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970.36224
Kraus,
Alain
Determination du poids et du conducteur associes aux representations
des points de $p$-torsion d'une courbe elliptique. (Determination of weight
and conductor associated to representations of $p$-torsion points of an
elliptic curve). (French)
[J] Diss. Math. 364, 39 p. (1997). [ISSN 0012-3862]
Review in preparation
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Classification:
-
*11G05
Elliptic curves over global fields
-
14H52
Elliptic curves
865.11050
Kraus,
Alain
Courbes elliptiques semi-stables et corps quadratiques. (Semi-stable
elliptic curves and quadratic fields). (French)
[J] J. Number Theory 60, No.2, 245-253, Art. No.0122 (1996). [ISSN
0022-314X]
This paper considers the Galois representations on the $p$-torsion
points $E\sb p$ of elliptic curves $E$ over a quadratic number field. For
a fixed elliptic curve $E$ over a number field $K$ without complex multiplication,
{\it J.-P. Serre} [Invent. Math. 15, 259-331 (1972; Zbl
235.14012)] proves that for $p$ large enough, the representation $\varphi\sb
p :G\sb K \to \Aut (E\sb p)$ is surjective $(G\sb K$ being the absolute
Galois group of $K)$. He asks if it is possible to have a bound that only
depends on $K$, but not on $E$. Nothing is known about the question in
this generality. However, it follows from results of {\it J.-P. Serre}
and {\it B. Mazur} [Invent. Math. 44, 129-162 (1978; Zbl
386.14009)] that for semi-stable elliptic curves over $\bbfQ$, $\varphi\sb
p$ is surjective as soon as $p\ge 11$.\par The author obtains some results
in this direction for quadratic number fields $K$. He proves that there
is some number $N(K)$ such that for $p>N(K)$, the Galois representation
$\varphi\sb p$ is irreducible for each semi-stable elliptic curve over
$K$. From this, he deduces that if $K=\bbfQ(\sqrt l)$ with a prime $l\equiv
1\bmod 4$, there is another number $N'(K)$ such that for $p>N'(K)$, the
representations $\varphi\sb p$ are surjective for all semi-stable elliptic
curves over $K$. For both results, explicit bounds are given if $K$ has
class number one.
[ M.Stoll
(Duesseldorf) ]
Citations: Zbl.235.14012;
Zbl.386.14009
Mots clés: $p$-torsion points of elliptic curves; Galois
representations; quadratic number field; semi-stable elliptic curves
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Classification:
-
*11G05
Elliptic curves over global fields
-
14H52
Elliptic curves
857.11027
Halberstadt,
Emmanuel;
Kraus, Alain
Sur la comparaison galoisienne des points de torsion des courbes
elliptiques. (On the Galois comparison of the torsion points of elliptic
curves). (French)
[J] C. R. Acad. Sci., Paris, Ser. I 322, No.4, 313-316 (1996). [ISSN
0764-4442]
Let $E$ and $E'$ be two elliptic curves defined over $\bbfQ$ which
are not isogenous over $\bbfQ$. Let $n \ge 1$ be an integer. Denote by
$E[n]$ and $E'[n]$ the group of $n$-torsion points of $E$ and $E'$ over
$\bbfQ$, respectively. Let $G\sb \bbfQ = \text {Gal} (\overline \bbfQ/
\bbfQ)$ be the absolute Galois group of $\bbfQ$ and $\rho\sb{E,n}:G\sb
\bbfQ \to \Aut (E[n])$ be the usual representation of $G$ on $\Aut (E[n])$.
{\it B. Mazur} [Invent. Math. 44, 129-162 (1978; Zbl
386.14009), p. 133] asks for examples of two elliptic curves such that
for $n\ge 7$ we have $\rho\sb{E,n} \cong \rho\sb{E',n}$. {\it A. Kraus}
and {\it J. Oesterle} give examples in the case $n=7$ [Math. Ann. 293,
259-275 (1992; Zbl
733.14017)]. This problem is closely related to the existence of rational
points on certain twists of the modular curves $Y(n)$ when these curves
have genus at least two and $n\ge 7$ [{\it A. Kraus}, {\it J. Oesterle},
loc. cit. and {\it B. Mazur}, loc. cit.].\par Let $p$ be a prime number.
The first result of the paper says that if $p\ge 11$, $p\ne 13$, the image
of $\rho\sb p$ is contained in the normalizer of a split Cartan subgroup
and there exists another elliptic curve $E'$ defined over $\bbfQ$ such
that $E[p] \cong E'[p]$ as $G$-modules, then $E$ and $E'$ have the same
conductor. As a corollary the authors show that if there exists an elliptic
$E'$ as above and $E$ and $E'$ are modular, then $E$ is isogenous to $E'$
over $\bbfQ$ if $p\ge 4 \sqrt N (1+\log \log N)\sp{1/2}$, where $N$ denotes
the conductor of $E$.\par The conditions of the theorem are satisfied if
$E$ has complex multiplication by an order of an imaginary quadratic field
$K$ and $p\ge 5$ is decomposed in $K$. In this case the authors' second
result is that for an elliptic curve $E$ defined over $\bbfQ$ with complex
multiplication as above such that its $j$-invariant is different from 0
and 1728, the conclusion of the corollary holds without any restriction
on the prime $p$ nor any hypothesis of modularity.
[ A.Pacheco
(Rio de Janeiro) ]
Citations: Zbl.386.14009;
Zbl.733.14017
Mots clés: Torsion subgroups; rational isogenies; $n$-torsion
points; elliptic curves; conductor
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Classification:
-
*11G05
Elliptic curves over global fields
-
14H52
Elliptic curves
855.11015
Kraus,
Alain
Sur les equations $a\sp p+b\sp p+15c\sp p=0$ et $a\sp p+3b\sp p+5c\sp
p=0$. (On the equations $a\sp p+b\sp p+15c\sp p=0$ and $a\sp p+3b\sp p+5c\sp
p=0$). (French)
[J] C. R. Acad. Sci., Paris, Ser. I 322, No.9, 809-812 (1996). [ISSN
0764-4442]
In the paper under review the author investigates the integer solutions
of the equations $$a\sp p+ b\sp p+ 15c\sp p= 0 \qquad \text {and} \qquad
a\sp p+ 3b\sp p+ 5c\sp p=0,$$ where $p$ is a prime $\geq 7$. He proves,
using Wiles' work on the Taniyama-Weil conjecture, that the above equations
have no solutions $(a, b, c)\in \bbfZ\sp 3$ with $abc \ne 0$ if $p$ divides
$abc$ or if one of the numbers $2p+1$ or $4p+1$ is a prime. In the case
where $p$ is $< 100$ these equations have no solutions $(a, b, c)\in
\bbfZ\sp 3$ with $abc \ne 0$.
[ D.Poulakis
(Thessaloniki) ]
Mots clés: higher degree diophantine equations; elliptic
curves; modular representations; Taniyama-Weil conjecture
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Classification:
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*11D41
Higher degree diophantine equations
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11G05
Elliptic curves over global fields
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14H52
Elliptic curves
853.11046
Kraus,
Alain
Sur les modules des points de 7-torsion d'une famille de courbes
elliptiques. (About the 7-torsion modules of points of a family of elliptic
curves). (French)
[J] Ann. Inst. Fourier 46, No.4, 899-907 (1996). [ISSN 0373-0956]
Do there exist two elliptic curves over $\bbfQ$, which are non isogenous
over $\bbfQ$ and an integer $n\geq 7$, such that the representations of
$\text{Gal} (\overline \bbfQ\slash \bbfQ)$ defined by their $n$-torsion
groups of points are symplectically isomorphic? This question has been
raised by B. Mazur in 1978. In the case $n=7$, we get infinitely many examples
giving a positive answer to that question.
Mots clés: elliptic curves; torsion points; Galois representations
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Classification:
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*11G05
Elliptic curves over global fields
-
14H52
Elliptic curves
862.11037
Kraus,
Alain
Une remarque sur les points de torsion des courbes elliptiques.
(A remark on the torsion points of elliptic curves). (French)
[J] C. R. Acad. Sci., Paris, Ser. I 321, No.9, 1143-1146 (1995). [ISSN
0764-4442]
Let $E$ be an elliptic curve defined over $\bbfQ$. We denote by $S\sb
E$ the set of primes $p$ such that $E$ has bad reduction at $p$. Put $N\sb
E = \prod\sb{p \in S\sb E} p$. Let $\ell$ be a prime and let $E\sb \ell$
be the subgroup of $\ell$-torsion points of $E(\overline \bbfQ)$. The Galois
group $\text {Gal} (\overline \bbfQ/ \bbfQ)$ acts on $E\sb \ell$ and thus
we have a homomorphism $\rho\sb \ell: \text {Gal} (\overline \bbfQ/ \bbfQ)
\to \Aut (E\sb \ell)$. In the case where $E$ has no complex multiplication
over $\overline \bbfQ$, {\it J. P. Serre} proved that the homomorphism
$\rho\sb \ell$ is surjective for every $\ell \ge N\sb E\sp c$, where $c$
is an absolute constant [Publ. Math., Inst. Hautes Etud. Sci. 54, 123-202
(1981; Zbl
496.12011)]. \par In the paper under review the author proves that
if the elliptic curve $E$ is modular and with no complex multiplication,
then $\rho\sb \ell$ is surjective for every prime $\ell$ with $$\ell \ge
68N\sb E (1 + \log \log N\sb E)\sp{1/2}.$$
[ D.Poulakis
(Thessaloniki) ]
Citations: Zbl.496.12011
Mots clés: modular elliptic curve; torsion points; $\ell$-adic
representation; elliptic curve
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Classification:
-
*11G05
Elliptic curves over global fields
-
14H52
Elliptic curves
792.14014
Kraus,
Alain
Sur le defaut de semi-stabilite des courbes elliptiques a reduction
additive. (On the failure of semistability of elliptic curves with additive
reduction). (French)
[J] Manuscr. Math. 69, No.4, 353-385 (1990).
Let $K$ be a field of characteristic 0 complete with respect to a discrete
valuation $v$, with a perfect residue field of characteristic $p>0$. Let
$\overline K$ be an algebraic closure of $K$ and $K\sb{nr}$ its maximal
unramified subextension. Let $E$ be an elliptic curve over $K$ with an
integral modular invariant. The curve $E$ has potentially good reduction
at $v$, and there exists a smallest extension $L$ of $K\sb{nr}$ over which
$E$ has good reduction at $v$. The Galois group $\text {Gal} (L/K\sb{nr})$
is known in the case $p \ge 5$. In this paper we give recipes to determine
this group in the cases $p=2$ and $p=3$.
Mots clés: characteristic $p$; Galois group of number
fields of curves; elliptic curve; potentially good reduction
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Classification:
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*14H25
Arithmetic ground fields (curves)
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14G20
p-adic ground fields
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11S20
Galois theory for local fields
628.14024
Kraus,
Alain
Quelques remarques a propos des invariants $c\sb 4$, $c\sb 6$ et
$\Delta$ d'une courbe elliptique. (Some remarks about the invariants $c\sb
4$, $c\sb 6$ and $\Delta$ of an elliptic curve). (French)
[J] Acta Arith. 54, No.1, 75-80 (1989).
Soit K un corps de caracteristique nulle muni d'une valuation discrete
v normalisee par $v(K\sp *)={\bbfZ}$. Soient ${\frak O}\sb K$ l'anneau
de valuation, p la caracteristique residuelle et $e=v(p)$. A une equation
de Weierstrass affine (W) a coefficients dans ${\frak O}\sb K$, $(W)\ y\sp
2+a\sb 1xy+a\sb 3y=x\sp 3+a\sb 2x\sp 2+a\sb 4x+a\sb 6,$ on associe des
invariants $c\sb 4$, $c\sb 6$ et $\Delta$ qu l'on calcule de maniere standard.
\par Etant donnes des elements $c\sb 4$, $c\sb 6$ et $\Delta$ de ${\frak
O}\sb K$ satisfaisant a $(*)\ c\sp 3\sb 4-c\sp 2\sb 6=1728\Delta$ et $\Delta
\ne 0$ on se propose de determiner des conditions necessaires et suffisantes
pour que $c\sb 4$, $c\sb 6$ et $\Delta$ puissent etre realises comme les
invariants $c\sb 4(W)$, $c\sb 6(W)$ et $\Delta$ (W) d'une equation de Weierstrass
(W) definie sur ${\frak O}\sb K.$ \par Sous l'hypothese (*), les resultats
sont les suivants: Si $p\ge 5:$ (W) existe toujours. - Sinon il est necessaire
et suffisant que l'une des conditions suivantes soit verifiee: \par Si
$p=3:$ (a) $c\sb 6$ est une unite de ${\frak O}\sb K$; $(b)\ on$ a $0<v(c\sb
6)<3e$ et il existe x dans ${\frak O}\sb K$ tel que $x\sp 3-3xc\sb 4-2c\sb
6\equiv 0 mod 27$; $(c)\ on$ a $c\sb 6\equiv 0 mod 27.$ \par Si $p=2:$
(a) $c\sb 4$ est une unite de ${\frak O}\sb K$ et $-c\sb 6$ est un carre
modulo 4; $(b)\ 0<v(c\sb 4)<4e$ et il existe x dans ${\frak O}\sb
K$ tel que si $P(X)=-X\sp 6+3X\sp 2c\sb 4+2c\sb 6$, on dit: $4x\sp 2P(x)\equiv
(x\sp 4-c\sb 4)\sp 2 mod 2\sp 8$; P(x) est multiple de 16; P(x)/16 est
un carre modulo 4; $(c)\ on$ a $c\sb 4\equiv 0 mod 16$ et il existe x dans
${\frak O}\sb Ktwl$ que $c\sb 6\equiv 8x\sp 2 mod 32$.
Citations: Zbl.628.14024
Mots clés: Chern class; Weierstrass curve
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Classification:
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*14H05
Algebraic functions
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11D25
Cubic and quartic diophantine equations
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14H52
Elliptic curves
-
14H45
Special curves and curves of low genus
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MATH Database, Zentralblatt für Mathematik / Mathematics Abstracts:
Copyright (c) 1997 European Mathematical Society, FIZ Karlsruhe
& Springer-Verlag.