A Remark on a Modular Analogue of the Sato–Tate Conjecture

2007 ◽  
Vol 50 (2) ◽  
pp. 234-242 ◽  
Author(s):  
Wentang Kuo
Keyword(s):  
Elliptic Curves ◽  
Angle Distribution ◽  
Hecke Eigenforms ◽  
Tate Conjecture ◽  
Central Characters ◽  
Holomorphic Hecke Eigenforms

AbstractThe original Sato–Tate Conjecture concerns the angle distribution of the eigenvalues arising from non-CM elliptic curves. In this paper, we formulate amodular analogue of the Sato–Tate Conjecture and prove that the angles arising from non-CM holomorphic Hecke eigenforms with non-trivial central characters are not distributed with respect to the Sate–Tatemeasure for non-CM elliptic curves. Furthermore, under a reasonable conjecture, we prove that the expected distribution is uniform.

A GENERALIZATION OF THE SATO–TATE CONJECTURE

2009 ◽  
Vol 05 (01) ◽  
pp. 173-184
Author(s):  
WENTANG KUO
Keyword(s):  
Elliptic Curves ◽  
Automorphic Forms ◽  
Angle Distribution ◽  
Tate Conjecture

The original Sato–Tate Conjecture concerns the angle distribution of the eigenvalues arisen from non-CM elliptic curves. In this paper, we formulate an analogue of the Sato–Tate Conjecture on automorphic forms of ( GL n) and, under a holomorphic assumption, prove that the distribution is either uniform or the generalized Sato–Tate measure.

scholarly journals ON SELMER RANK PARITY OF TWISTS

2016 ◽  
Vol 102 (3) ◽  
pp. 316-330 ◽  
Author(s):  
MAJID HADIAN ◽  
MATTHEW WEIDNER
Keyword(s):  
Elliptic Curves ◽  
Abelian Variety ◽  
Number Field ◽  
Arbitrary Number ◽  
Hyperelliptic Curve ◽  
Abelian Varieties ◽  
Hyperelliptic Curves ◽  
Torsion Subgroup ◽  
Tate Conjecture ◽  
Quadratic Twists

In this paper we study the variation of the $p$-Selmer rank parities of $p$-twists of a principally polarized Abelian variety over an arbitrary number field $K$ and show, under certain assumptions, that this parity is periodic with an explicit period. Our result applies in particular to principally polarized Abelian varieties with full $K$-rational $p$-torsion subgroup, arbitrary elliptic curves, and Jacobians of hyperelliptic curves. Assuming the Shafarevich–Tate conjecture, our result allows one to classify the rank parities of all quadratic twists of an elliptic or hyperelliptic curve after a finite calculation.

The Shafarevich-Tate conjecture for pencils of elliptic curves onK3 surfaces

1973 ◽  
Vol 20 (3) ◽  
pp. 249-266 ◽  
Author(s):  
M. Artin ◽  
H. P. F. Swinnerton-Dyer
Keyword(s):  
Elliptic Curves ◽  
Tate Conjecture

Arithmetic of weakly holomorphic Hecke eigenforms

2021 ◽  
Vol 384 ◽  
pp. 107750
Author(s):  
Jihyun Hwang ◽  
Chang Heon Kim
Keyword(s):  
Hecke Eigenforms ◽  
Holomorphic Hecke Eigenforms

The Bombieri–Vinogradov Theorem on Higher Rank Groups and its Applications

2019 ◽  
Vol 72 (4) ◽  
pp. 928-966
Author(s):  
Yujiao Jiang ◽  
Guangshi Lü
Keyword(s):  
Fourier Coefficients ◽  
Arithmetic Function ◽  
Maass Form ◽  
Hecke Eigenforms ◽  
Hecke Eigenform ◽  
Symmetric Square ◽  
Ramanujan Conjecture ◽  
Higher Rank ◽  
Image Position ◽  
Holomorphic Hecke Eigenforms

AbstractWe study the analogue of the Bombieri–Vinogradov theorem for $\operatorname{SL}_{m}(\mathbb{Z})$ Hecke–Maass form $F(z)$. In particular, for $\operatorname{SL}_{2}(\mathbb{Z})$ holomorphic or Maass Hecke eigenforms, symmetric-square lifts of holomorphic Hecke eigenforms on $\operatorname{SL}_{2}(\mathbb{Z})$, and $\operatorname{SL}_{3}(\mathbb{Z})$ Maass Hecke eigenforms under the Ramanujan conjecture, the levels of distribution are all equal to $1/2,$ which is as strong as the Bombieri–Vinogradov theorem. As an application, we study an automorphic version of Titchmarch’s divisor problem; namely for $a\neq 0,$$$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}\unicode[STIX]{x1D6EC}(n)\unicode[STIX]{x1D70C}(n)d(n-a)\ll x\log \log x,\end{eqnarray}$$ where $\unicode[STIX]{x1D70C}(n)$ are Fourier coefficients $\unicode[STIX]{x1D706}_{f}(n)$ of a holomorphic Hecke eigenform $f$ for $\operatorname{SL}_{2}(\mathbb{Z})$ or Fourier coefficients $A_{F}(n,1)$ of its symmetric-square lift $F$. Further, as a consequence, we get an asymptotic formula $$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}\unicode[STIX]{x1D6EC}(n)\unicode[STIX]{x1D706}_{f}^{2}(n)d(n-a)=E_{1}(a)x\log x+O(x\log \log x),\end{eqnarray}$$ where $E_{1}(a)$ is a constant depending on $a$. Moreover, we also consider the asymptotic orthogonality of the Möbius function against the arithmetic function $\unicode[STIX]{x1D70C}(n)d(n-a)$.

RATIONAL TORSION ON OPTIMAL CURVES

2005 ◽  
Vol 01 (04) ◽  
pp. 513-531 ◽  
Author(s):  
NEIL DUMMIGAN
Keyword(s):  
Elliptic Curves ◽  
Prime Order ◽  
Rational Points ◽  
Root Number ◽  
Hecke Eigenforms ◽  
Modular Parametrization ◽  
Local Root

Vatsal has proved recently a result which has consequences for the existence of rational points of odd prime order ℓ on optimal elliptic curves over ℚ. When the conductor N is squarefree, ℓ ∤ N and the local root number wp= -1 for at least one prime p | N, we offer a somewhat different proof, starting from an explicit cuspidal divisor on X0(N). We also prove some results linking the vanishing of L(E,1) with the divisibility by ℓ of the modular parametrization degree, fitting well with the Bloch–Kato conjecture for L( Sym2E,2), and with an earlier construction of elements in Shafarevich–Tate groups. Finally (following Faltings and Jordan) we prove an analogue of the result on ℓ-torsion for cuspidal Hecke eigenforms of level one (and higher weight), thereby strengthening some existing evidence for another case of the Bloch–Kato conjecture.

scholarly journals One level density of low-lying zeros of families of L-functions

2010 ◽  
Vol 147 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Peng Gao ◽  
Liangyi Zhao
Keyword(s):  
Level Density ◽  
Hecke Eigenforms ◽  
Dirichlet Characters ◽  
Level 1 ◽  
Holomorphic Hecke Eigenforms ◽  
Density Results

AbstractIn this paper, we prove some one level density results for the low-lying zeros of families of L-functions. More specifically, the families under consideration are that of L-functions of holomorphic Hecke eigenforms of level 1 and weight k twisted with quadratic Dirichlet characters and that of cubic and quartic Dirichlet L-functions.

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