Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userModular iterated integrals associated with cusp forms
88
0
0.0
(
0
)
Authors
Nikolaos Diamantis
Ask ChatGPT about the research
No Arabic abstract
We construct an explicit family of modular iterated integrals which involves cusp forms. This leads to a new method of producing invaria
rate research
Read More
We continue the analysis of modular invariant functions, subject to inhomogeneous Laplace eigenvalue equations, that were determined in terms of Poincare series in a companion paper. The source term of the Laplace equation is a product of (derivatives of) two non-holomorphic Eisenstein series whence the modular invariants are assigned depth two. These modular invariant functions can sometimes be expressed in terms of single-valued iterated integrals of holomorphic Eisenstein series as they appear in generating series of modular graph forms. We show that the set of iterated integrals of Eisenstein series has to be extended to include also iterated integrals of holomorphic cusp forms to find expressions for all modular invariant functions of depth two. The coefficients of these cusp forms are identified as ratios of their L-values inside and outside the critical strip.
We study generating series of torus integrals that contain all so-called modular graph forms relevant for massless one-loop closed-string amplitudes. By analysing the differential equation of the generating series we construct a solution for its low-energy expansion to all orders in the inverse string tension $alpha$. Our solution is expressed through initial data involving multiple zeta values and certain real-analytic functions of the modular parameter of the torus. These functions are built from real and imaginary parts of holomorphic iterated Eisenstein integrals and should be closely related to Browns recent construction of real-analytic modular forms. We study the properties of our real-analytic objects in detail and give explicit examples to a fixed order in the $alpha$-expansion. In particular, our solution allows for a counting of linearly independent modular graph forms at a given weight, confirming previous partial results and giving predictions for higher, hitherto unexplored weights. It also sheds new light on the topic of uniform transcendentality of the $alpha$-expansion.
We define L-functions for the class of real-analytic modular forms recently introduced by F. Brown. We establish their main properties and construct the analogue of period polynomial in cases of special interest, including those of modular iterated integrals.
A generalized Riemann hypothesis states that all zeros of the completed Hecke $L$-function $L^*(f,s)$ of a normalized Hecke eigenform $f$ on the full modular group should lie on the vertical line $Re(s)=frac{k}{2}.$ It was shown by Kohnen that there exists a Hecke eigenform $f$ of weight $k$ such that $L^*(f,s) eq 0$ for sufficiently large $k$ and any point on the line segments $Im(s)=t_0, frac{k-1}{2} < Re(s) < frac{k}{2}-epsilon, frac{k }{2}+epsilon < Re(s) < frac{k+1}{2},$ for any given real number $t_0$ and a positive real number $epsilon.$ This paper concerns the non-vanishing of the product $L^*(f,s)L^*(f,w)$ $(s,win mathbb{C})$ on average.
In this paper we explicitly compute mod-l Galois representations associated to modular forms. To be precise, we look at cases with l<=23 and the modular forms considered will be cusp forms of level 1 and weight up to 22. We present the result in terms of polynomials associated to the projectivised representations. As an application, we will improve a known result on Lehmers non-vanishing conjecture for Ramanujans tau function.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
