Let R be a complete rank-1 valuation ring of mixed characteristic (0, p), and let K be its field of fractions. A g-dimensional truncated Barsotti–Tate group G of level n over R is said to have a level-n canonical subgroup if there is a K-subgroup of G ⊗RK with geometric structure (Z/pnZ)g consisting of points ‘closest to zero’. We give a non-trivial condition on the Hasse invariant of G that guarantees the existence of the canonical subgroup, analogous to a result of Katz and Lubin for elliptic curves. The bound is independent of the height and dimension of G.

We prove that the Newton polygons of Frobenius on the crystalline cohomology of proper smooth varieties satisfy a symmetry that results, in the case of projective smooth varieties, from Poincaré duality and the hard Lefschetz theorem. As a corollary, we deduce that the Betti numbers in odd degrees of any proper smooth variety over a field are even (a consequence of Hodge symmetry in characteristic zero), answering an old question of Serre. Then we give a generalization and a refinement for arbitrary varieties over finite fields, in response to later questions of Serre and of Katz.

We describe an algorithm to prove the Birch and Swinnerton-Dyer conjectural formula for any given elliptic curve defined over the rational numbers of analytic rank zero or one. With computer assistance we rigorously prove the formula for 16714 of the 16725 such curves of conductor less than 5000.

Let G be a subgroup of the symmetric group Sn, and let δG=∣Sn/G∣−1 where ∣Sn/G∣ is the index of G in Sn. Then there are at most On,ϵ(Hn−1+δG+ϵ) monic integer polynomials of degree n that have Galois group G and height not exceeding H, so there are only a “few” polynomials having a “small” Galois group.

In this work, we study the intersection cohomology of Siegel modular varieties. The goal is to express the trace of a Hecke operator composed with a power of the Frobenius endomorphism (at a good place) on this cohomology in terms of the geometric side of Arthur’s invariant trace formula for well-chosen test functions. Our main tools are the results of Kottwitz about the contribution of the cohomology with compact support and about the stabilization of the trace formula, Arthur’s L2 trace formula and the fixed point formula of Morel [Complexes pondérés sur les compactifications de Baily–Borel. Le cas des variétés de Siegel, J. Amer. Math. Soc. 21 (2008), 23–61]. We ‘stabilize’ this last formula, i.e. express it as a sum of stable distributions on the general symplectic groups and its endoscopic groups, and obtain the formula conjectured by Kottwitz in [Shimura varieties and λ-adic representations, in Automorphic forms, Shimura varieties and L-functions, Part I, Perspectives in Mathematics, vol. 10 (Academic Press, San Diego, CA, 1990), 161–209]. Applications of the results of this article have already been given by Kottwitz, assuming Arthur’s conjectures. Here, we give weaker unconditional applications in the cases of the groups GSp4 and GSp6.

According to the André–Oort conjecture, an algebraic curve in Y (1)n that is not equal to a special subvariety contains only finitely many points which correspond to ann-tuple of elliptic curves with complex multiplication. Pink’s conjecture generalizes the André–Oort conjecture to the extent that if the curve is not contained in a special subvariety of positive codimension, then it is expected to meet the union of all special subvarieties of codimension two in only finitely many points. We prove this for a large class of curves in Y (1)n. When restricting to special subvarieties of codimension two that are not strongly special we obtain finiteness for all curves defined over . Finally, we formulate and prove a variant of the Mordell–Lang conjecture for subvarieties of Y (1)n.

We construct six infinite series of families of pairs of curves (X,Y ) of arbitrarily high genus, defined over number fields, together with an explicit isogeny from the Jacobian of X to the Jacobian of Y splitting multiplication by 2, 3 or 4. For each family, we compute the isomorphism type of the isogeny kernel and the dimension of the image of the family in the appropriate moduli space. The families are derived from Cassou-Noguès and Couveignes’ explicit classification of pairs (f,g) of polynomials such that f(x1)−g(x2) is reducible.

Supplementary materials are available with this article.

We present p-adic algorithms for computing Hecke polynomials and Hecke eigenforms associated to spaces of classical modular forms, using the theory of overconvergent modular forms. The algorithms have a running time which grows linearly with the logarithm of the weight and are well suited to investigating the dimension variation of certain p-adically defined spaces of classical modular forms.

We consider exponential sums with x-coordinates of points qG and q−1G where G is a point of order T on an elliptic curve modulo a prime p and q runs through all primes up to N (with gcd (q,T)=1in the case of the points q−1G). We obtain a new bound on exponential sums with q−1G and correct an imprecision in the work of W. D. Banks, J. B. Friedlander, M. Z. Garaev and I. E. Shparlinski on exponential sums with qG. We also note that similar sums with g1/q for an integer g with gcd (g,p)=1have been estimated by J. Bourgain and I. E. Shparlinski.

We associate to certain filtrations of a graded linear series of a big line bundle a concave function on its Okounkov body, whose law with respect to the Lebesgue measure describes the asymptotic distribution of the jumps of the filtration. As a consequence, we obtain a Fujita-type approximation theorem in this general filtered setting. We then specialize these results to the filtrations by minima in the usual context of Arakelov geometry (and for more general adelically normed graded linear series), thereby obtaining in a simple way a natural construction of an arithmetic Okounkov body, the existence of the arithmetic volume as a limit and an arithmetic Fujita approximation theorem for adelically normed graded linear series. We also obtain an easy proof of the existence of the sectional capacity previously obtained by Lau, Rumely and Varley.

We prove the p-parity conjecture for elliptic curves over global fields of characteristic p>3. We also present partial results on the ℓ-parity conjecture for primes ℓ≠p.

We develop a new p-adic algorithm to compute the minimal polynomial of a class invariant. Our approach works for virtually any modular function yielding class invariants. The main algorithmic tool is modular polynomials, a concept which we generalize to functions of higher level.

We prove vanishing of the μ-invariant of the p-adic Katz L-function in N. M. Katz [p-adic L-functions for CM fields, Invent. Math. 49 (1978), 199–297].

For p=3 and p=5, we exhibit a finite nonsolvable extension of ℚ which is ramified only at p, proving in the affirmative a conjecture of Gross. Our construction involves explicit computations with Hilbert modular forms.

Let E/ℚ be an elliptic curve and let D<0 be a sufficiently large fundamental discriminant. If contains Heegner points of discriminant D, those points generate a subgroup of rank at least |D|δ, where δ>0 is an absolute constant. This result is compatible with the Birch and Swinnerton-Dyer conjecture.

We introduce a ‘limiting Frobenius structure’ attached to any degeneration of projective varieties over a finite field of characteristic p which satisfies a p-adic lifting assumption. Our limiting Frobenius structure is shown to be effectively computable in an appropriate sense for a degeneration of projective hypersurfaces. We conjecture that the limiting Frobenius structure relates to the rigid cohomology of a semistable limit of the degeneration through an analogue of the Clemens–Schmidt exact sequence. Our construction is illustrated, and conjecture supported, by a selection of explicit examples.

Cassels has described a pairing on the 2-Selmer group of an elliptic curve which shares some properties with the Cassels–Tate pairing. In this article, we prove that the two pairings are the same.

We determine the smallest possible canonical height for a non-torsion point P of an elliptic curve E over a function field ℂ(t) of discriminant degree 12n with a 2-torsion point for n=1,2,3, and with a 3-torsion point for n=1,2. For each m=2,3, we parametrize the set of triples (E,P,T) of an elliptic curve E/ℚ with a rational point P and m-torsion point T that satisfy certain integrality conditions by an open subset of ℙ2. We recover explicit equations for all elliptic surfaces (E,P,T)attaining each minimum by locating them as curves in our projective models. We also prove that for n=1,2 , these heights are minimal for elliptic curves over a function field of any genus. In each case, the optimal (E,P,T)are characterized by their patterns of integral points.

Soit A une variété abélienne définie sur un corps de nombres K, le nombre de points de torsion définis sur une extension finie L est borné polynomialement en terme du degré [L : K]. Lorsque A est isogène à un produit de variétés abéliennes simples de type GSp, c'est-à-dire dont le groupe de Mumford–Tate est « générique » (isomorphe au groupe des similitudes symplectiques) et vérifiant la conjecture de Mumford–Tate, nous calculons l'exposant optimal dans cette borne, en terme de la dimension des sous-variétés abéliennes de A. Le résultat est inconditionnel pour un produit de variétés abéliennes simples dont l'anneau d'endomorphismes est ℤ et la dimension n'appartient pas à un ensemble exceptionnel explicite S = {4, 10, 16, 32, …}. Par ailleurs nous prouvons, suivant une stratégie de Serre, que si la conjecture de Mumford–Tate est vraie pour des variétés abéliennes de type GSp, alors la conjecture de Mumford–Tate est vraie pour un produit de telles variétés abéliennes.

We discuss the Mordell–Weil sieve as a general technique for proving results concerning rational points on a given curve. In the special case of curves of genus 2, we describe quite explicitly how the relevant local information can be obtained if one does not want to restrict to mod p information at primes of good reduction. We describe our implementation of the Mordell–Weil sieve algorithm and discuss its efficiency.

Supplementary materials are available with this article.