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Search: MSC category 11F11 ( Holomorphic modular forms of integral weight )

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1. CJM 2012 (vol 66 pp. 170)

Guitart, Xavier; Quer, Jordi
Modular Abelian Varieties Over Number Fields
The main result of this paper is a characterization of the abelian varieties $B/K$ defined over Galois number fields with the property that the $L$-function $L(B/K;s)$ is a product of $L$-functions of non-CM newforms over $\mathbb Q$ for congruence subgroups of the form $\Gamma_1(N)$. The characterization involves the structure of $\operatorname{End}(B)$, isogenies between the Galois conjugates of $B$, and a Galois cohomology class attached to $B/K$. We call the varieties having this property strongly modular. The last section is devoted to the study of a family of abelian surfaces with quaternionic multiplication. As an illustration of the ways in which the general results of the paper can be applied we prove the strong modularity of some particular abelian surfaces belonging to that family, and we show how to find nontrivial examples of strongly modular varieties by twisting.

Keywords:Modular abelian varieties, $GL_2$-type varieties, modular forms
Categories:11G10, 11G18, 11F11

2. CJM 2011 (vol 64 pp. 282)

Dahmen, Sander R.; Yazdani, Soroosh
Level Lowering Modulo Prime Powers and Twisted Fermat Equations
We discuss a clean level lowering theorem modulo prime powers for weight $2$ cusp forms. Furthermore, we illustrate how this can be used to completely solve certain twisted Fermat equations $ax^n+by^n+cz^n=0$.

Keywords:modular forms, level lowering, Diophantine equations
Categories:11D41, 11F33, 11F11, 11F80, 11G05

3. CJM 2011 (vol 63 pp. 1328)

Gun, Sanoli; Murty, M. Ram; Rath, Purusottam
On a Conjecture of Chowla and Milnor
In this paper, we investigate a conjecture due to S. and P. Chowla and its generalization by Milnor. These are related to the delicate question of non-vanishing of $L$-functions associated to periodic functions at integers greater than $1$. We report on some progress in relation to these conjectures. In a different vein, we link them to a conjecture of Zagier on multiple zeta values and also to linear independence of polylogarithms.

Categories:11F20, 11F11

4. CJM 2011 (vol 63 pp. 634)

Lü, Guangshi
On Higher Moments of Fourier Coefficients of Holomorphic Cusp Forms
Let $S_{k}(\Gamma)$ be the space of holomorphic cusp forms of even integral weight $k$ for the full modular group. Let $\lambda_f(n)$ and $\lambda_g(n)$ be the $n$-th normalized Fourier coefficients of two holomorphic Hecke eigencuspforms $f(z), g(z) \in S_{k}(\Gamma)$, respectively. In this paper we are able to show the following results about higher moments of Fourier coefficients of holomorphic cusp forms.\newline (i) For any $\varepsilon>0$, we have \begin{equation*} \sum_{n\leq x}\lambda_f^5(n) \ll_{f,\varepsilon}x^{\frac{15}{16}+\varepsilon} \quad\text{and}\quad\sum_{n\leq x}\lambda_f^7(n) \ll_{f,\varepsilon}x^{\frac{63}{64}+\varepsilon}. \end{equation*} (ii) If $\operatorname{sym}^3\pi_f \ncong \operatorname{sym}^3\pi_g$, then for any $\varepsilon>0$, we have \begin{equation*} \sum_{n \leq x}\lambda_f^3(n)\lambda_g^3(n)\ll_{f,\varepsilon}x^{\frac{31}{32}+\varepsilon}; \end{equation*} If $\operatorname{sym}^2\pi_f \ncong \operatorname{sym}^2\pi_g$, then for any $\varepsilon>0$, we have \[ \sum_{n \leq x}\lambda_f^4(n)\lambda_g^2(n)=cx\log x+c'x+O_{f,\varepsilon}\bigl(x^{\frac{31}{32}+\varepsilon}\bigr); \] If $\operatorname{sym}^2\pi_f \ncong \operatorname{sym}^2\pi_g$ and $\operatorname{sym}^4\pi_f \ncong \operatorname{sym}^4\pi_g$, then for any $\varepsilon>0$, we have \[ \sum_{n \leq x}\lambda_f^4(n)\lambda_g^4(n)=xP(\log x)+O_{f,\varepsilon}\bigl(x^{\frac{127}{128}+\varepsilon}\bigr), \] where $P(x)$ is a polynomial of degree $3$.

Keywords: Fourier coefficients of cusp forms, symmetric power $L$-function
Categories:11F30, , , , 11F11, 11F66

5. CJM 2011 (vol 63 pp. 298)

Gun, Sanoli; Murty, V. Kumar
A Variant of Lehmer's Conjecture, II: The CM-case
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.

Categories:11F11, 11F30

6. CJM 2004 (vol 56 pp. 373)

Orton, Louisa
An Elementary Proof of a Weak Exceptional Zero Conjecture
In this paper we extend Darmon's theory of ``integration on $\uh_p\times \uh$'' to cusp forms $f$ of higher even weight. This enables us to prove a ``weak exceptional zero conjecture'': that when the $p$-adic $L$-function of $f$ has an exceptional zero at the central point, the $\mathcal{L}$-invariant arising is independent of a twist by certain Dirichlet characters.

Categories:11F11, 11F67

7. CJM 2004 (vol 56 pp. 23)

Bennett, Michael A.; Skinner, Chris M.
Ternary Diophantine Equations via Galois Representations and Modular Forms
In this paper, we develop techniques for solving ternary Diophantine equations of the shape $Ax^n + By^n = Cz^2$, based upon the theory of Galois representations and modular forms. We subsequently utilize these methods to completely solve such equations for various choices of the parameters $A$, $B$ and $C$. We conclude with an application of our results to certain classical polynomial-exponential equations, such as those of Ramanujan--Nagell type.

Categories:11D41, 11F11, 11G05

8. CJM 2000 (vol 52 pp. 31)

Chan, Heng Huat; Liaw, Wen-Chin
On Russell-Type Modular Equations
In this paper, we revisit Russell-type modular equations, a collection of modular equations first studied systematically by R.~Russell in 1887. We give a proof of Russell's main theorem and indicate the relations between such equations and the constructions of Hilbert class fields of imaginary quadratic fields. Motivated by Russell's theorem, we state and prove its cubic analogue which allows us to construct Russell-type modular equations in the theory of signature~$3$.

Categories:33D10, 33C05, 11F11

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