We prove new results on the distribution of rational points on ramified covers of abelian varieties over finitely generated fields $k$ of characteristic zero. For example, given a ramified cover $π: X \to A$, where $A$ is an abelian variety over $k$ with a dense set of $k$-rational points, we prove that there is a finite-index coset $C \subset A(k)$ such that $π(X(k))$ is disjoint from $C$.
Our results do not seem to be in the range of other methods available at present; they confirm predictions coming from Lang's conjectures on rational points, and also go in the direction of an issue raised by Serre regarding possible applications to the Inverse Galois Problem. Finally, the conclusions of our work may be seen as a sharp version of Hilbert's irreducibility theorem for abelian varieties.
We continue our study of the reduction of PEL Shimura varieties with parahoric level structure at primes p at which the group that defines the Shimura variety ramifies. We describe "good" $p$-adic integral models of these Shimura varieties and study their 'etale local structure. In this paper we mainly concentrate on the case of unitary groups for a ramified quadratic extension. Some of our results are applications of the theory of twisted affine flag varieties in our previous paper math.AG/0607130.
Corvaja and Zannier conjectured that an abelian variety over a number field satisfies a modified version of the Hilbert property. We investigate their conjecture for products of elliptic curves using Kawamata's structure result for ramified covers of abelian varieties, and Faltings's finiteness theorem for rational points on higher genus curves.
In this article, we prove results about the cohomology of compact unitary group Shimura varieties at split places. In nonendoscopic cases, we are able to give a full description of the cohomology, after restricting to integral Hecke operators at p on the automorphic side. We allow arbitrary ramification at p; even the PEL data may be ramified. This gives a description of the semisimple local Hasse-Weil zeta function in these cases. We also treat cases of nontrivial endoscopy. For this purpose, we give a general stabilization of the expression given in previous work, following the stabilization…
In this paper we prove that there exists a Zariski dense open subset U defined over the rationals Q in the space of all one-variable rational functions with arbitrary k poles of prescribed orders, such that for every geometric point f in U(Qbar)$, the L-function of the exponential sum of f at a prime p has Newton polygon approaching the Hodge polygon as p approaches infinity. As an application to algebraic geometry, we prove that the p-adic Newton polygon of the zeta function of a p-cover of the projective line totally ramified at arbitrary k points of prescribed orders has an asymptotic gener…
A standard observation in algebraic geometry and number theory is that a ramified cover of an algebraic variety $\widetilde{X}\rightarrow X$ over a finite field $F_q$ furnishes the rational points $x\in X(F_q)$ with additional arithmetic structure: the Frobenius action on the fiber over $x$. For example, in the case of the Vieta cover of polynomials over $F_q$ this structure describes a polynomial's irreducible decomposition type. Furthermore, the distribution of these Frobenius actions is encoded in the cohomology of $\widetilde{X}$ via the Grothendieck-Lefschetz trace formula. This note pres…
For p = 3 and p = 5, we exhibit a finite nonsolvable extension of the rational numbers which is ramified only at p via explicit computations with Hilbert modular forms.
Let $K$ be a complete discrete valuation field. Let $\mathcal{O}_K$ be its ring of integers. Let $k$ be its residue field which we assume to be algebraically closed of characteristic exponent $p\geq1$. Let $G/K$ be a semi-abelian variety. Let $\mathcal{G}/\mathcal{O}_K$ be its Néron model. The special fiber $\mathcal{G}_k/k$ is an extension of the identity component $\mathcal{G}_k^0/k$ by the group of components $Φ(G)$. We say that $G/K$ has split reduction if this extension is split. Whereas $G/K$ has always split reduction if $p=1$ we prove that it is no longer the case if $p>1$ even if $G/K…
The Brauer-Manin obstruction is used to explain the failure of the local-global principle for algebraic varieties. In 1999 Skorobogatov gave the first example of a variety that does not satisfy the local-global principle which is not explained by the Brauer-Manin obstruction. He did so by applying the Brauer-Manin obstruction to étale covers of the variety, and thus defining a finer obstruction. In 2008 Poonen gave the first example of failure of the local-global principle which cannot be explained for by Skorobogatov's étale-Brauer obstruction. However, Poonen's construction was not accompani…
We prove the geometrical Satake isomorphism for a reductive group defined over F=k((t)), and split over a tamely ramified extension. As an application, we give a description of the nearby cycles on certain Shimura varieties via the Rapoport-Zink-Pappas local models.
We construct local charts for the ramified cusps of the Hilbert modular variety of level $Γ_1(\mathfrak{c},\mathfrak{n})$. This allows us, following closely M. Rapoport and C.-L. Chai, to construct arithmetic toroidal and minimal compactifications of these modular varieties. Some applications to modular forms are given. See our joint work with J. Tilouine for other applications.
We report on progress in the qualitative study of rational points on rationally connected varieties over number fields, also examining integral points, zero-cycles, and non-rationally connected varieties. One of the main objectives is to highlight and explain the many recent interactions with analytic number theory.
We investigate Fano varieties defined over a number field that contain subvarieties whose number of rational points of bounded height is comparable to the total number on the variety.
For an odd prime p, we construct integral models over p for Shimura varieties with parahoric level structure, attached to Shimura data (G,X) of abelian type, such that G splits over a tamely ramified extension of Q_p. The local structure of these integral models is related to certain "local models", which are defined group theoretically. Under some additional assumptions, we show that these integral models satisfy a conjecture of Kottwitz which gives an explicit description for the trace of Frobenius action on their sheaf of nearby cycles.
We compute new polynomials with Galois group $M_{11}$ over $\mathbb{Q}(t)$. These polynomials stem from various families of covers of $\mathbb{P}^1\mathbb{C}$ ramified over at least 4 points. Each of these families has features that make a detailed study interesting. Some of the polynomials lead, via specialization, to number fields with very small discriminant or few ramified primes.
We prove a large finite field version of the Boston--Markin conjecture on counting Galois extensions of the rational function field with a given Galois group and the smallest possible number of ramified primes. Our proof involves a study of structure groups of (direct products of) racks.
We use birational geometry to show that the existence of rational points on proper rationally connected varieties over fields of characteristic $0$ is a consequence of the existence of rational points on terminal Fano varieties. We discuss several consequences of this result, especially in relation to the $C_1$-conjecture. We also provide evidence that supports the conjecture in dimension $3$ for $C_1$ fields of characteristic $0$.
Let $Z$ be a projective geometrically integral algebraic variety. This paper is concerned with estimating the number of rational points on $Z$ which have height at most $B$. The bounds obtained are uniform in varieties of fixed degree and fixed dimension, and are essentially best possible for varieties of degree at least six.
In this paper, we obtain bounds for the Mordell-Weil ranks over cyclotomic extensions of a wide range of abelian varieties defined over a number field $F$ whose primes above $p$ are totally ramified over $F/\mathbb{Q}$. We assume that the abelian varieties may have good non-ordinary reduction at those primes. Our work is a generalization of \cite{Kim}, in which the second author generalized Perrin-Riou's Iwasawa theory for elliptic curves over $\mathbb{Q}$ with supersingular reduction (\cite{Perrin-Riou}) to elliptic curves defined over the above-mentioned number field $F$. On top of non-ordin…
We consider the problem of counting the number of varieties in a family over a number field which contain a rational point. In particular, for products of Brauer-Severi varieties and closely related counting functions associated to Brauer group elements. Using harmonic analysis on toric varieties, we provide a positive answer to a question of Serre on such counting functions in some cases. We also formulate some conjectures on generalisations of Serre's problem.
We show that the number of rational points on the fibres of a proper morphism of smooth varieties over a finite field k whose generic fibre has a ``trival'' Chow group of zero cycles is congruent to 1 mod |k|. As a consequence we prove that there is a rational point on any degeneration of a smooth proper rationally chain connected variety over a finite field. We also obtain a generalisation of the Chevalley-Warning theorem.
Wan conjectures that if $X$ and $Y$ form a strong mirror pair of Calabi-Yau varieties over a finite field $F_q$ with $q$ elements, then X and Y have the same number of $F_{q^k}$-rational points modulo $q^k$. We prove this conjecture under the condition that $Y$ can be obtained from $X$ through quotient construction.
Let p be an odd prime, and k_\infty the cyclotomic Z_p-extension of an abelian field k. For a finite set S of rational primes which does not include p, we will consider the maximal S-ramified abelian pro-p extension M_S(k_\infty) over k_\infty. We shall give a formula of the Z_p-rank of Gal(M_S(k_\infty)/k_\infty). In the proof of this formula, we also show that M_{q}(k_\infty)/L(k_\infty) is a finite extension for every real abelian field k and every rational prime q distinct from p, where L(k_\infty) is the maximal unramified abelian pro-p extension over k_\infty.
In this paper we prove a formula for the number of rational points of bounded height relative to all the generators of the cone of effective divisor for a toric variety over a number field.
We study zero cycles on rationally connected varieties defined over characteristic zero Laurent fields with algebraically closed residue fields. We show that the degree map induces an isomorphism for rationally connected threefolds defined over such fields. In general, the degree map is an isomorphism if rationally connected varieties defined over algebraically closed fields of characteristic zero satisfy the integral Hodge/Tate conjecture for one cycles, or if the Tate conjecture is true for divisor classes on surfaces defined over finite fields. To prove these results, we introduce technique…
This is a survey of recent examples of varieties that are not stably rational. We review the specialization method based on properties of the Chow group of zero-cycles used in these examples and explain the point of view of unramified cohomology for the construction of nontrivial stable invariants of the special fiber. In particular, we find an explicit formula for the Brauer group of fourfolds fibered in quadrics of dimension two over a rational surface.
We construct a concrete example of a 1-parameter family of smooth projective geometrically integral varieties over an open subscheme of P^1_Q such that there is exactly one rational fiber with no rational points. This makes explicit a construction of Poonen.
We prove new upper bounds for the sup-norm of Hecke Maaß newforms on $GL(2)$ over a number field. Our newforms are more general than those considered in a recent paper by Blomer, Harcos, Maga, and Milicevic: we do not require square free level. Furthermore, we allow for non-trivial central character. Over the rationals we cover the best bounds as obtained by Saha.
In this paper, we prove a conjecture of Kottwitz and Rapoport on a union of (generalized) affine Deligne-Lusztig varieties $X(μ, b)_J$ for any tamely ramified group $G$ and its parahoric subgroup $P_J$. We show that $X(μ, b)_J \neq \emptyset$ if and only if the group-theoretic version of Mazur's inequality is satisfied. In the process, we obtain a generalization of Grothendieck's conjecture on the closure relation of $\s$-conjugacy classes of a twisted loop group.
In this paper we make an initial study on type D moduli spaces in positive characteristic $p\neq 2$, where we allow $p$ ramified in the definite quaternion algebra. We classify the isogeny classes of $p$-divisible groups with additional structures in question. We also study the reduction of the type $D$ moduli spaces of minimal rank.
Let $A$ be an abelian variety defined over a number field $F$. Suppose its dual abelian variety $A'$ has good non-ordinary reduction at the primes above $p$. Let $F_{\infty}/F$ be a $\mathbb Z_p$-extension, and for simplicity, assume that there is only one prime $\mathfrak p$ of $F_{\infty}$ above $p$, and $F_{\infty, \mathfrak p}/\mathbb Q_p$ is totally ramified and abelian. (For example, we can take $F=\mathbb Q(ζ_{p^N})$ for some $N$, and $F_{\infty}=\mathbb Q(ζ_{p^{\infty}})$.) As Perrin-Riou did, we use Fontaine's theory of group schemes to construct series of points over each $F_{n, \mat…
We prove Kawaguchi-Silverman conjecture for all surjective endomorphisms on every smooth rationally connected variety admitting an int-amplified endomorphism.
We give an upper bound on the number of rational points of an arbitrary Zariski closed subset of a projective space over a finite field. This bound depends only on the dimensions and degrees of the irreducible components and holds for very general varieties, even reducible and non equidimensional. As a consequence, we prove a conjecture of Ghorpade and Lachaud on the maximal number of rational points of an equidimensional projective variety.
A sieve for rational points on suitable varieties is developed, together with applications to counting rational points in thin sets, the number of varieties in a family which are everywhere locally soluble, and to the notion of friable rational points with respect to divisors. In the special case of quadrics, sharper estimates are obtained by developing a version of the Selberg sieve for rational points.
Let l be a prime. We show that there do not exist any non-zero semi-stable abelian varieties over Q with good reduction outside l if and only if l=2, 3, 5, 7 or 13. We show that any semi-stable abelian variety over Q with good reduction outside l=11 is isogenous to a power of the Jacobian variety of the modular curve X_0(11). In addition we show that there do not exist any non-zero abelian varieties over Q with good reduction outside l acquiring semi-stable reduction at l over a tamely ramified extension if and only if l \le 5.
We prove several results on the number of rational points on open subsets of Kummer varieties of arbitrary dimension. Some of our results are unconditional, and others depend on the Parity Conjecture (a corollary of the Conjecture of Birch and Swinnerton-Dyer). As examples, we show that (conditional on the Parity Conjecture) all odd-dimension Kummer varieties over Q which are quotients of absolutely simple abelian varieties have dense rational points, and we construct an infinite family of K3 surfaces with dense rational points.
We study the minimal number of ramified primes in Galois extensions of rational function fields over finite fields with prescribed finite Galois group. In particular, we obtain a general conjecture in analogy with the well studied case of number fields, which we establish for abelian, symmetric and alternating groups in many cases.
In this paper, we consider Abelian varieties over function fields that arise as twists of Abelian varieties by cyclic covers of irreducible quasi-projective varieties. Then, in terms of Prym varieties associated to the cyclic covers, we prove a structure theorem on their Mordell-Weil group. Our results give an explicit method for construction of elliptic curves, hyper- and super-elliptic Jacobians that have large ranks over function fields of certain varieties.
