Symmetric Power L-functions Research Articles

ON THE SPINOR L-FUNCTION OF MIYAWAKI–IKEDA LIFTS

We consider lifts from two elliptic modular forms to Siegel modular forms of odd degrees which are special cases of Miyawaki–Ikeda lifts. Assuming non-vanishing of these Miyawaki–Ikeda lifts, we show that the spinor L-functions of these Miyawaki–Ikeda lifts are products of certain symmetric power L-functions determined by original two elliptic modular forms.

On the Riesz means of coefficients of mth symmetric power L-functions

Let cn be the Fourier coefficients of L(symm f, s), and Δρ(x; symm f) be the error term in the asymptotic formula for ∑n≪xcn. In this paper, we study the Riesz means of Δρ(x; symm f) and obtain a truncated Voronoi-type formula under the hypothesis Nice(m, f).

Twisted moments of automorphic L-functions

We study the moments of the symmetric power L-functions of primitive forms at the edge of the critical strip twisted by the square of the value of the standard L-function at the center of the critical strip. We give a precise expansion of the moments as the order goes to infinity.

Statistics for low-lying zeros of symmetric power L-functions in the level aspect

We study one-level and two-level densities for low-lying zeros of symmetric power L-functions in the level aspect. This allows us to completely determine the symmetry types of some families of symmetric power L-functions with prescribed sign of functional equation. We also compute the moments of one-level density and exhibit mock-Gaussian behavior discovered by Hughes & Rudnick.

On an open problem of Sankaranarayanan

Let Λ(n) be the n-th normalized Fourier coefficient of a holomorphic Hecke eigenform f(z) ∈ Sk(Γ) for the full modular group. In one of his papers, Sankaranarayanan mentioned that it is an open problem to give a non-trivial estimate for the sum of Λ(n) over cubes, i.e. Σn⩽x Λ(n3). In this paper, we are able to use the analytic properties of symmetric power L-functions to solve his problem. More precisely, we prove that $$ \sum\limits_{n \leqslant x} {\lambda (n^3 ) \ll x^{\frac{3} {4} + \varepsilon } } ,\sum\limits_{n \leqslant x} {\lambda (n^4 ) \ll x^{\frac{7} {9} + \varepsilon } } . $$

Lower order terms for the one-level densities of symmetric power L-functions in the level aspect

In a previous paper, the authors determined, among other things, the main terms for the one-level densities for low-lying zeros of symmetric power L-functions in the level aspect. In this paper, the lower order terms of these one-level densities are found. The combinatorial difficulties, which should arise in such context, are drastically reduced thanks to Chebyshev polynomials, which are the characters of the irreducible representations of SU(2). %

On the Special Values of Certain Rankin–Selberg L-Functions and Applications to Odd Symmetric Power L-Functions of Modular Forms

We prove an algebraicity result for the central critical value of certain Rankin-Selberg L-functions for GL n × GL n−1 . This is a generalization and refinement of the results of Harder [14], Kazhdan, Mazur, and Schmidt [23], and Mahnkopf [29]. As an application of this result, we prove algebraicity results for certain critical values of the fifth and the seventh symmetric power L-functions attached to a holomorphic cusp form. Assuming Langlands' functoriality, one can prove similar algebraicity results for the special values of any odd symmetric power L-function. We also prove a conjecture of Blasius and Panchishkin on twisted L-values in some cases. These results, as in the above works, are, in general, based on a nonvanishing hypothesis on certain archimedean integrals.

Mean Square Estimates for Coefficients of Symmetric Power L-Functions

Let L(symjf,s) be the jth symmetric power L-function attached to a holomorphic Hecke eigencuspform f(z) for the full modular group, and \(\lambda_{\mathrm{sym}^{j}f}(n)\) denote its nth coefficient. In this paper we are able to prove that $$\int_{1}^{x}\bigg|\sum_{n\leq y}\lambda_{\mathrm{sym}^{3}f}(n)\bigg|^{2}dy=O\bigl(x^{2}\bigr),$$ and $$\int_{1}^{x}\bigg|\sum_{n\leq y}\lambda_{\mathrm{sym}^{4}f}(n)\bigg|^{2}dy=O\bigl(x^{\frac{11}{5}}\log x\bigr).$$

The average behavior of Fourier coefficients of cusp forms over sparse sequences

Let λ(n) be the nth normalized Fourier coefficient of a holomorphic Hecke eigenform f(z) ∈ S k (Γ). In this paper we are interested in the average behavior of λ 2 (n) over sparse sequences. By using the properties of symmetric power L-functions and their Rankin-Selberg L-functions, we are able to establish that for any e > 0, formula math. where j = 2, 3, 4.

Critical Values of Symmetric Power L-functions

We consider the critical values of symmetric power L-functions attached to elliptic curves over Q. We show how to calculate a canonical Deligne period, and in several numerical examples, especially for sixth and tenth powers, we examine the factorisation of the rational number apparently obtained when one divides the critical value by the canonical period. This seems to provide some support for the Bloch-Kato conjecture, when we com- pare it with calculations and bounds for Tamagawa factors and global torsion terms. For large odd powers (5th-9th), we see several examples fitting well with the squareness of the order of the Shafarevich-Tate group.

ON THE SIEGEL-TATUZAWA THEOREM FOR A CLASS OF L-FUNCTIONS

We consider an effective lower bound of the Siegel-Tatuzawa type for general L-functions with three standard assumptions. We further assume three hypotheses in this paper that are essential in developing our argument.Under these assumptions and hypotheses, we prove a theorem of Siegel-Tatuzawa type for general L-functions. In particular, we prove such a theorem for symmetric power L-functions under certain assumptions.

The universality of symmetric power L-functions and their Rankin-Selberg L-functions

We establish the universality theorem for the first four symmetric power L -functions of automorphic forms and their associated Rankin-Selberg L -functions. This generalizes some results of Laurinčikas and Matsumoto and of Matsumoto, respectively.

Extreme values of symmetric power L-functions at 1

We obtain an almost all result on the size of the symmetric mth power L-functions (m = 1, 2, 3, 4) for the normalized Hecke eigencuspforms at s = 1, which extends results of Elliott and Montgomery & Vaughan on Dirichlet L-functions to higher degree L-functions.

Holomorphy of the 9th symmetric power L-functions for Re(s) > 1

Holomorphy of the 9th symmetric power L-functions for Re(s) > 1

Low-lying zeroes of symmetric power L-functions

We study the distribution of low lying zeroes assuming the generalized Riemann hypothesis of symmetric power L-functions associated to cusp forms of weight k for SL2 (ℤ) which are Hecke eigenforms. Following the theory of Katz and Sarnak and the approach of Iwaniec, Luo, and Sarnak (2000), we determine the symmetry type of the corresponding family of L-functions, thereby providing further evidence to the truth of density conjecture.

On the complex moments of symmetric power $L$-functions at $s = 1$

The study of the distribution values of L-functions at the point s = 1 is a classical topic in analytic number theory which goes back to Chowla and Erdös. For some time, this question has been limited to the case of Euler products of degree 1, and it is only recently that the case of L-functions of higher degree was investigated, firstly by Luo and then by Royer et al., namely, for the standard and the symmetric square L-functions of modular forms. In this paper we propose a conceptual approach to analyzing the distribution of values at s = 1 of automorphic L-functions for appropriate families of automorphic forms. We illustrate this by computing arbitrary complex moments of arbitrary symmetric power L-functions of holomorphic forms of large (prime) level at s = 1 (assuming that the corresponding L-functions are automorphic, as is predicted by the Langlands functoriality conjectures and effectively proved for the symmetric powers up to 4). In addition, we provide a natural probabilistic interpretation of these computations. Our approach should generalize to quite arbitrary L-functions of appropriate families of automorphic forms.

Nonvanishing of the central critical value of the third symmetric power L-functions

Nonvanishing of the central critical value of the third symmetric power L-functions

Holomorphy of Rankin tripleL-functions; special values and root numbers for symmetric cubeL-functions

In this paper we prove the holomorphy of Rankin tripleL-functions for three cusp forms on GL(2) on the entire complex plane, if at least one of them is non-monomial. We conclude the paper by proving the equality of our root numbers for the third and the fourth symmetric powerL-functions with those of Artin through the local Langlands correspondence. We also revisit Deligne’s conjecture on special values of symmetric cubeL-functions.