The Langlands program plays an important role in Number theory and Representation Theory. A crucial aspect of this program is the functoriality conjecture, expressed in a letter of Langlands to Weil in 1967. Let F be a global field with ring of ad`eles AF and
let LG ! LH, be a given L-homomorphism between the L-groups of two connected (quasi-split) reductive groups G and H over F. Then, according to this conjecture, for every cuspidal automorphic representation = N x x of G(AF ), there exists an automorphic representation = N x x of H(AF ) such that, at almost all places x where x is unramified, x is unramified and its Satake parameter corresponds to the image under of the Satake parameter of x. Such representation will be called a weak lift or transfer of .
Furthermore that transfer process should respect arithmetic information coming from -factors, L-functions and "-factors, and leads to a local version of functoriality at the ramified places as well. When G is a classical group, LG has a natural representation into LH for a specific general linear group H, and that case has been studied by many people. When F is a number field, two main tools have been used: converse theorem and trace formulas.
The former was used by Cogdell, Kim, Piateski-Shapiro and Shahidi in combination with the Langlands-Shahidi method to prove the conjecture for a globally generic automorphic representation when G is a quasi-split symplectic, unitary or special orthogonal
group. For the latter, Arthur and his continuators used trace formulas to get more complete results, not restricted to quasi-split groups in characteristic zero. Lomel´ı extended the converse theorem method to global function fields, getting functoriality for globally generic automorphic representations of split classical groups and unitary groups. The present thesis further extends the converse theorem method, over a function field F, to establish the functoriality conjecture when G is a quasi-split non-split even special orthogonal group, and a globally generic representation

Thesis in co-tutelle

The Langlands program plays an important role in Number theory and Representation Theory. A crucial aspect of this program is the functoriality conjecture, expressed in a letter of Langlands to Weil in 1967. Let F be a global field with ring of ad`eles AF and
let LG ! LH, be a given L-homomorphism between the L-groups of two connected (quasi-split) reductive groups G and H over F. Then, according to this conjecture, for every cuspidal automorphic representation = N x x of G(AF ), there exists an automorphic representation = N x x of H(AF ) such that, at almost all places x where x is unramified, x is unramified and its Satake parameter corresponds to the image under of the Satake parameter of x. Such representation will be called a weak lift or transfer of .
Furthermore that transfer process should respect arithmetic information coming from -factors, L-functions and "-factors, and leads to a local version of functoriality at the ramified places as well. When G is a classical group, LG has a natural representation into LH for a specific general linear group H, and that case has been studied by many people. When F is a number field, two main tools have been used: converse theorem and trace formulas.
The former was used by Cogdell, Kim, Piateski-Shapiro and Shahidi in combination with the Langlands-Shahidi method to prove the conjecture for a globally generic automorphic representation when G is a quasi-split symplectic, unitary or special orthogonal
group. For the latter, Arthur and his continuators used trace formulas to get more complete results, not restricted to quasi-split groups in characteristic zero. Lomel´ı extended the converse theorem method to global function fields, getting functoriality for globally generic automorphic representations of split classical groups and unitary groups. The present thesis further extends the converse theorem method, over a function field F, to establish the functoriality conjecture when G is a quasi-split non-split even special orthogonal group, and a globally generic representation

Doctorado en Matemática