6
$\begingroup$

The question below originates from another post on MathOverflow and a (successful) attempt to prove the Chudnovsky algorithm with modular forms arising from quaternion algebra.

Motivation: As is stated in the former post, a left ideal $I$ of a maximal order $\mathcal{O}$ in a quaternion algebra ramified at $p$ ($p$ is a prime) and $\infty$ can be used to construct modular form $\theta_I$ of weight 2 with Fricke eigenvalue $-1:$

$$\theta_I(\tau)=\sum_{x\in I}e^{2\pi i\tau\frac{N(x)}{N(I)}}$$ One can use the quotient of different linear combinations of theta functions $\theta_I$ to construct explicit equations of modular curves $X_0^{+}(p)$. It is of great interest to investigate the exact positions of the poles of certain quotients (or the zeros of certain linear combination of theta functions from quaternion algebra).

Experiment: Let $p=37$ (a case which is known to E. Hecke). Suppose the maximal order $\mathcal{O}$ in $A(37)$ is the order given by the Proposition 5.2 in A. Pizer's paper, then one can find the class number of left-$\mathcal{O}$ ideals in $A(37)$ is $3$, and there are two different theta functions associated to these ideals:

$$\theta_{I_i}(\tau)=\sum_{x\in\mathbb{Z}^4}q^{x^{T}M_ix},q=e^{\pi i\tau},i=1,2$$

while the $M_1$ and $M_2$ are two Gram matrices attached to two even lattices of dimension $4$:

$$M_1=\left(\begin{matrix}2 & 0 & 1 & 1\\ 0 & 4 & 1 &2\\ 1 & 1 & 10 & 1\\ 1 & 2 & 1 & 20 \end{matrix}\right),M_2=\left(\begin{matrix}4 & 1 & 0 & 1\\ 1 & 6 & 3 &1\\ 0 & 3 & 8 & 2\\ 1 & 1 & 2 & 10 \end{matrix}\right).$$

Then $\phi=(\theta_{I_1}-\theta_{I_2})/2$ is a cusp form on $\Gamma_0(p)$ with Fricke eigenvalue $-1$($\phi$ is such cusp form with zeros at cusps of the possible highest order).

We note that $$\Phi(\tau)=\phi(\tau)\prod_{k=0}^{p-1}\phi\left(\frac{\tau+k}{p}\right)$$ is a modular form of weight $2(p+1)$ on $\Gamma(1)$. As $\Gamma_0(37)$ has two elliptic point of order $2$ and two elliptic points of order $3$, then $$\varphi=\frac{\Phi}{E_4^4E_6^2\Delta^4}$$ is a modular function on $\Gamma(1)$ (everywhere holomorphic except a double pole at $\infty$), $E_4,E_6,\Delta$ are Eisenstein series and modular discriminant, respectively. It is possible to determine numerically the first few coefficients of the $q$-expansion of $\varphi$, and one can determine that $$\varphi(\tau)=(j(\tau)-66^3)^2,$$ which leads to the conclusion that $\phi(\tau)$ vanishes at $(\pm12+2i)/37$.

Experiment Results: One can extend these calculations to some other primes. I tried to construct modular forms of weight $2$ on $\Gamma_0(p)$ with Fricke eigenvalue $-1$ and zeros of highest possible order at the cusp of $\Gamma_0(p)$ for a few primes, and all the zeros of such cusp forms are quadratic irrationals. The results are listed in the table below.

Notations: We denote $\#0$ to be the highest possible order of zeros at the cusp of $\Gamma_0(p)$ for the linear combinations of theta functions. Let $-D,D>0$, $(a_i,b_i,c_i)$ be all the possible reduced binary quadratic forms with discriminant $-D$ and $\tau_{(i,-D)}=(-b_i+\sqrt{-D})/(2a_i)$. We denote a zero of those modular forms by $(\tau_{(i,-D)},k),k\in\mathbb{Z}$, since those modular forms vanish at $(\tau_{(i,-D)}+k)/p$.

\begin{array}{|c|r|c|c|c|c|c|c|} \hline p & \#0 & 1 & 2 & 3 & 4 & 5 & 6\\ \hline 37&1& (\tau_{(1,-16)},12) & & & & & \\ 43&2& (\tau_{(1,-12)},13)& & & & & \\ 67& 3& (\tau_{(1,-12)},8) & (\tau_{(1,-7)},12)& & & & \\ 73& 4& (\tau_{(1,-27)},26)& & & & & \\ 97& 6& (\tau_{(1,-12)},26)& & & & & \\ 163& 7& (\tau_{(1,-7)},26)& (\tau_{(1,-11)},29)& (\tau_{(1,-27)},13)& (\tau_{(1,-44)},57)& (\tau_{(2,-44)},35)& (\tau_{(3,-44)},90)\\ 193& 9& (\tau_{(1,-8)},34)& (\tau_{(1,-16)},31) & (\tau_{(1,-24)},90)& (\tau_{(2,-24)},45)& (\tau_{(1,-75)},37)& (\tau_{(2,-75)},77)\\ \hline \end{array}

Question: What is the reason that all the zeros of these modular forms are quadratic irrationals?

$\endgroup$
2
  • $\begingroup$ Does this condition on order of zeroes at the cusps give you a unique (up to scaling) cusp form? Is it eigen? Is it ever a linear combination of more than 2 theta series? $\endgroup$
    – Kimball
    Mar 31, 2018 at 13:08
  • $\begingroup$ @Kimball: I think the condition on the order of zeros provides the uniqueness of such cusp forms because all the different theta functions span the space of weight 2 modular forms on $\Gamma_0(p)$ with Fricke eigenvalue 1(when one limits oneself to the primes given in the table above). I just calculated those example with $\mathbb{Q}(\sqrt{-p})$ having small class numbers, and I did not find a linear combination of more than 2 theta series. I also did not test whether it is a Hecke eigenform or not. $\endgroup$
    – Y. Zhao
    Apr 1, 2018 at 20:12

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.