main result of this paper is that, over a non-commutative Koszul algebra, high truncations of finitely generated graded modules have linear free resolutions.

We obtain a decomposition formula of a representation of Sp(p, q) or SO*(2n) unitarily induced from a derived functor module, which enables us to reduce the problem of irreducible decompositions to the study of derived functor modules. In particular, we show that such an induced representation is decomposed into a direct sum of irreducible unitarily induced modules from derived functor modules under some regularity condition on the parameters. In particular, representations of SO*(2n) and Sp(p, q) induced from one-dimensional unitary representations of their parabolic subgroups are irreducible.

For any integer $m\geq 4$ we construct twistor spaces of algebraic dimension a = 2 associated to some self-dual structures on mP2, the connected sum of m complex projective planes P2. Together with previously known results this implies that for any $m\geq 5$ all the possible values for algebraic dimension $0\leq a \leq 3$ are attained for some such twistor spaces. Our method is to construct such twistor spaces by small deformations of Joyce twistor spaces.

In the Mathematical Review of ‘On a conjecture of Kontsevich and variants of Castelnuovo's lemma’ [Compositio Mathematica 115 (1999), 205–230], Gizatullin pointed out that the proof as written is incomplete for the complex case because the possibility of a certain equation vanishing identically was ignored. This corrigendum addresses the missing case. Note that it is simply another iteration of the argument already present in the article.