Rational eigenvectors in spaces of ternary forms

We describe the explicit computation of linear combinations of ternary quadratic forms which are eigenvectors, with rational eigenvalues, under all Hecke operators. We use this process to construct, for each elliptic curve of rank zero and conductor for which or is squarefree, a weight 3/2 cusp form which is (potentially) a preimage of the weight two newform under the Shimura correspondence.

     

Twists of elliptic curves

If E is an elliptic curve over , then let E(D) denote theD-quadratic twist of E. It is conjectured that there are infinitely many primesp for which E(p) has rank 0, and that there are infinitely many primes for which has positive rank. For some special curvesE we show that there is a set S of primes p with density for which if is a squarefree integer where , then E(D) has rank 0. In particular E(p) has rank 0 for every . As an example let E1 denote the curve .Then its associated set of primes S₁ consists of the prime11 and the primes p for which the order of the reduction ofX₀(11) modulo p is odd. To obtain the general result we show for primes that the rational factor of L(E(p),1) is nonzero which implies thatE(p) has rank 0. These special values are related to surjective Galois representations that are attached to modularforms. Another example of this result is given, and we conclude with someremarks regarding the existence of positive rank prime twists via polynomialidentities.     

Representations of integers by certain positive definite binary quadratic forms

We prove part of a conjecture of Borwein and Choi concerning an estimate on the square of the number of solutions to n=x ²+Ny ² for a squarefree integer N.      

Iwasawa Theory for Elliptic Curves at Unstable Primes

In this paper we examine the Iwasawa theory of modular elliptic curves E defined over Q without semi-stable reduction at p. By constructing p-adic L-functions at primes of additive reduction, we formulate a "Main Conjecture" linking this L-function with a certain Selmer group for E over the Zp-extension. Thus the leading term is expressible in terms of III_E, E(Q)_tors and a p-adic regulator term.     

The eligible numbers of positive definite ternary forms

Several nontrivial genera of positive ternary forms with small discriminants have been studied in this paper. Especially we prove that there are only finitely many, square-free eligible numbers in the genus of and in the genus of which cannot be represented by and , respectively. Our method is to use modular forms of weight 3/2. Received February 18, 1999; in final form July 29, 1999 / Published online September 14, 2000     

Congruences between Modular Forms, Cyclic Isogenies of Modular Elliptic Curves, and Integrality of -Functions

Let be a congruence subgroup of type and of level . We study congruences between weight 2 normalized newforms and Eisenstein series on modulo a prime above a rational prime . Assume that , is a common eigenfunction for all Hecke operators and is ordinary at . We show that the abelian variety associated to and the cuspidal subgroup associated to intersect non-trivially in their -torsion points. Let be a modular elliptic curve over with good ordinary reduction at . We apply the above result to show that an isogeny of degree divisible by from the optimal curve in the -isogeny class of elliptic curves containing to extends to an étale morphism of Néron models over if . We use this to show that -adic distributions associated to the -adic -functions of are -valued.

     

Elliptic Apostol sums and their reciprocity laws

We introduce an elliptic analogue of the Apostol sums, which we call elliptic Apostol sums. These sums are defined by means of certain elliptic functions with a complex parameter having positive imaginary part. When , these elliptic Apostol sums represent the well-known Apostol generalized Dedekind sums. Also these elliptic Apostol sums are modular forms in the variable . We obtain a reciprocity law for these sums, which gives rise to new relations between certain modular forms (of one variable).

     

Elliptic subcovers of hyperelliptic curves

The main result of this paper states that if C is a hyperelliptic curve of even genus over an arbitrary field K , then there is a natural bijection between the set of equivalence classes of elliptic subcovers of and the set of elliptic subgroups of its Jacobian .     

On positive definite quadratic forms in correlatedt variables

Summary In this paper we extend Ruben's [4] result for quadratic forms in normal variables. He represented the distribution function of the quadratic form in normal variables as an infinite mixture of chi-square distribution functions. In the central case, we show that the distribution function of a quadratic form int-variables can be represented as a mixture of beta distribution functions. In the noncentral case, the distribution function presented is an infinite series in beta distribution functions. An application to quadratic discrimination is given.     

Eta-quotients and elliptic curves

In this paper we list all the weight newforms that are products and quotients of the Dedekind eta-function

where There are twelve such and we give a model for the strong Weil curve whose Hasse-Weil function is the Mellin transform for each of them. Five of the have complex multiplication, and we give elementary formulae for their Fourier coefficients which are sums of Hecke Grössencharacter values. These formulae follow easily from well known series infinite product identities.

     

Finiteness theorems for positive definite -regular quadratic forms

An integral quadratic form of variables is said to be -regular if globally represents all quadratic forms of variables that are represented by the genus of . For any , it is shown that up to equivalence, there are only finitely many primitive positive definite integral quadratic forms of variables that are -regular. We also investigate similar finiteness results for almost -regular and spinor -regular quadratic forms. It is shown that for any , there are only finitely many equivalence classes of primitive positive definite spinor or almost -regular quadratic forms of variables. These generalize the finiteness result for 2-regular quaternary quadratic forms proved by Earnest (1994).

     

Elliptic curves retaining their rank in finite extensions and Hilbert's Tenth Problem for rings of algebraic numbers

Using Poonen's version of the ``weak vertical method' we produce new examples of ``large' and ``small' rings of algebraic numbers (including rings of integers) where and/or the ring of integers of a subfield are existentially definable and/or where the ring version of Mazur's conjecture on the topology of rational points does not hold.

     

On the Density of Elliptic Curves

We show that 17.9% of all elliptic curves over Q, ordered by their exponential height, are semistable, and that there is a positive density subset of elliptic curves for which the root numbers are uniformly distributed. Moreover, for any > 1/6 (resp. > 1/12) the set of Frey curves (resp. all elliptic curves) for which the generalized Szpiro Conjecture |(E)| N _E ¹² is false has density zero. This implies that the ABC Conjecture holds for almost all Frey triples. These results remain true if we use the logarithmic or the Faltings height. The proofs make use of the fibering argument in the square-free sieve of Gouvêa and Mazur. We also obtain conditional as well as unconditional lower bounds for the number of curves with Mordell–Weil rank 0 and 2, respectively.     

Generating More MNT Elliptic Curves

In their seminal paper, Miyaji et al. [13] describe a simple method for the creation of elliptic curves of prime order with embedding degree 3, 4, or 6. Such curves are important for the realisation of pairing-based cryptosystems on ordinary (non-supersingular) elliptic curves. We provide an alternative derivation of their results, and extend them to allow for the generation of many more suitable curves. Research supported by Enterprise Ireland grant IF/2002/0312/N.     

A note on a question of J. Nekovár and the Birch and Swinnerton-Dyer conjecture

If is a square-free integer, then let denote the elliptic curve over given by the equation

Let denote the Hasse-Weil -function of , and let denote the `algebraic part' of the central critical value . Using a theorem of Sturm, we verify a congruence conjectured by J. Neková\v{r}. By his work, if denotes the 3-Selmer group of and is a square-free integer with , then we find that

     

强正定二次型与Weyl-Heisenberg 框架的构造

本文研究了一类具有特殊结构的无限维二次型, 得到这类二次型的对称矩阵是符号为多项式的模的平方的Laurent 矩阵, 进一步得到了这类二次型是强正定的判断标准以及一类Weyl-Heisenberg 框架的构造. 本文还研究了这类二次型的矩阵的所有有限维主对角子矩阵的强正定性, 并由此得到一类子空间Weyl-Heisenberg 框架的构造. 最后举例说明本文的主要结果及其应用. 本文建立了两个看似不相关的领域间的联系.     

Elliptic polylogarithms: An analytic theory

In this article we introduce a natural elliptic generalization of theclassical polylogarithms, study the properties of these functions and theirrelations with Eisenstein series.     

Multiplicities of non-arithmetic ternary quadratic forms and elliptic curves of positive rank

In this note we will characterize all non-arithmetic ternary quadratic forms with unbounded multiplicities. It turns out that their existence is closely related with certain classes of elliptic curves whose investigation is interesting on its own. Received March 3, 1997; in final form November 13, 1997