This question is a follow-up of an old question posted on MathOverflow.

Motivation: The exact equations of modular curves $X_0^{+}(p)=X_0(p)/w_p$($p>13$ is a prime number,$w_p$ is the Fricke involution) are intriguing objects. One can construct exact equations from modular forms of weight 2 on $\Gamma_0(p)$ from a quaternion algebra $A(p)$ over $\mathbb{Q}$ ramified at $p$ and $\infty$. Let $O$ be a maximal order of $A(p)$ and $I$ be a left $O$-ideal. The theta function determined by $I$ $$\theta_I(\tau)=\sum_{x\in I}e^{2\pi i\tau\frac{N(x)}{N(I)}}$$ is always a modular form of weight 2 on $\Gamma_0(p)$ with Fricke eigenvalue $-1$(see A. Pizer's paper).

Experiment: According to the Proposition 2.17 of Pizer's paper, the number of different theta functions determined by left ideals of $A(p)$ is bounded by the type number $T(p)$, which can be computed explicitly in another paper of Pizer (p. 94). A numerical computation with MAGMA seems to suggest that the number of distinct theta functions associated to the left $O$-ideals of $A(p)$ always reaches the upper bound, and all these theta functions are linear independent.

Question: Is there any example that the linear independence of theta functions fails for a certain prime number $p>13$? If not, is there a reference for the proof of this fact?

Update: The conjecture is likely to be false for $p=227$.


Hecke conjectured that $\theta_I$ form a basis for the space $S_2(p)$, but this was found to fail for $p=37$ by Eichler. In fact, Gross realized that whenever you get vanishing central $L$-values you get a linear relation among theta series. This happens in $S_2(37)$ since there is an elliptic curve with root number $-1$ of conductor 37.

However, Eichler showed you can find a basis for the space of modular forms among the theta series attached to lattices $J^{-1}I$ as $I, J$ vary over left $O$-ideals (so you range over left $O'$-ideals where $O'$ ranges over all maximal orders). The precise linear relations are mysterious however. See for instance the introduction to Hijikata, Pizer and Shemanske's Memoirs article on the basis problem.


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