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Acta Arithmetica

A remark on Tate's algorithm and Kodaira types

Acta Arithmetica

We remark that Tate’s algorithm to determine the minimal model of an elliptic curve can be stated in a way that characterises Kodaira types from the minimum of $v\left({a}_{i}\right)/i$. As an application, we deduce the behaviour of Kodaira types in tame extensions of local fields.

Acta Arithmetica

Algebraic values of G-functions.

Journal für die reine und angewandte Mathematik

An alternative way to classify some Generalized Elliptic Curves and their isotopic loops

Commentationes Mathematicae Universitatis Carolinae

The Generalized Elliptic Curves $\left(GECs\right)$ are pairs $\left(Q,T\right)$, where $T$ is a family of triples $\left(x,y,z\right)$ of “points” from the set $Q$ characterized by equalities of the form $x.y=z$, where the law $x.y$ makes $Q$ into a totally symmetric quasigroup. Isotopic loops arise by setting $x*y=u.\left(x.y\right)$. When $\left(x.y\right).\left(a.b\right)=\left(x.a\right).\left(y.b\right)$, identically $\left(Q,T\right)$ is an entropic $GEC$ and $\left(Q,*\right)$ is an abelian group. Similarly, a terentropic $GEC$ may be characterized by ${x}^{2}.\left(a.b\right)=\left(x.a\right)\left(x.b\right)$ and $\left(Q,*\right)$ is then a Commutative Moufang Loop $\left(CML\right)$. If in addition ${x}^{2}=x$, we have Hall $GECs$ and $\left(Q,*\right)$ is an exponent $3$

Acta Arithmetica

An analytic construction of degenerating curves over complete local rings

Compositio Mathematica

An elementary proof of the Mazur-Tate-Teitelbaum conjecture for elliptic curves.

Documenta Mathematica

Arithmetic differential equations in several variables

Annales de l’institut Fourier

We survey recent work on arithmetic analogues of ordinary and partial differential equations.

Beyond two criteria for supersingularity: coefficients of division polynomials

Journal de Théorie des Nombres de Bordeaux

Let $f\left(x\right)$ be a cubic, monic and separable polynomial over a field of characteristic $p\ge 3$ and let $E$ be the elliptic curve given by ${y}^{2}=f\left(x\right)$. In this paper we prove that the coefficient at ${x}^{\frac{1}{2}p\left(p-1\right)}$ in the $p$–th division polynomial of $E$ equals the coefficient at ${x}^{p-1}$ in $f{\left(x\right)}^{\frac{1}{2}\left(p-1\right)}$. For elliptic curves over a finite field of characteristic $p$, the first coefficient is zero if and only if $E$ is supersingular, which by a classical criterion of Deuring (1941) is also equivalent to the vanishing of the second coefficient. So the zero loci...

Computer-aided serendipity

Rendiconti del Seminario Matematico della Università di Padova

Descente infinie et hauteur p-adique sur les courbes elliptiques à multiplication complexe.

Inventiones mathematicae

Dynamical systems arising from elliptic curves

Colloquium Mathematicae

We exhibit a family of dynamical systems arising from rational points on elliptic curves in an attempt to mimic the familiar toral automorphisms. At the non-archimedean primes, a continuous map is constructed on the local elliptic curve whose topological entropy is given by the local canonical height. Also, a precise formula for the periodic points is given. There follows a discussion of how these local results may be glued together to give a map on the adelic curve. We are able to give a map whose...

Explicit local heights.

The New York Journal of Mathematics [electronic only]

Fake CM and the stable model of ${X}_{0}\left(N{p}^{3}\right)$.

Documenta Mathematica

Formes de jacobi et formule de Weber $p$-adique

Journal de théorie des nombres de Bordeaux

Dans ce texte, on construit sur un corps local de caractéristique strictement positive, un analogue $p$-adique aux formes de Jacobi méromorphes complexes ${D}_{L}\left(z;\varphi \right)$, étudiées dans  et . Le théorème principal établit que les formes de Jacobi $p$-adiques obtenues satisfont deux relations de distribution et d’inversion additives. L’analogue $p$-adique à une formule de Weber généralisée est prouvé comme corollaire du théorème principal.

Généralités sur les courbes elliptiques

Séminaire Delange-Pisot-Poitou. Théorie des nombres

Groupe de Selmer d'une courbe elliptique à multiplication complexe

Compositio Mathematica

Indices of double coverings of genus 1 over $p$-adic fields

Annales de la Faculté des sciences de Toulouse : Mathématiques

Invariance of the parity conjecture for $p$-Selmer groups of elliptic curves in a ${D}_{2{p}^{n}}$-extension

Bulletin de la Société Mathématique de France

We show a $p$-parity result in a ${D}_{2{p}^{n}}$-extension of number fields $L/K$ ($p\ge 5$) for the twist $1\oplus \eta \oplus \tau$: $W\left(E/K,1\oplus \eta \oplus \tau \right)={\left(-1\right)}^{〈1\oplus \eta \oplus \tau ,{X}_{p}\left(E/L\right)〉}$, where $E$ is an elliptic curve over $K$, $\eta$ and $\tau$ are respectively the quadratic character and an irreductible representation of degree $2$ of $\mathrm{Gal}\left(L/K\right)={D}_{2{p}^{n}}$, and ${X}_{p}\left(E/L\right)$ is the $p$-Selmer group. The main novelty is that we use a congruence result between ${\epsilon }_{0}$-factors (due to Deligne) for the determination of local root numbers in bad cases (places of additive reduction above 2 and 3). We also give applications to the $p$-parity conjecture (using...

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