scholarly journals On the gaps between nonzero Fourier coefficients of eigenforms with CM

2017 ◽  
Vol 14 (01) ◽  
pp. 95-101
Author(s):  
Surjeet Kaushik ◽  
Narasimha Kumar
Keyword(s):  
Elliptic Curve ◽  
Fourier Coefficient ◽  
Fourier Coefficients ◽  
Short Interval ◽  
Complex Multiplication ◽  
Hecke Eigenform

Suppose [Formula: see text] is an elliptic curve over [Formula: see text] of conductor [Formula: see text] with complex multiplication (CM) by [Formula: see text], and [Formula: see text] is the corresponding cuspidal Hecke eigenform in [Formula: see text]. Then [Formula: see text]th Fourier coefficient of [Formula: see text] is nonzero in the short interval [Formula: see text] for all [Formula: see text] and for some [Formula: see text]. As a consequence, we produce infinitely many cuspidal CM eigenforms [Formula: see text] level [Formula: see text] and weight [Formula: see text] for which [Formula: see text] holds, for all [Formula: see text].

scholarly journals POWER RESIDUES OF FOURIER COEFFICIENTS OF ELLIPTIC CURVES WITH COMPLEX MULTIPLICATION

2009 ◽  
Vol 05 (01) ◽  
pp. 109-124
Author(s):  
TOM WESTON ◽  
ELENA ZAUROVA
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Fourier Coefficient ◽  
Fourier Coefficients ◽  
Complex Multiplication ◽  
Roots Of Unity ◽  
Modulo P

Fix m greater than one and let E be an elliptic curve over Q with complex multiplication. We formulate conjectures on the density of primes p (congruent to one modulo m) for which the pth Fourier coefficient of E is an mth power modulo p; often these densities differ from the naive expectation of 1/m. We also prove our conjectures for m dividing the number of roots of unity lying in the CM field of E; the most involved case is m = 4 and complex multiplication by Q(i).

A Variant of Lehmer’s Conjecture, II: The CM-case

2011 ◽  
Vol 63 (2) ◽  
pp. 298-326 ◽  
Author(s):  
Sanoli Gun ◽  
V. Kumar Murty
Keyword(s):  
Asymptotic Formula ◽  
Fourier Coefficient ◽  
Fourier Coefficients ◽  
Complex Multiplication ◽  
Rational Integer ◽  
Know How ◽  
Lehmer's Conjecture ◽  
Hecke Eigenform

Abstract Let f be a normalized Hecke eigenform with rational integer Fourier coefficients. It is an interesting question to know how often an integer n has a factor common with the n-th Fourier coefficient of f. It has been shown in previous papers that this happens very often. In this paper, we give an asymptotic formula for the number of integers n for which (n, a(n)) = 1, where a(n) is the n-th Fourier coefficient of a normalized Hecke eigenform f of weight 2 with rational integer Fourier coefficients and having complex multiplication.

scholarly journals Fourier coefficients of the overconvergent generalized eigenform associated to a CM form

2020 ◽  
Vol 16 (06) ◽  
pp. 1185-1197
Author(s):  
Chi-Yun Hsu
Keyword(s):  
Modular Form ◽  
Fourier Coefficients ◽  
Complex Multiplication ◽  
Hecke Eigenvalues ◽  
Hecke Eigenform ◽  
Critical Slope

Let [Formula: see text] be a modular form with complex multiplication. If [Formula: see text] has critical slope, then Coleman’s classicality theorem implies that there is a [Formula: see text]-adic overconvergent generalized Hecke eigenform with the same Hecke eigenvalues as [Formula: see text]. We give a formula for the Fourier coefficients of this generalized Hecke eigenform. We also investigate the dimension of the generalized Hecke eigenspace of [Formula: see text]-adic overconvergent forms containing [Formula: see text].

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 On Fourier Coefficients of the Symmetric Square L-Function at Piatetski-Shapiro Prime Twins

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1254
Author(s):  
Xue Han ◽  
Xiaofei Yan ◽  
Deyu Zhang
Keyword(s):  
Asymptotic Formula ◽  
Fourier Coefficient ◽  
Cusp Form ◽  
Riemann Hypothesis ◽  
Modular Group ◽  
Fourier Coefficients ◽  
Mean Value ◽  
Symmetric Square ◽  
The Mean ◽  
Almost All

Let Pc(x)={p≤x|p,[pc]areprimes},c∈R+∖N and λsym2f(n) be the n-th Fourier coefficient associated with the symmetric square L-function L(s,sym2f). For any A>0, we prove that the mean value of λsym2f(n) over Pc(x) is ≪xlog−A−2x for almost all c∈ε,(5+3)/8−ε in the sense of Lebesgue measure. Furthermore, it holds for all c∈(0,1) under the Riemann Hypothesis. Furthermore, we obtain that asymptotic formula for λf2(n) over Pc(x) is ∑p,qprimep≤x,q=[pc]λf2(p)=xclog2x(1+o(1)), for almost all c∈ε,(5+3)/8−ε, where λf(n) is the normalized n-th Fourier coefficient associated with a holomorphic cusp form f for the full modular group.

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 Symmetric power functoriality for holomorphic modular forms, II

2021 ◽  
Author(s):  
James Newton ◽  
Jack A. Thorne
Keyword(s):  
Modular Forms ◽  
Complex Multiplication ◽  
Symmetric Power ◽  
Hecke Eigenform ◽  
Holomorphic Modular Forms

AbstractLet $f$ f be a cuspidal Hecke eigenform without complex multiplication. We prove the automorphy of the symmetric power lifting $\operatorname{Sym}^{n} f$ Sym n f for every $n \geq 1$ n ≥ 1 .

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