Algebraic points of small height missing a union of varieties

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Let K be a number field, , or the field of rational functions on a smooth projective curve over a perfect field, and let V be a subspace of KN, N?2. Let ZK be a union of varieties defined over K such that V?ZK. We prove the existence of a point of small height in V?ZK, providing an explicit upper bound on the height of such a point in terms of the height of V and the degree of a hypersurface containing ZK, where dependence on both is optimal. This generalizes and improves upon the results of Fukshansky (2006) [6] and [7]. As a part of our argument, we provide a basic extension of the function field version of Siegel's lemma (Thunder, 1995) [21] to an inequality with inhomogeneous heights. As a corollary of the method, we derive an explicit lower bound for the number of algebraic integers of bounded height in a fixed number field.

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Siegel's lemma with additional conditions

Let K be a number field, and let W be a subspace of KN, N?1. Let V₁,…,VM be subspaces of KN of dimension less than dimension of W. We prove the existence of a point of small height in , providing an explicit upper bound on the height of such a point in terms of heights of W and V₁,…,VM. Our main tool is a counting estimate we prove for the number of points of a subspace of KN inside of an adelic cube. As corollaries to our main result we derive an explicit bound on the height of a nonvanishing point for a decomposable form and an effective subspace extension lemma.     

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Let E be an elliptic curve over F=F_q(t) having conductor (p)·∞, where (p) is a prime ideal in F_q[t]. Let d∈F_q[t] be an irreducible polynomial of odd degree, and let . Assume (p) remains prime in K. We prove the analogue of the formula of Gross for the special value L(E⊗_FK,1). As a consequence, we obtain a formula for the order of the Tate-Shafarevich group Ш(E/K) when L(E⊗_FK,1)≠0.     

On the Ono invariants of imaginary quadratic number fields

J. Cohen, J. Sonn, F. Sairaiji and K. Shimizu proved that there are only finitely many imaginary quadratic number fields K whose Ono invariants OnoK are equal to their class numbers hK. Assuming a Restricted Riemann Hypothesis, namely that the Dedekind zeta functions of imaginary quadratic number fields K have no Siegel zeros, we determine all these K's. There are 114 such K's. We also prove that we are missing at most one such K. M. Ishibashi proved that if OnoK is large enough compared with hK, then the ideal class groups of K is cyclic. We give a short proof and a precision of Ishibashi's result. We prove that there are only finitely many imaginary quadratic number fields K satisfying Ishibashi's sufficient condition. Assuming our Restricted Riemann Hypothesis, we prove that the absolute values dK of their discriminants are less than 2.3⋅¹⁰⁹. We determine all these K's with dK?¹⁰⁶. There are 76 such K's. We prove that there is at most one such K with dK?1.8⋅¹⁰¹¹.     

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A lower bound for the canonical height on elliptic curves over abelian extensions

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Adelic openness for Drinfeld modules in generic characteristic

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K3 surfaces, rational curves, and rational points

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Prescribing valuations of the order of a point in the reductions of abelian varieties and tori

Let G be the product of an abelian variety and a torus defined over a number field K. Let R be a K-rational point on G of infinite order. Call nR the number of connected components of the smallest algebraic K-subgroup of G to which R belongs. We prove that nR is the greatest positive integer which divides the order of for all but finitely many primes p of K. Furthermore, let m>0 be a multiple of nR and let S be a finite set of rational primes. Then there exists a positive Dirichlet density of primes p of K such that for every ? in S the ?-adic valuation of the order of equals v?(m).     

Moving subvarieties by endomorphisms

Let K be an algebraically closed field of characteristic zero and let A be a semiabelian variety defined over K. Let End(A) be the ring of endomorphisms of A. Let X ⊂ A be a subvariety of smaller dimension. We show that does not equal A(K). In particular, we may take K to be countable.     

Bernoulli-Hurwitz numbers, Wieferich primes and Galois representations

Let K be a quadratic imaginary number field with discriminant DK≠−3,−4 and class number one. Fix a prime p?7 which is unramified in K. Given an elliptic curve A/Q with complex multiplication by K, let be the representation which arises from the action of Galois on the Tate module. Herein it is shown that, for all but finitely many inert primes p, the image of a certain deformation of is “as large as possible”, that is, it is the full inverse image of a Cartan subgroup of SL(2,Zp). If p splits in K, then the same result holds as long as a certain Bernoulli-Hurwitz number is a p-adic unit which, in turn, is equivalent to a prime ideal not being a Wieferich place. The proof rests on the theory of elliptic units of Robert and Kubert-Lang, and on the two-variable main conjecture of Iwasawa theory for quadratic imaginary fields.     

A formula for the central value of certain Hecke L-functions

Let be the negative of a prime, and O_K its ring of integers. Let D be a prime ideal in O_K of prime norm congruent to . Under these assumptions, there exists Hecke characters ψ_D of K with conductor (D) and infinite type (1,0). Their L-series L(ψ_D,s) are associated to a CM elliptic curve A(N,D) defined over the Hilbert class field of K. We will prove a Waldspurger-type formula for L(ψ_D,s) of the form L(ψ_D,1)=Ω∑_[A],Ir(D,[A],I)m_[A],I([D]) where the sum is over class ideal representatives I of a maximal order in the quaternion algebra ramified at |N| and infinity and [A] are class group representatives of K. An application of this formula for the case N=-7 will allow us to prove the non-vanishing of a family of L-series of level 7|D| over K.     

The Galois representations associated to a Drinfeld module in special characteristic—I: Zariski density

Let ? be a Drinfeld A-module of rank r over a finitely generated field K. Assume that ? has special characteristic p₀ and consider any prime p≠p₀ of A. If End_K^sep(?)=A, we prove that the image of Gal(K^sep/K) in its representation on the p-adic Tate module of ? is Zariski dense in GL_r.     

On the independence of Heegner points in the function field case

Let ∞ be a fixed place of a global function field k. Let E be an elliptic curve defined over k which has split multiplicative reduction at ∞ and fix a modular parametrization ΦE:X₀(N)→E. Let be Heegner points associated to the rings of integers of distinct quadratic “imaginary” fields K₁,…,Kr over (k,∞). We prove that if the “prime-to-2p” part of the ideal class numbers of ring of integers of K₁,…,Kr are larger than a constant C=C(E,ΦE) depending only on E and ΦE, then the points P₁,…,Pr are independent in . Moreover, when k is rational, we show that there are infinitely many imaginary quadratic fields for which the prime-to-2p part of the class numbers are larger than C.     

A note on the Mordell-Weil rank modulo n

Conjecturally, the parity of the Mordell-Weil rank of an elliptic curve over a number field K is determined by its root number. The root number is a product of local root numbers, so the rank modulo 2 is (conjecturally) the sum over all places of K of a function of elliptic curves over local fields. This note shows that there can be no analogue for the rank modulo 3, 4 or 5, or for the rank itself. In fact, standard conjectures for elliptic curves imply that there is no analogue modulo n for any n>2, so this is purely a parity phenomenon.     

Densité de points et minoration de hauteur

We obtain a lower bound for the normalised height of a non-torsion subvariety V of a C.M. abelian variety. This lower bound is optimal in terms of the geometric degree of V, up to a power of a “log”. We thus extend the results of Amoroso and David on the same problem on a multiplicative group . We prove furthermore that the optimal lower bound (conjectured by David and Philippon) is a corollary of the conjecture of David and Hindry on the abelian Lehmer's problem. We deduce these results from a density theorem on the non-torsion points of V.     

Borne uniforme pour les homothéties dans l?image de Galois associée aux courbes elliptiques

Let K be a fixed number field and GK its absolute Galois group. We give a bound C(K), depending only on the degree, the class number and the discriminant of K, such that for any elliptic curve E defined over K and any prime number p strictly larger than C(K), the image of the representation of GK attached to the p-torsion points of E contains a subgroup of homotheties of index smaller than 12.     

Visualizing elements of order two in the Weil-Châtelet group

Let E be an elliptic curve over an infinite field K with characteristic ≠2, and σ∈H¹(G_K,E)[2] a two-torsion element of its Weil-Châtelet group. We prove that σ is always visible in infinitely many abelian surfaces up to isomorphism, in the sense put forward by Cremona and Mazur in their article (J. Exp. Math. 9(1) (2000) 13). Our argument is a variant of Mazur's proof, given in (Asian J. Math. 3(1) (1999) 221), for the analogous statement about three-torsion elements of the Shafarevich-Tate group in the setting where K is a number field. In particular, instead of the universal elliptic curve with full level-three-structure, our proof makes use of the universal elliptic curve with full level-two-structure and an invariant differential.