Automorphic Forms, L-functions and Galois Representations

AUTOMORPHIC FORMS, L-FUNCTIONS AND GALOIS REPRESENTATIONS The princial investigator proposes to do the following: (i) attachGalois representations to cusp forms of weight k 1 over any CM field K (with totally real subfield F) by transferring certain associated forms on GL(4)/F to suitable unitary groups by making use of L-functions, trace formula, congruences, restrictions of Hasse invariant forms and pseudo-representations; (ii) construct certain special holomorphic forms on GSp(4)/Q, study their lifting to GL(4)/Q, and derive consequences for certain Galois representations; and (iii) to continue ongoing work with D. Prasad on a refinement of the local Langlands correspondence for self-dual representations of GL(n). The field of research of the P.I. is Automorphic Forms. The simplest, yet not so simple, instance of the basic problem of the field is the following: Start with a sequence of numbers {a_0, a_1 , a_2, a_3, .., a_n, ...} and consider the "generating function" f(q) = a_0 + a_1q + a_2q^2 + ... + a_nq^n + ...., where q is a dummy variable. A fundamental question is to know when f(q) satisfies a "hidden symmetry". To elaborate, write q = exp(2\pi iz), with z a complex number of positive imaginary part, and set q* = exp(-2\pi i/z). What one is often looking for, and this shows up in disparate fields like string Physics and combinatorics, is a relationship between the pair (f(q), f(q*)). One says that f has weight k if f(q*) =(-log q/2\pi i)^k f(q). The existence of such a symmetry implies that the sequence {a_n} we started with has miraculous properties. For example, when the a_n are multiplicative, i.e., when a_{mn} = a_ma_n for m,n relatively prime, with a_0=0 and a_1=1, then there is an associated 2-dimensional Galois representation R coming from geometry whose associated "L-function" equals 1 + a_2/2^s + a_3/3^s + ..., implying that for each prime p, a_p =u_p + 1/u_p with u_p an algebraic integer of absolute value p^{(k-1)/2}; in particular, |a_p| is bounded by 2p^{(k-1)/2}, which is not provable by an apriori analytic estimate. A key example to keep in mind is the ubiquitous Delta function q{(1-q)(1-q^2)(1-q^3)...)}^{24} = q+tau_2q^2 + tau_3q^3 + ..., which has weight 12. The general "Langlands program" envisions many such occurrances, and they involve a family of symmetries ("modularity") which are complicated to write down explicitly, but are nevertheless very important to pursue due to their far-reaching consequences. For example, one key ingredient of the celebrated proof of Fermat's last theorem by Wiles makes use of a result obtained in this program. In his work related to hiscurrent (about to become preious) NSF proposal, the P.I. proved that given two functions f(q), g(q) as above attached to {a_n}, {b_n} respectively, admitting hidden symmetries of some weights, the product sequence {a_nb_n} is associated to a modular object of degree 4. This has the following consequence. Suppose f, g have the same weights, and suppose further that a_p^2 equals b_p^2 for almost all primes p. Then f equals g.

AUTOMORPHIC FORMS, L-FUNCTIONS AND GALOIS REPRESENTATIONS The princial investigator proposes to do the following: (i) attachGalois representations to cusp forms of weight k 1 over any CM field K (with totally real subfield F) by transferring certain associated forms on GL(4)/F to suitable unitary groups by making use of L-functions, trace formula, congruences, restrictions of Hasse invariant forms and pseudo-representations; (ii) construct certain special holomorphic forms on GSp(4)/Q, study their lifting to GL(4)/Q, and derive consequences for certain Galois representations; and (iii) to continue ongoing work with D. Prasad on a refinement of the local Langlands correspondence for self-dual representations of GL(n). The field of research of the P.I. is Automorphic Forms. The simplest, yet not so simple, instance of the basic problem of the field is the following: Start with a sequence of numbers {a_0, a_1 , a_2, a_3, .., a_n, ...} and consider the "generating function" f(q) = a_0 + a_1q + a_2q^2 + ... + a_nq^n + ...., where q is a dummy variable. A fundamental question is to know when f(q) satisfies a "hidden symmetry". To elaborate, write q = exp(2\pi iz), with z a complex number of positive imaginary part, and set q* = exp(-2\pi i/z). What one is often looking for, and this shows up in disparate fields like string Physics and combinatorics, is a relationship between the pair (f(q), f(q*)). One says that f has weight k if f(q*) =(-log q/2\pi i)^k f(q). The existence of such a symmetry implies that the sequence {a_n} we started with has miraculous properties. For example, when the a_n are multiplicative, i.e., when a_{mn} = a_ma_n for m,n relatively prime, with a_0=0 and a_1=1, then there is an associated 2-dimensional Galois representation R coming from geometry whose associated "L-function" equals 1 + a_2/2^s + a_3/3^s + ..., implying that for each prime p, a_p =u_p + 1/u_p with u_p an algebraic integer of absolute value p^{(k-1)/2}; in particular, |a_p| is bounded by 2p^{(k-1)/2}, which is not provable by an apriori analytic estimate. A key example to keep in mind is the ubiquitous Delta function q{(1-q)(1-q^2)(1-q^3)...)}^{24} = q+tau_2q^2 + tau_3q^3 + ..., which has weight 12. The general "Langlands program" envisions many such occurrances, and they involve a family of symmetries ("modularity") which are complicated to write down explicitly, but are nevertheless very important to pursue due to their far-reaching consequences. For example, one key ingredient of the celebrated proof of Fermat's last theorem by Wiles makes use of a result obtained in this program. In his work related to hiscurrent (about to become preious) NSF proposal, the P.I. proved that given two functions f(q), g(q) as above attached to {a_n}, {b_n} respectively, admitting hidden symmetries of some weights, the product sequence {a_nb_n} is associated to a modular object of degree 4. This has the following consequence. Suppose f, g have the same weights, and suppose further that a_p^2 equals b_p^2 for almost all primes p. Then f equals g.