scholarly journals Six unlikely intersection problems in search of effectivity

2016 ◽  
Vol 162 (3) ◽  
pp. 447-477 ◽  
Author(s):  
P. HABEGGER ◽  
G. JONES ◽  
D. MASSER
Keyword(s):  
Elliptic Curve ◽  
Finite Order ◽  
Complex Multiplication ◽  
Root Of Unity ◽  
Legendre Form

AbstractWe investigate four properties related to an elliptic curve Et in Legendre form with parameter t: the curve Et has complex multiplication, E−t has complex multiplication, a point on Et with abscissa 2 is of finite order, and t is a root of unity. Combining all pairs of properties leads to six problems on unlikely intersections. Using a variety of techniques we solve these problems with varying degrees of effectivity (and for three of them we even present the list of all possible t).

scholarly journals Primitive divisors of sequences associated to elliptic curves with complex multiplication

2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Matteo Verzobio
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Complex Multiplication ◽  
Dedekind Domain ◽  
Torsion Point ◽  
Integral Ideal ◽  
Primitive Divisors ◽  
Primitive Divisor ◽  
The Ideal

AbstractLet P and Q be two points on an elliptic curve defined over a number field K. For $$\alpha \in {\text {End}}(E)$$ α ∈ End ( E ) , define $$B_\alpha $$ B α to be the $$\mathcal {O}_K$$ O K -integral ideal generated by the denominator of $$x(\alpha (P)+Q)$$ x ( α ( P ) + Q ) . Let $$\mathcal {O}$$ O be a subring of $${\text {End}}(E)$$ End ( E ) , that is a Dedekind domain. We will study the sequence $$\{B_\alpha \}_{\alpha \in \mathcal {O}}$$ { B α } α ∈ O . We will show that, for all but finitely many $$\alpha \in \mathcal {O}$$ α ∈ O , the ideal $$B_\alpha $$ B α has a primitive divisor when P is a non-torsion point and there exist two endomorphisms $$g\ne 0$$ g ≠ 0 and f so that $$f(P)= g(Q)$$ f ( P ) = g ( Q ) . This is a generalization of previous results on elliptic divisibility sequences.

scholarly journals Algebraic functional equations and completely faithful Selmer groups

2015 ◽  
Vol 11 (04) ◽  
pp. 1233-1257
Author(s):  
Tibor Backhausz ◽  
Gergely Zábrádi
Keyword(s):  
Functional Equation ◽  
Elliptic Curve ◽  
Galois Group ◽  
Number Field ◽  
Functional Equations ◽  
Complex Multiplication ◽  
Selmer Groups ◽  
Selmer Group ◽  
Torsion Module ◽  
Ordinary Reduction

Let E be an elliptic curve — defined over a number field K — without complex multiplication and with good ordinary reduction at all the primes above a rational prime p ≥ 5. We construct a pairing on the dual p∞-Selmer group of E over any strongly admissible p-adic Lie extension K∞/K under the assumption that it is a torsion module over the Iwasawa algebra of the Galois group G = Gal(K∞/K). Under some mild additional hypotheses, this gives an algebraic functional equation of the conjectured p-adic L-function. As an application, we construct completely faithful Selmer groups in case the p-adic Lie extension is obtained by adjoining the p-power division points of another non-CM elliptic curve A.

AVERAGE SIZE OF GAPS IN THE FOURIER EXPANSION OF MODULAR FORMS

2007 ◽  
Vol 03 (02) ◽  
pp. 207-215 ◽  
Author(s):  
EMRE ALKAN
Keyword(s):  
Elliptic Curve ◽  
Modular Forms ◽  
Fourier Expansion ◽  
Complex Multiplication ◽  
Gap Function ◽  
Quantitative Results ◽  
Average Size

We prove that certain powers of the gap function for the newform associated to an elliptic curve without complex multiplication are "finite" on average. In particular we obtain quantitative results on the number of large values of the gap function.

scholarly journals Explicit computations of Serre’s obstruction for genus-3 curves and application to optimal curves

2010 ◽  
Vol 13 ◽  
pp. 192-207 ◽  
Author(s):  
Christophe Ritzenthaler
Keyword(s):  
Modular Form ◽  
Elliptic Curve ◽  
Complex Multiplication ◽  
Numerical Results ◽  
Siegel Modular Form ◽  
Square Root ◽  
The Third

AbstractLetkbe a field of characteristic other than 2. There can be an obstruction to a principally polarized abelian threefold (A,a) overk, which is a Jacobian over$\bar {k}$, being a Jacobian over k; this can be computed in terms of the rationality of the square root of the value of a certain Siegel modular form. We show how to do this explicitly for principally polarized abelian threefolds which are the third power of an elliptic curve with complex multiplication. We use our numerical results to prove or refute the existence of some optimal curves of genus 3.

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 No Singular Modulus Is a Unit

2018 ◽  
Vol 2020 (24) ◽  
pp. 10005-10041 ◽  
Author(s):  
Yuri Bilu ◽  
Philipp Habegger ◽  
Lars Kühne
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Endomorphism Ring ◽  
Complex Multiplication ◽  
Computer Assisted ◽  
Algebraic Unit ◽  
Special Points ◽  
Effective Proof ◽  
Rule Out

Abstract A result of the 2nd-named author states that there are only finitely many complex multiplication (CM)-elliptic curves over $\mathbb{C}$ whose $j$-invariant is an algebraic unit. His proof depends on Duke’s equidistribution theorem and is hence noneffective. In this article, we give a completely effective proof of this result. To be precise, we show that every singular modulus that is an algebraic unit is associated with a CM-elliptic curve whose endomorphism ring has discriminant less than $10^{15}$. Through further refinements and computer-assisted arguments, we eventually rule out all remaining cases, showing that no singular modulus is an algebraic unit. This allows us to exhibit classes of subvarieties in ${\mathbb{C}}^n$ not containing any special points.

A Titchmarsh divisor problem for elliptic curves

2015 ◽  
Vol 160 (1) ◽  
pp. 167-189 ◽  
Author(s):  
PAUL POLLACK
Keyword(s):  
Elliptic Curve ◽  
Complex Multiplication ◽  
Zeta Functions ◽  
Dirichlet Character ◽  
Good Reduction ◽  
Heuristic Argument ◽  
Mean Values ◽  
Average Size ◽  
Image Position ◽  
Ordinary Reduction

AbstractLet E/Q be an elliptic curve with complex multiplication. We study the average size of τ(#E(Fp)) as p varies over primes of good ordinary reduction. We work out in detail the case of E: y2 = x3 − x, where we prove that $$\begin{equation} \sum_{\substack{p \leq x \\p \equiv 1\pmod{4}}} \tau(\#E({\bf{F}}_p)) \sim \left(\frac{5\pi}{16} \prod_{p > 2} \frac{p^4-\chi(p)}{p^2(p^2-1)}\right)x, \quad\text{as $x\to\infty$}. \end{equation}$$ Here χ is the nontrivial Dirichlet character modulo 4. The proof uses number field analogues of the Brun–Titchmarsh and Bombieri–Vinogradov theorems, along with a theorem of Wirsing on mean values of nonnegative multiplicative functions.Now suppose that E/Q is a non-CM elliptic curve. We conjecture that the sum of τ(#E(Fp)), taken over p ⩽ x of good reduction, is ~cEx for some cE > 0, and we give a heuristic argument suggesting the precise value of cE. Assuming the Generalized Riemann Hypothesis for Dedekind zeta functions, we prove that this sum is ≍Ex. The proof uses combinatorial ideas of Erdős.

Sign in / Sign up

Export Citation Format

Share Document