We prove that if q is a power of an odd prime, then there is no genus-2 curve over $\mathbf{F}_q$ whose Jacobian has characteristic polynomial of Frobenius equal to $x^4 + (2 - 2q)x^2 + q^2$. Our proof uses the Brauer relations in a biquadratic extension of $\mathbb{Q}$ to show that every principally polarized abelian surface over $\mathbf{F}_q$ with the given characteristic polynomial splits over $\mathbf{F}_{q^2}$ as a product of polarized elliptic curves.

For the reductive dual pair (U(n, n), U(n, n)) over a p-adic field, we study the local theta correspondence for certain tempered representations in terms of R-groups. In the case we consider, the Langlands parameter is preserved, but, a twist occurs in the L-packet. Moreover this twist is determined by root numbers.

In Bull. Soc. Math. France 115 (1987), 399–456, Perrin-Riou formulates a form of the Iwasawa main conjecture which relates Heegner points to the Selmer group of an elliptic curve defined over $\mathbb{Q}$, as one goes up the anticyclotomic $\mathbb{Z}_p$-extension of a quadratic imaginary field K. Building on the earlier work of Bertolini on this conjecture, and making use of the recent work of Mazur and Rubin on Kolyvagin's theory of Euler systems, we prove one divisibility of Perrin-Riou's conjectured equality. As a consequence, one obtains an upper bound on the rank of the Mordell–Weil group E(K) in terms of Heegner points.

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.

In this paper, we construct a theory of geometric Euler systems, complementary to the arithmetic theory of Rubin, Kato and Perrin-Riou. We show that geometric Euler systems can be used to prove the finiteness of certain Galois representations of weight zero and we discuss a conjectural framework for the existence of geometric Euler systems for motivic Galois representations. We give applications to adjoint Selmer groups of certain classical and Drinfeld modular forms.

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.

Building on the work of Davenport and Schmidt, we mainly prove two results. The first one is a version of Gel'fond's transcendence criterion which provides a sufficient condition for a complex or p-adic number $\xi$ to be algebraic in terms of the existence of polynomials of bounded degree taking small values at $\xi$ together with most of their derivatives. The second one, which follows from this criterion by an argument of duality, is a result of simultaneous approximation by conjugate algebraic integers for a fixed number $\xi$ that is either transcendental or algebraic of sufficiently large degree. We also present several constructions showing that these results are essentially optimal.

Following the analogy between primes and knots, we introduce the refined Milnor invariants for prime numbers and establish their connection with certain Massey products in Galois cohomology. This generalizes the well-known relation between the power residue symbol and cup product and gives a cohomological interpretation of L. Rédei's triple symbol.

Let F be the fraction field of the ring of Witt vectors over a perfect field of characteristic p (for example $F=\mathbb{Q}_p$), and let GF be the absolute Galois group of F. The main result of this article is the following: a p-adic representation of GF, which is a limit of subquotients of crystalline representations with Hodge–Tate weights in an interval [a; b], is itself crystalline with Hodge–Tate weights in [a; b]. In order to show this, we study the $(\phi,\Gamma)$-modules attached to crystalline representations, which allows us to improve some results of Fontaine, Wach and Colmez.

In this work we study the moments of central values of Hecke L-functions associated with cubic characters, and establish quantitative non-vanishing result for the L-values.

The aim of this paper is to give an application of p-adic Hodge theory to stringy Hodge numbers introduced by V. Batyrev for a mathematical formulation of mirror symmetry. Since the stringy Hodge numbers of an algebraic variety are defined by choosing a resolution of singularities, the well-definedness is not clear from the definition. We give a proof of the well-definedness by using arithmetic techniques such as p-adic integration and p-adic Hodge theory. Note that another proof of the well-definedness was obtained by V. Batyrev himself by motivic integration.

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.

Let p be a prime number and let F be a local field with finite residual field of characteristic p. The Langlands local correspondence modulo $\ell \neq p $ for GL(n, F) is known for all integers $n\geq 1$ but the case $\ell=p$ is still mysterious even when n = 2 (the case n = 1 is given by the local class field theory). Any irreducible $\overline{\bf F}_p$-representation of GL(n, F) has a non-zero vector invariant by the pro-p-Iwahori subgroup I(1) and the pro-p-Iwahori–Hecke $\overline{\bf F}_p$-algebra ${\cal H}_{ \overline{\bf F}_p}(GL(n,F),I(1))$ plays a fundamental role in the theory of $\overline{\bf F}_p$-representations of G. We get when n = 2: (i) A bijection between the irreducible $\overline{\bf F}_p$-representations of dimension 2 of the Weil group $W(\overline F/ F)$ and the simple supersingular modules of the pro-p-Iwahori–Hecke $\overline{\bf F}_p$-algebra ${\cal H}_{\overline{\bf F}_p}(GL(2,F),I(1))$. (ii) A new proof of the Barthel–Livne classification of the irreducible non-supersingular $\overline{\bf F}_p$-representations of GL(2, F) using the I(1)-invariant functor. (iii) A bijection between the irreducible $\overline{\bf F}_p$-representations of GL(2, Qp) and the simple right ${\cal H}_{\overline{\bf F}_p}(GL(2,{\bf Q}_p),I(1))$-modules given by the I(1)-invariant functor, using the recent results of Breuil.

We construct a finite-dimensional vector space of functions of two complex variables attached to a smooth algebraic curve C over a finite field $\mathbb{F}_q$, q odd, and a level. These functions collect the analytic information about the cohomology of the curve and its quadratic twists that is encoded in the corresponding L-functions; they are double Dirichlet series in two independent complex variables s and w. We prove that these series satisfy a finite, non-abelian group of functional equations in the two complex variables (s, w) and are rational functions in q-s and q-w with a specified denominator. The group is D6, the dihedral group of order 12.

Estimates are established for the number of integers of size N, in intervals of size $N^\theta$, that fail to admit a representation as the sum of s cubes (s = 5, 6). Thereby it is shown that almost all such integers are represented in the proposed manner. When s = 5 one may take $\theta = 10/21$, and when s = 6 one may take any $\theta >17/63$. Similar such conclusions are also established for the related problem associated with the expected asymptotic formula.

We generalize a theorem of Tate and show that the second cohomology of the Weil group of a global or local field with coefficients in $\mathbb{C}^*$ (or, more generally, with coefficients in the complex points of an algebraic torus over $\mathbb{C}$) vanish, where the cohomology groups are defined using measurable cochains in the sense of Moore. We recover a theorem of Labesse stating that the admissible homomorphisms of a Weil group to the Langlands dual group of a reductive group can be lifted to an extension of the Langlands dual group by a torus.

We show that the extrapolation to the case of global fields of characteristic p of a question posed by Stark in 1980, regarding abelian L-functions of order of vanishing 2 at s = 0, has a negative answer. We provide links between various versions of Stark's question and a natural refinement of Brumer's conjecture, in the general context of global fields of arbitrary characteristic. As a consequence, we show that the refinement of Brumer's conjecture is, in general, false for characteristic p global fields.

We set up the basic theory of $\mathcal P$-adic modular forms over certain unitary PEL Shimura curves M′K′. For any PEL abelian scheme classified by M′K′, which is not ‘too supersingular’, we construct a canonical subgroup which is essentially a lifting of the kernel of Frobenius from characteristic p. Using this construction we define the U and Frob operators in this context. Following Coleman, we study the spectral theory of the action of U on families of overconvergent $\mathcal P$-adic modular forms and prove that the dimension of overconvergent eigenforms of U of a given slope is a locally constant function of the weight.

This corrigendum corrects a mistake in K. Belabas, On the mean 3-rank of quadratic fields, Compositio Math. 118 (1999), 1–9.

Gross and Rubin have made conjectures about special values of equivariant L-functions associated to abelian extensions of global fields. We describe a common refinement, due to Burns, and give evidence in favour of this conjecture for quadratic extensions and cyclotomic fields. We also note that the statement provides a new interpretation of further conjectures of Darmon and Gross.