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A new form of the circle method, and its application to quadratic forms.

Journal für die reine und angewandte Mathematik

Acta Arithmetica

A note on the representability of quaternary quadratic forms as sums of squares of two linear forms

Portugaliae mathematica

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 $p\le 37$. 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....

An Asymmetric Inequality for non-homogeneous ternary quadratic forms.

Monatshefte für Mathematik

Applications des formules générales qui donnent la solution complète, en nombres entiers, de l'équation homogène du second degré contenant un nombre quelconque d'inconnues

Nouvelles annales de mathématiques : journal des candidats aux écoles polytechnique et normale

Acta Arithmetica

Séminaire Dubreil. Algèbre et théorie des nombres

Class numbers of positive definite quaternary quadratic forms.

Journal für die reine und angewandte Mathematik

CM liftings of supersingular elliptic curves

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 $|D|<{D}_{p}$.

Correspondencia entre formas ternarias enteras y órdenes cuatemiónicos.

Revista de la Real Academia de Ciencias Exactas Físicas y Naturales

Commentarii mathematici Helvetici

De la réduction des formes quadratiques ternaires positives et de son application aux irrationnelles du troisième degré

Annales scientifiques de l'École Normale Supérieure

Deformations of Diophantine systems for quadratic forms of the root lattices ${A}_{n}$.

Zapiski Nauchnykh Seminarov POMI

Diskontinuitätsbereich für arithmetische Äquivalenz.

Journal für die reine und angewandte Mathematik

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

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.

Erratum “On the computation of quadratic $2$-class groups”

Journal de théorie des nombres de Bordeaux

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

Open Mathematics

The convolution sum, [...] ∑(l,m)∈N02αl+βm=nσ(l)σ(m), $\begin{array}{c}\sum _{\genfrac{}{}{0pt}{}{\left(l\phantom{\rule{0.166667em}{0ex}},m\right)\in {ℕ}_{0}^{2}}{\alpha \phantom{\rule{0.166667em}{0ex}}l+\beta \phantom{\rule{0.166667em}{0ex}}m=n}}\sigma \left(l\right)\sigma \left(m\right),\end{array}$ 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

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

Acta Arithmetica

1. Introduction. In a recent article , 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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