scholarly journals Constructing elliptic curves from Galois representations

2018 ◽  
Vol 154 (10) ◽  
pp. 2045-2054
Author(s):  
Andrew Snowden ◽  
Jacob Tsimerman
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Galois Representations ◽  
Good Reduction ◽  
Arithmetic Surface

Given a non-isotrivial elliptic curve over an arithmetic surface, one obtains a lisse $\ell$-adic sheaf of rank two over the surface. This lisse sheaf has a number of straightforward properties: cyclotomic determinant, finite ramification, rational traces of Frobenius elements, and somewhere not potentially good reduction. We prove that any lisse sheaf of rank two possessing these properties comes from an elliptic curve.

scholarly journals p-adic Eisenstein-Kronecker series for CM elliptic curves and the Kronecker limit formulas

2015 ◽  
Vol 219 ◽  
pp. 269-302
Author(s):  
Kenichi Bannai ◽  
Hidekazu Furusho ◽  
Shinichi Kobayashi
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Theta Function ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Imaginary Quadratic Field ◽  
Good Reduction ◽  
Ring Of Integers

AbstractConsider an elliptic curve defined over an imaginary quadratic fieldKwith good reduction at the primes abovep≥ 5 and with complex multiplication by the full ring of integersof K. In this paper, we constructp-adic analogues of the Eisenstein-Kronecker series for such an elliptic curve as Coleman functions on the elliptic curve. We then provep-adic analogues of the first and second Kronecker limit formulas by using the distribution relation of the Kronecker theta function.

On the Surjectivity of the Galois Representations Associated to Non-CM Elliptic Curves

2005 ◽  
Vol 48 (1) ◽  
pp. 16-31 ◽  
Author(s):  
Alina Carmen Cojocaru ◽  
Ernst Kani
Keyword(s):  
Positive Integer ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Galois Representations ◽  
Complex Multiplication ◽  
Galois Representation ◽  
Upper Bounds

AbstractLet E be an elliptic curve defined over ℚ, of conductor N and without complex multiplication. For any positive integer l, let ϕl be the Galois representation associated to the l-division points of E. From a celebrated 1972 result of Serre we know that ϕl is surjective for any sufficiently large prime l. In this paper we find conditional and unconditional upper bounds in terms of N for the primes l for which ϕl is not surjective.

scholarly journals Wild Galois representations: Elliptic curves over a 2-adic field with non-abelian inertia action

2020 ◽  
Vol 16 (06) ◽  
pp. 1199-1208
Author(s):  
Nirvana Coppola
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Galois Representations ◽  
Galois Representation

In this paper, we present a description of the [Formula: see text]-adic Galois representation attached to an elliptic curve defined over a [Formula: see text]-adic field [Formula: see text], in the case where the image of inertia is non-abelian. There are two possibilities for the image of inertia, namely [Formula: see text] and [Formula: see text], and in each case, we need to distinguish whether the inertia degree of [Formula: see text] over [Formula: see text] is even or odd. The results presented here are being implemented in an algorithm to compute explicitly the Galois representation in these four cases.

scholarly journals COMPUTING IMAGES OF GALOIS REPRESENTATIONS ATTACHED TO ELLIPTIC CURVES

2016 ◽  
Vol 4 ◽  
Author(s):  
ANDREW V. SUTHERLAND
Keyword(s):  
Monte Carlo ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Galois Representations ◽  
Complex Multiplication ◽  
Monte Carlo Algorithm ◽  
Quadratic Fields ◽  
Complete List ◽  
Torsion Subgroup ◽  
Probabilistic Algorithms

Let $E$ be an elliptic curve without complex multiplication (CM) over a number field $K$, and let $G_{E}(\ell )$ be the image of the Galois representation induced by the action of the absolute Galois group of $K$ on the $\ell$-torsion subgroup of $E$. We present two probabilistic algorithms to simultaneously determine $G_{E}(\ell )$ up to local conjugacy for all primes $\ell$ by sampling images of Frobenius elements; one is of Las Vegas type and the other is a Monte Carlo algorithm. They determine $G_{E}(\ell )$ up to one of at most two isomorphic conjugacy classes of subgroups of $\mathbf{GL}_{2}(\mathbf{Z}/\ell \mathbf{Z})$ that have the same semisimplification, each of which occurs for an elliptic curve isogenous to $E$. Under the GRH, their running times are polynomial in the bit-size $n$ of an integral Weierstrass equation for $E$, and for our Monte Carlo algorithm, quasilinear in $n$. We have applied our algorithms to the non-CM elliptic curves in Cremona’s tables and the Stein–Watkins database, some 140 million curves of conductor up to $10^{10}$, thereby obtaining a conjecturally complete list of 63 exceptional Galois images $G_{E}(\ell )$ that arise for $E/\mathbf{Q}$ without CM. Under this conjecture, we determine a complete list of 160 exceptional Galois images $G_{E}(\ell )$ that arise for non-CM elliptic curves over quadratic fields with rational $j$-invariants. We also give examples of exceptional Galois images that arise for non-CM elliptic curves over quadratic fields only when the $j$-invariant is irrational.

scholarly journals Pseudoprime Reductions of Elliptic Curves

2012 ◽  
Vol 64 (1) ◽  
pp. 81-101 ◽  
Author(s):  
C. David ◽  
J. Wu
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Complex Multiplication ◽  
Upper Bounds ◽  
Good Reduction ◽  
Elliptic Pseudoprimes

Abstract Let E be an elliptic curve over ℚ without complex multiplication, and for each prime p of good reduction, let nE(p) = |E(𝔽p)|. For any integer b, we consider elliptic pseudoprimes to the base b. More precisely, let QE,b(x) be the number of primes p ⩽ x such that bnE(p) ≡ b (mod nE(p)), and let πpseuE,b (x) be the number of compositive nE(p) such that bnE(p) ≡ b (mod nE(p)) (also called elliptic curve pseudoprimes). Motivated by cryptography applications, we address the problem of finding upper bounds for QE,b(x) and πpseuE,b (x), generalising some of the literature for the classical pseudoprimes to this new setting.

scholarly journals Mordell–Weil ranks and Tate–Shafarevich groups of elliptic curves with mixed-reduction type over cyclotomic extensions

2021 ◽  
pp. 1-28
Author(s):  
Antonio Lei ◽  
Meng Fai Lim
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Finite Extension ◽  
Selmer Groups ◽  
Good Reduction ◽  
Cyclotomic Extensions ◽  
Iwasawa Algebra

Let [Formula: see text] be an elliptic curve defined over a number field [Formula: see text] where [Formula: see text] splits completely. Suppose that [Formula: see text] has good reduction at all primes above [Formula: see text]. Generalizing previous works of Kobayashi and Sprung, we define multiply signed Selmer groups over the cyclotomic [Formula: see text]-extension of a finite extension [Formula: see text] of [Formula: see text] where [Formula: see text] is unramified. Under the hypothesis that the Pontryagin duals of these Selmer groups are torsion over the corresponding Iwasawa algebra, we show that the Mordell–Weil ranks of [Formula: see text] over a subextension of the cyclotomic [Formula: see text]-extension are bounded. Furthermore, we derive an aysmptotic formula of the growth of the [Formula: see text]-parts of the Tate–Shafarevich groups of [Formula: see text] over these extensions.

scholarly journals Integral Tate modules and splitting of primes in torsion fields of elliptic curves

2016 ◽  
Vol 12 (01) ◽  
pp. 237-248 ◽  
Author(s):  
Tommaso Giorgio Centeleghe
Keyword(s):  
Finite Field ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Prime Number ◽  
Galois Representations ◽  
Number Fields ◽  
Tate Module ◽  
Explicit Procedure ◽  
Polynomial Formula ◽  
Class Polynomials

Let [Formula: see text] be an elliptic curve over a finite field [Formula: see text], and [Formula: see text] a prime number different from the characteristic of [Formula: see text]. In this paper, we consider the problem of finding the structure of the Tate module [Formula: see text] as an integral Galois representations of [Formula: see text]. We indicate an explicit procedure to solve this problem starting from the characteristic polynomial [Formula: see text] and the [Formula: see text]-invariant [Formula: see text] of [Formula: see text]. Hilbert Class Polynomials of imaginary quadratic orders play an important role here. We give a global application to the study of prime-splitting in torsion fields of elliptic curves over number fields.

scholarly journals Criteria for p-ordinarity of families of elliptic curves over infinitely many number fields

2014 ◽  
Vol 11 (01) ◽  
pp. 81-87
Author(s):  
Nuno Freitas ◽  
Panagiotis Tsaknias
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Sufficient Conditions ◽  
Number Fields ◽  
Good Reduction ◽  
Ordinary Reduction

Let Ki be a number field for all i ∈ ℤ>0 and let ℰ be a family of elliptic curves containing infinitely many members defined over Ki for all i. Fix a rational prime p. We give sufficient conditions for the existence of an integer i0 such that, for all i > i0 and all elliptic curve E ∈ ℰ having good reduction at all 𝔭 | p in Ki, we have that E has good ordinary reduction at all primes 𝔭 | p. We illustrate our criteria by applying it to certain Frey curves in [Recipes to Fermat-type equations of the form xr + yr = Czp, to appear in Math. Z.; http://arXiv.org/abs/1203.3371 ] attached to Fermat-type equations of signature (r, r, p).

scholarly journals Irreducibility of mod p Galois representations of elliptic curves with multiplicative reduction over number fields

2021 ◽  
pp. 1-10
Author(s):  
Filip Najman ◽  
George C. Ţurcaş
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Galois Representations ◽  
Galois Representation ◽  
Number Fields ◽  
Fermat's Last Theorem ◽  
Modular Method ◽  
Explicit Constant ◽  
Mod P

In this paper we prove that for every integer [Formula: see text], there exists an explicit constant [Formula: see text] such that the following holds. Let [Formula: see text] be a number field of degree [Formula: see text], let [Formula: see text] be any rational prime that is totally inert in [Formula: see text] and [Formula: see text] any elliptic curve defined over [Formula: see text] such that [Formula: see text] has potentially multiplicative reduction at the prime [Formula: see text] above [Formula: see text]. Then for every rational prime [Formula: see text], [Formula: see text] has an irreducible mod [Formula: see text] Galois representation. This result has Diophantine applications within the “modular method”. We present one such application in the form of an Asymptotic version of Fermat’s Last Theorem that has not been covered in the existing literature.

scholarly journals p-adic Eisenstein-Kronecker series for CM elliptic curves and the Kronecker limit formulas

2015 ◽  
Vol 219 ◽  
pp. 269-302
Author(s):  
Kenichi Bannai ◽  
Hidekazu Furusho ◽  
Shinichi Kobayashi
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Theta Function ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Imaginary Quadratic Field ◽  
Good Reduction ◽  
Ring Of Integers

AbstractConsider an elliptic curve defined over an imaginary quadratic field K with good reduction at the primes above p ≥ 5 and with complex multiplication by the full ring of integers of K. In this paper, we construct p-adic analogues of the Eisenstein-Kronecker series for such an elliptic curve as Coleman functions on the elliptic curve. We then prove p-adic analogues of the first and second Kronecker limit formulas by using the distribution relation of the Kronecker theta function.

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