\newcommand{\A}{\mathbf{A}}
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<<<<<<<, Blasius >>>>>>>>>

(Be careful with these notes.  Blasius is imprecise, changes things from time to 
time, and speaks in incomplete sentences.  I've extended things to complete sentences,
which adds a *lot* of meaning.) 

Langlands Reciprocity Conj.


Iso classes of pure    <----------------> Algebraic cusp forms
simple motives M           1:1 corr         on some GL.

        \                                      /
         \                                    /
 
                         good families of   
                         lambda-adic reps. (in sense of Fontaine-Mazur)


   M/F <-------------------------------------> GL/F

Coefficients

 C \supset T \hookrightarrow End(M)

           T is a maximal commutative s.s. subalgebra of End(M)

          dim_T M_B = n        <---------> GL_n / F

--------------

Z = center of End(M)

M1 \isom M2 if there exists lambda M_1 ---> M_2 and (i_1, T_1), (i_2, T_2) with ????? i don't understand what he means.

---------------

M_B \tensor C has a Hodge structure with F_\infty action at each v \mid \infty.
         /|\
          |
         \|/
        Pi_v,

etc., for finite places.

---------------


For today, n=2, F totally real. 

Then dim_T M = 2.                          cusp forms, GL_2(\A_F).

    Z = T if 
    Tr(rho_v) = 0 (odd)
       rho_v := complex conjugation

         /|\                    lately people have gone in the direction \/, but today I'll go in the direction /\!
          |                     this could have been done years ago, but nobody bothered
          |
          |                     [[ this looks like a version of Shimura's "f|---> A_f"]]
         \|/

   Holomorphic Hilbert modular forms \f of weight k, which is constant mod 2. 

----------------
 
The "R-Hodge datum" attached to M:
          
   Let L be a field.
   J_L := Hom(L,C) in category of fields   (just the embeddings of L into C!)
   p : J_F x J_T --> Z     (just gives the Hodge-Tate weights coming from a particular choice of pair of embeddings)
   
   M |----------> (\tau M) \tensor_{\tau,\sigma} \C 
                         <==== Hodge-Tate weights ===> 
                                   \{ (p(\tau,\sigma), q(\tau,\sigma)), (q(\tau,\sigma), p(\tau,\sigma)) \}
   
Now ASSUME: min_{\sigma} q(\tau,\sigma) = 0 

Note: p(\eta \tau, \eta\sigma) = p(\tau,\sigma)


----------------

\f = holomorphic Hilbert modular newform
k(\f) : J_f --> Z_{\geq 1}   ("multi-weight" of the Hilbert modular form)
k_{\max} := max k(\f)(\sigma), over all \sigma.

Going in the "other direction":

                             HODGE DATA of sought for motive

   \f |----------------> P(\f)(\sigma,1_T) = \frac{k_{\max} + k(\sigma) - 2}{ 2 } 

                     also, if w=k_{\max} - 1, then q(\f)(\sigma) = w-p(\f)(\sigma).

WANT M(\f), a motive with coefficients in \T_f (Hecke eigenvalue field) and:

  1. p(M(\f)) = p(\f)
  2. L_{aa}(M(\f),T,s) = L_{aa}(\f,s).

These motives could be called "Hilbert-Blumenthal motives.".


Is there hope??  I wouldn't give this talk if there weren't hope.  For example, we have

Situation:
---------

  1. k \cong 1, wt. 1,   totally odd Galois, Rogawski-Tunnell  (generalization of Deligne-Serre)

  2. k(\sigma) \geq 2 for all \sigma.

       (a) k(\sigma) > 2 for some \sigma.  [Blasius-Rogawski]   lifting to U(3), M(\f) \subset H^*(Sh(U(3))). 

       (b) Case (qc) "quanternionic curve"  (k \cong 2)   [[WHAT DOES "k \cong 2" mean?!]]

              (qc):   [K:\Q] is odd or \f is discrete series at some finite place.
            
           THEOREM (Shimura, HIda, Zhang).  There exists a Hilbert-Blumenthal abelian variety A such that 
                        T_\f \hookrightarrow End(A)
                and     dim A = [T_\f : \Q], and 
                        M = H^1(A). 
            
           Note: The constructions of this theorem involve constructions in the Jacobians of Shimura curves.

REMARK: What if (qc) is not satisfied???   
E.g., "not (qc)" means the following:
 
       In the case [F:\Q]=even, and \f unramified principal series, k=2, level 1

Question from audience: are there any examples?   Elliptic curves over real quadratic fields, having
everywhere good reduction give rise to such examples.   Also, it is easy to construct CM examples. 

        1. Negative "folk Theorem":
 
            There is no A_\f in the abelian part of H^1(Sh) for \f, for any Shimura variety Sh. 

        2. Generalized Hodge conjecture:

               =========>  if M_\f exists then A_\f exists. 

           Really, one finds M_\f(-\alpha) \subset H^t(Sh).   Then M_{\f} is an ineffective motive:
                            M_\f = H^t(Sh)(\alpha)

           The motives we make are frequently not effective!

        3. If some k(\sigma) = 1 and [??] some k(\sigma) > 1. 

           Problem seems very hard!

                      a) F. Jarvis has attached \ell-adic reps for them. 
                      b) They exist!  (I don't know any non-CM examples...)
                      c) No geom. realization of M_\f's is likely using Sh. varieties. 
    
THEOREM. If the Hodge conjecture is true, then there ixists HBAV's for all \f's of weight 2. 

REMARKS. 
       1. By "Hodge conjecture" mean that the following functor is fully faithful:

            H : (Motives/C) --------------> (rational hodge structures).

       2. Hodge classes of the following are algebraic:   (qs) x {abelian varieties}
 

Proof of theorem: 
  First, we replace F by F_0, where \f "lives on" \GL_2(\A_{F_0}). 

  [R:\Q] = 2 real.       F=F_0\cdot R
  [I:\Q] = 2 imaginary

Let \f_F denote the base change of \f to F. 

Consider the quanternion algebra B/F such that
   (1) unramified at v
   (2) split at 1_F, \tau  (inert at other places)

B|----------> S_B = shimura surface

LEMMA: S_B is deifned over F_0.
proof: Trivial to people who have thought about Shimura varieties.


Jaquet-Langlands connects forms like \f to forms on quaternion algebras!

\f_F |----------- Jaquet-Langlands -> \f_B on "B". 


H^2(S_B) \supset H^2(\f_B) = (B\tensor \A_\f)^*.   
                  this is a rank 4 motive, which is twice the dimension we want. 

B has involution of second kind; let's call it \tau.  Decompose using this involution:
 
    H^2(\f_B) = H^2(\f_B)^{-} \oplus H^2(f_B)^{+}

                    d=3                d=1

Here we are assuming, for simplicity, that \T_\f = \Q.

Let's henceforth call the first factor M_B^{-}; it has rank 3.

M_B^{-}\tensor \C = H^{20} \oplus H^{11} \oplus H^{02}

Cup product: M_B x M_B -----> \Q_B(-2). 

-----------------------------

LEMMA:  MT(M^{-}_B)  \subset \GO(3).       (I don't know what "MT" is; maybe something to do with Tate.)

--------------------------

   exists lifting
     htilde /---> GL_2(\R)
     /-----        |
    /              |
   /              \|/
     h
\S --------->  GO(3)(\R)   V
         

M_{B}^{-} \isom Sym^{2}((\Q^2;\htilde)). 

Can make the right hand side over C, and an E such that H^1(E) \isom (\Q^2, \tildeh).

1. E is definable / \Qbar.

   \tau E

   \tau|_{F_0} = 1. 

Key point:   M^{-} \isom Sym^2(H^1(E)) is an isomorphism of motives over C. 
Proof. Follows from the Hodge conjecture. 

\tau E ,  H^1(\tau E),   
         
   Sym^2(H^1(\tau E)) = \tau Sym^2 (H^1(E)) = \tau M^{-} = M^{-}. 

    \tau E is isogeneous to E

Thus [E] has countably many conjugates  ==> [E] has only finitely many conjugates.

Thus j(E)/\qbar ===> E/\Qbar.

------------------

Assume E/L, where L contains F_0, where L is a number field. 

Consider the restriction of scalars 

          R_{L/F_0} E. 

By Taylor, we have an \ell-adic representations \rho_{\ell}^{T}|L \isom \rho_{\ell}(E). 

The \ell-adic repn
            \rho_ell ( R_{L/F_0} E )

is isomorphic to Taylor's representation. 


  LEMMA: Any central idempotent \calE of End(rho_{\ell}^T\tensor Reg) satisfies

                  N\calE in End(Rest of scalars of E). 

Finally, put: (N\calE)(RE) = E_1 is sought for elliptic curve.