Purusottam Rath

ALGEBRAIC INDEPENDENCE OF VALUES OF MODULAR FORMS

We investigate values of modular forms with algebraic Fourier coefficients at algebraic arguments. As a consequence, we conclude about the nature of zeros of such modular forms. In particular, the singular values of modular forms (that is, values at CM points) are related to the recent work of Nesterenko. As an application, we deduce the transcendence of critical values of certain Hecke L-series. We also discuss how these investigations generalize to the case of quasi-modular forms with algebraic Fourier coefficients.

Transcendental Nature of Special Values of

In this paper, we study the non-vanishing and transcendence of special values of a varying class of L-functions and their derivatives. This allows us to investigate the transcendence of Petersson norms of certain weight one modular forms.

L

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 peri- odic 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.

-Functions

In this paper, we link the nature of special values of certain Dirichlet L-functions to those of multiple gamma values.

On a Conjecture of Chowla and Milnor

Schanuel’s Conjecture: Suppose α 1, …, α n are complex numbers which are linearly independent over \(\mathbb{Q}\). Then the transcendence degree of the field

Schanuel’s Conjecture

\displaystyle{\mathbb{Q}(\alpha _{1},\ldots,\alpha _{n},e^{\alpha _{1}},\ldots,e^{\alpha _{n}})}

More Elliptic Integrals

over \(\mathbb{Q}\) is at least n.

Hermite’s Theorem

We will now prove that π is transcendental. This was first proved by F. Lindemann in 1882 by modifying Hermite’s methods. The proof proceeds by contradiction. Before we begin the proof, we recall two facts from algebraic number theory.