Related papers: Endomorphisms of polynomial rings and Jacobians

This paper has been withdrawn by the author due to a crucial sign error in Theorem 3.4.

The paper has been withdrawn due to a crucial error in section 3.

This paper has been withdrawn by the author, due to an error in the proof of Theorem 3.8.

This paper has been withdrawn by author due to an error in the proof.

This paper has been withdrawn by the authors, due a crucial mistake in Lemma 2

This paper has been withdrawn by the author due to a crucial sign error in equation 1.

This paper has been withdrawn by the author due to a crucial error in eq.59. I apologize for the inconveniences.

This paper has been withdrawn by the author due to a gap in the proof of Lemma 3.4

The paper was withdrawn because of its significant overlap with a paper appeared recently.

This paper has been withdrawn by the authors, due to a mistake pointed out by Lenny Ng and Josh Sabloff.

This paper has been withdrawn by the author due to an error.

This paper is cancelled

This paper has been withdrawn due to a crucial theoretical and experimental error.

This paper has been withdrawn by the author due to a crucial error in the definition of homomorphism.

This article has been withdrawn due to an error in a proof of the main result.

This paper is withdrawn. We found a mistake in Lemma 4.1

The paper has been withdrawn due to numerical error.

This paper is withdrawn from submission due to a critical error in the proof of main theorem.

This paper has been withdrawn by the author due to extremely unscientific errors.

This paper has been withdrawn by the author due to serious flaws in certain proofs. For instance, the method used to construct certain automorphic representations is flawed.