scholarly journals Explicit calculation of the mod 4 Galois representation associated with the Fermat quartic

2019 ◽  
Vol 16 (04) ◽  
pp. 881-905
Author(s):  
Yasuhiro Ishitsuka ◽  
Tetsushi Ito ◽  
Tatsuya Ohshita
Keyword(s):  
Dihedral Group ◽  
Cyclotomic Field ◽  
Galois Representation ◽  
Explicit Calculation ◽  
Computer Algebra Systems ◽  
Jacobian Variety ◽  
Torsion Points ◽  
Quadratic Extensions ◽  
Galois Action ◽  
Weil Group

We use explicit methods to study the [Formula: see text]-torsion points on the Jacobian variety of the Fermat quartic. With the aid of computer algebra systems, we explicitly give a basis of the group of [Formula: see text]-torsion points. We calculate the Galois action, and show that the image of the mod [Formula: see text] Galois representation is isomorphic to the dihedral group of order [Formula: see text]. As applications, we calculate the Mordell–Weil group of the Jacobian variety of the Fermat quartic over each subfield of the [Formula: see text]th cyclotomic field. We determine all of the points on the Fermat quartic defined over quadratic extensions of the [Formula: see text]th cyclotomic field. Thus, we complete Faddeev’s work in 1960.

ARITHMETIC ON A QUINTIC THREEFOLD

2001 ◽  
Vol 12 (08) ◽  
pp. 943-972 ◽  
Author(s):  
CATERINA CONSANI ◽  
JASPER SCHOLTEN
Keyword(s):  
Modular Form ◽  
Theta Series ◽  
Continuous System ◽  
Galois Representation ◽  
Hilbert Modular Form ◽  
Galois Action ◽  
Hilbert Modular ◽  
Common Eigenvector ◽  
Quintic Threefold ◽  
Real Quadratic

This paper investigates some aspects of the arithmetic of a quintic threefold in Pr 4 with double points singularities. Particular emphasis is given to the study of the L-function of the Galois action ρ on the middle ℓ-adic cohomology. The main result of the paper is the proof of the existence of a Hilbert modular form of weight (2, 4) and conductor 30, on the real quadratic field [Formula: see text], whose associated (continuous system of) Galois representation(s) appears to be the most likely candidate to induce the scalar extension [Formula: see text]. The Hilbert modular form is interpreted as a common eigenvector of the Brandt matrices which describe the action of the Hecke operators on a space of theta series associated to the norm form of a quaternion algebra over [Formula: see text] and a related Eichler order.

scholarly journals On Modular Ball-Quotient Surfaces of Kodaira Dimension One

ISRN Geometry ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-5
Author(s):  
Aleksander Momot
Keyword(s):  
Short Note ◽  
Open Unit ◽  
Bergman Metric ◽  
Modular Surface ◽  
Open Unit Ball ◽  
Torsion Points ◽  
New Class ◽  
Ball Quotients ◽  
Weil Group ◽  
Dimension One

Let be a lattice which is not co-ompact, of finite covolume with respect to the Bergman metric and acting freely on the open unit ball . Then the toroidal compactification is a projective smooth surface with elliptic compactification divisor . In this short note we discover a new class of unramifed ball quotients . We consider ball quotients with kod and . We prove that each minimal surface with finite Mordell-Weil group in the class described admits an étale covering which is a pull-back of . Here denotes the elliptic modular surface parametrizing elliptic curves with 6-torsion points which generate [6].

scholarly journals Hyperelliptic curves with maximal Galois action on the torsion points of their Jacobians

2020 ◽  
Vol 69 (7) ◽  
pp. 2461-2492
Author(s):  
Aaron Landesman ◽  
Ashvin Swaminathan ◽  
James Tao ◽  
Yujie Xu
Keyword(s):  
Hyperelliptic Curves ◽  
Torsion Points ◽  
Galois Action

scholarly journals Abelian varieties over fields of finite characteristic

2014 ◽  
Vol 12 (5) ◽  
Author(s):  
Yuri Zarhin
Keyword(s):  
Abelian Varieties ◽  
Finitely Generated ◽  
Torsion Points ◽  
Finite Characteristic ◽  
Galois Action ◽  
Characteristic 2

AbstractThe aim of this paper is to extend our previous results about Galois action on the torsion points of abelian varieties to the case of (finitely generated) fields of characteristic 2.

scholarly journals Descending Rational Points on Elliptic Curves to Smaller Fields

2001 ◽  
Vol 53 (3) ◽  
pp. 449-469
Author(s):  
Amir Akbary ◽  
V. Kumar Murty
Keyword(s):  
Elliptic Curve ◽  
Galois Group ◽  
Galois Extension ◽  
Dihedral Group ◽  
Infinite Order ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Galois Module ◽  
Ring Of Integers ◽  
Weil Group

AbstractIn this paper, we study the Mordell-Weil group of an elliptic curve as a Galois module. We consider an elliptic curve E defined over a number field K whose Mordell-Weil rank over a Galois extension F is 1, 2 or 3. We show that E acquires a point (points) of infinite order over a field whose Galois group is one of Cn×Cm (n = 1, 2, 3, 4, 6, m = 1, 2), Dn×Cm (n = 2, 3, 4, 6, m = 1, 2), A4×Cm (m = 1, 2), S4 × Cm (m = 1, 2). Next, we consider the case where E has complex multiplication by the ring of integers of an imaginary quadratic field contained in K. Suppose that the -rank over a Galois extension F is 1 or 2. If ≠ and and h (class number of ) is odd, we show that E acquires positive -rank over a cyclic extension of K or over a field whose Galois group is one of SL2(/3), an extension of SL2(/3) by /2, or a central extension by the dihedral group. Finally, we discuss the relation of the above results to the vanishing of L-functions.

COURBES ELLIPTIQUES SEMI-STABLES SUR LES CORPS DE NOMBRES

2007 ◽  
Vol 03 (04) ◽  
pp. 611-633 ◽  
Author(s):  
ALAIN KRAUS
Keyword(s):  
Elliptic Curve ◽  
Prime Number ◽  
Number Field ◽  
Galois Representation ◽  
Torsion Points

Let K be a number field. In this paper, we are interested in the following problem: does there exist a constant cK, which depends only on K, such that for any semi-stable elliptic curve defined over K, the Galois representation in its p-torsion points is irreducible whenever p is a prime number greater than cK? In case the answer is positive, how can we get such a constant? We prove that if a certain condition is satisfied by K, the answer is positive and we obtain cK explicitly. Furthermore, we prove that this condition is realized in many situations.

scholarly journals Galois action on some ideal section points of the abelian variety associated with a modular form and its application

1983 ◽  
Vol 91 ◽  
pp. 19-36 ◽  
Author(s):  
Fumiyuki Momose
Keyword(s):  
Modular Form ◽  
Abelian Variety ◽  
Cusp Form ◽  
Modular Group ◽  
Jacobian Variety ◽  
Modular Curve ◽  
Galois Action ◽  
New Form

For an integer N let X1(N) be the modular curve defined over Q which corresponds to the modular group Γ1(N) To each primitive cusp form f ═ Σ amqm, a1═1, (Γ normalized new form in the sense of [1]) on Γ1(N) of weight 2, there corresponds a factor Jf of the jacobian variety of X1(N) (cf. Shimura [19]).

EXPLICIT SURJECTIVITY RESULTS FOR DRINFELD MODULES OF RANK 2

2017 ◽  
Vol 234 ◽  
pp. 17-45 ◽  
Author(s):  
IMIN CHEN ◽  
YOONJIN LEE
Keyword(s):  
Elliptic Curves ◽  
Upper Bound ◽  
Riemann Hypothesis ◽  
Galois Representations ◽  
Complex Multiplication ◽  
Galois Representation ◽  
Drinfeld Module ◽  
Torsion Points ◽  
Rank 2 ◽  
Invent Math

Let $K=\mathbb{F}_{q}(T)$ and $A=\mathbb{F}_{q}[T]$. Suppose that $\unicode[STIX]{x1D719}$ is a Drinfeld $A$-module of rank $2$ over $K$ which does not have complex multiplication. We obtain an explicit upper bound (dependent on $\unicode[STIX]{x1D719}$) on the degree of primes ${\wp}$ of $K$ such that the image of the Galois representation on the ${\wp}$-torsion points of $\unicode[STIX]{x1D719}$ is not surjective, in the case of $q$ odd. Our results are a Drinfeld module analogue of Serre’s explicit large image results for the Galois representations on $p$-torsion points of elliptic curves (Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259–331; Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Etudes Sci. Publ. Math. 54 (1981), 323–401.) and are unconditional because the generalized Riemann hypothesis for function fields holds. An explicit isogeny theorem for Drinfeld $A$-modules of rank $2$ over $K$ is also proven.

Images of mod p Galois Representations Associated to Elliptic Curves

2001 ◽  
Vol 44 (3) ◽  
pp. 313-322 ◽  
Author(s):  
Amadeu Reverter ◽  
Núria Vila
Keyword(s):  
Elliptic Curves ◽  
Prime Number ◽  
Galois Representations ◽  
Complex Multiplication ◽  
Torsion Points ◽  
Galois Action ◽  
Mod P

AbstractWe give an explicit recipe for the determination of the images associated to the Galois action on p-torsion points of elliptic curves. We present a table listing the image for all the elliptic curves defined over without complex multiplication with conductor less than 200 and for each prime number p.

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