121B is tricky, it seems, as it has odd functional equation (and is the first example of such). The paper of Pacetti and Tornaría simply says that the standard \Theta-function is zero on the 121B-fixed subspace. However, as with other cases, we might be able to recover the desired lift via using sublattices and weighting functions (see the third symmetric power of 49A).
Via the use of Waldspurger's formula, we have isolated (in the space of modular forms) the Shimura lifts for both genera of 121B for both the first and third symmetric powers. However, the identifcation using ternary quadratic forms is yet to be achieved.
The initial segments for some powers are given here.
There are six forms in the plus genus, and four in the minus genus.
Q_1^+=Matrix(3,3,[1,0,0,0,44,-22,0,-22,132])
Q_2^+=Matrix(3,3,[16,-6,-2,-6,16,-2,-2,-2,25])
Q_3^+=Matrix(3,3,[5,2,2,2,36,14,2,14,36])
Q_4^+=Matrix(3,3,[4,-2,0,-2,12,0,0,0,121])
Q_5^+=Matrix(3,3,[5,-1,-2,-1,9,-4,-2,-4,124])
Q_6^+=Matrix(3,3,[4,0,-2,0,33,-11,-2,-11,45])
Q_1^-=Matrix(3,3,[8,4,2,4,13,1,2,1,61])
Q_2^-=Matrix(3,3,[13,-3,-3,-3,21,-1,-3,-1,21])
Q_3^-=Matrix(3,3,[17,-7,-7,-7,21,-1,-7,-1,21])
Q_4^-=Matrix(3,3,[13,-3,-2,-3,21,-8,-2,-8,24])
After some work, we found the following for the lifts of the first few symmetric powers.
First Symmetric Power
\mathop{\sum\sum\sum}_{x,y,z} w_3(x,y,z)w_{11}(x,y,z)q^{Q_5^+(x,y,z)/3}=w_3 \circ w_11 \circ Q_5^+
\mathop{\sum\sum\sum}_{x,y,z} -w_3(x,y,z)w_{11}(x,y,z)q^{Q_4^-(x,y,z)/3}=-w_3 \circ w_11 \circ Q_4^-
where w_{11} is of the first kind, and w_3 is of the second. Explicitly, w_p of the first kind can be defined (in general for each form Q in the sum) by taking any vector \vec v_0 whose norm (for the form in question) is not p-divisible and any odd periodic function \psi_p of period~p, with then
w_p(\vec v)=\psi_p\bigl(2\langle \vec v_0,\vec v\rangle_Q\bigr)=\psi_p\bigl(2\cdot \vec v_0\cdot Q\cdot \vec v\bigr).
Similarly, the weights w_l of the second kind are defined on vectors whose Q-norm is p-divisible. Taking a primitive such as \vec v_0, we have w_l(\vec v)=\chi_l\bigl(2D\cdot\langle \vec v_0,\vec v\rangle_Q\bigr) where D is the discriminant (11 in our case) and \chi_l is the quadratic character mod~l. When the argument is l-divisible, we instead have w_l(\vec v)=\chi_l\bigl({\vec v\over \vec v_0}\bigr).
Third Symmetric Power
Here we have
\bigl(x\circ w_{11}\circ Q_1^+\bigr) \oplus \bigl([8y-6z]\circ w_{11}\circ Q_2^+\bigr) \oplus \bigl([-2x+2y]\circ w_{11}\circ Q_5^+\bigr)
and
\bigl([2x-5z]\circ w_{11}\circ Q_1^-\bigr) \oplus \bigl([3x+2z]\circ w_{11}\circ Q_2^-\bigr) \oplus \bigl([x-6z]\circ w_{11}\circ Q_3^-\bigr) \oplus \bigl([2x-2y+4z]\circ w_{11}\circ Q_4^-\bigr)
Fifth Symmetric Power
Here we have
\bigl([11xy/2]\circ w_3 \circ w_{11}\circ Q_1^+\bigr) \oplus \bigl([22y^2-33yz]\circ w_3\circ w_{11}\circ Q_2^+\bigr) \oplus \bigl([3x^2-10xy+y^2-118yz]\circ w_3\circ w_{11}\circ Q_5^+\bigr)
and
\bigl([-11xy/2+55yz/4]\circ w_3\circ w_{11}\circ Q_1^-\bigr) \oplus \bigl([-33xz/2-11z^2/2]\circ w_3 \circ w_{11}\circ Q_2^-\bigr) \oplus \bigl([11xz/2-33z^2/2]\circ w_{11}\circ Q_3^-\bigr) \oplus \bigl([x^2+8xy-24xz+5y^2+14yz-10z^2]\circ w_{11}\circ Q_4^-\bigr)
LfunctionsAndModularFormsII/CentralValues/121B (last edited 2009-03-01 00:55:50 by localhost)
