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Almost regular quaternary quadratic forms

Jacek Bochnak, Byeong-Kweon Oh (2008)

Annales de l’institut Fourier

We investigate the almost regular positive definite integral quaternary quadratic forms. In particular, we show that every such form is p -anisotropic for at most one prime number p . Moreover, for a prime p there is an almost regular p -anisotropic quaternary quadratic form if and only if p 37 . We also study the genera containing some almost regular p -anisotropic quaternary form. We show several finiteness results concerning the families of these genera and give effective criteria for almost regularity....

CM liftings of supersingular elliptic curves

Ben Kane (2009)

Journal de Théorie des Nombres de Bordeaux

Assuming GRH, we present an algorithm which inputs a prime p and outputs the set of fundamental discriminants D < 0 such that the reduction map modulo a prime above p from elliptic curves with CM by 𝒪 D to supersingular elliptic curves in characteristic p is surjective. In the algorithm we first determine an explicit constant D p so that | D | > D p implies that the map is necessarily surjective and then we compute explicitly the cases | D | < D p .

Domaines de Voronoï et algorithme de réduction des formes quadratiques définies positives

David-Olivier Jaquet (1990)

Journal de théorie des nombres de Bordeaux

J’illustre la situation générale par un exemple simple, qui permet de mieux comprendre la géométrie de l’espace des domaines de Voronoï. Ensuite, je donne des résultats généraux sur les arêtes d’un domaine de Voronoï. Finalement, pour les représentants des 15 classes connues de formes parfaites à 7 variables, non équivalentes à E 7 et qui possèdent plus de 28 vecteurs minimaux, je fournis une description détaillée de leurs orbites de voisines.

Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52

Ebénézer Ntienjem (2017)

Open Mathematics

The convolution sum, [...] ∑(l,m)∈N02αl+βm=nσ(l)σ(m), ( l , m ) 0 2 α l + β m = n σ ( l ) σ ( m ) , where αβ = 22, 44, 52, is evaluated for all natural numbers n. Modular forms are used to achieve these evaluations. Since the modular space of level 22 is contained in that of level 44, we almost completely use the basis elements of the modular space of level 44 to carry out the evaluation of the convolution sums for αβ = 22. We then use these convolution sums to determine formulae for the number of representations of a positive integer by...

Faithfully quadratic rings - a summary of results

M. Dickmann, F. Miraglia (2016)

Banach Center Publications

This is a summary of some of the main results in the monograph Faithfully Ordered Rings (Mem. Amer. Math. Soc. 2015), presented by the first author at the ALANT conference, Będlewo, Poland, June 8-13, 2014. The notions involved and the results are stated in detail, the techniques employed briefly outlined, but proofs are omitted. We focus on those aspects of the cited monograph concerning (diagonal) quadratic forms over preordered rings.

Five regular or nearly-regular ternary quadratic forms

William C. Jagy (1996)

Acta Arithmetica

1. Introduction. In a recent article [6], the positive definite ternary quadratic forms that can possibly represent all odd positive integers were found. There are only twenty-three such forms (up to equivalence). Of these, the first nineteen were proven to represent all odd numbers. The next four are listed as "candidates". The aim of the present paper is to show that one of the candidate forms h = x² + 3y² + 11z² + xy + 7yz does represent all odd (positive) integers, and that it is regular in...

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