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edited Mar 17, 2017 at 19:19

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You can construct some by (pluri-) harmonic theta series on even-dimensional quadratic spaces, imposing congruence conditions to assure not-identical-vanishing. For example, $$ \theta(z_1,z_2) \;=\; \sum_{m,n\in \mathfrak o,\,m=m_o,\,n=n_o\!\!\!\mod\!\! N} P_1(m_1,n_1)\,P_2(m_2,n_2)\,e^{2\pi i ((m_1^2+n_1^2)z_1+(m_2^2+n_2^2)z_2)} $$ where $P_1,P_2$ are homogeneous, harmonic, both of degree $d$, $m_1,m_2$ are the two real imbeddings of $m\in\mathfrak o$, is a cuspform of weight $(1+d,1+d)$. Sufficiently increasing $N$ will guarantee non-vanishing on general principles. Specific numerical choices are surely computable, if not by hand, in Sage or similar. Likewise, with quadratic spaces of dimension $2n$ and harmonic polynomials of degrees $d,d,\ldots,d$ (with $d>0$), you can make cuspforms of weight (n+d,n+d,\ldots,n+d)$$(n+d,n+d,\ldots,n+d)$, with non-vanishing assured by a sufficiently strong congruence condition.

You can construct some by (pluri-) harmonic theta series on even-dimensional quadratic spaces, imposing congruence conditions to assure not-identical-vanishing. For example, $$ \theta(z_1,z_2) \;=\; \sum_{m,n\in \mathfrak o,\,m=m_o,\,n=n_o\!\!\!\mod\!\! N} P_1(m_1,n_1)\,P_2(m_2,n_2)\,e^{2\pi i ((m_1^2+n_1^2)z_1+(m_2^2+n_2^2)z_2)} $$ where $P_1,P_2$ are homogeneous, harmonic, both of degree $d$, $m_1,m_2$ are the two real imbeddings of $m\in\mathfrak o$, is a cuspform of weight $(1+d,1+d)$. Sufficiently increasing $N$ will guarantee non-vanishing on general principles. Specific numerical choices are surely computable, if not by hand, in Sage or similar. Likewise, with quadratic spaces of dimension $2n$ and harmonic polynomials of degrees $d,d,\ldots,d$ (with $d>0$), you can make cuspforms of weight (n+d,n+d,\ldots,n+d)$, with non-vanishing assured by a sufficiently strong congruence condition.

You can construct some by (pluri-) harmonic theta series on even-dimensional quadratic spaces, imposing congruence conditions to assure not-identical-vanishing. For example, $$ \theta(z_1,z_2) \;=\; \sum_{m,n\in \mathfrak o,\,m=m_o,\,n=n_o\!\!\!\mod\!\! N} P_1(m_1,n_1)\,P_2(m_2,n_2)\,e^{2\pi i ((m_1^2+n_1^2)z_1+(m_2^2+n_2^2)z_2)} $$ where $P_1,P_2$ are homogeneous, harmonic, both of degree $d$, $m_1,m_2$ are the two real imbeddings of $m\in\mathfrak o$, is a cuspform of weight $(1+d,1+d)$. Sufficiently increasing $N$ will guarantee non-vanishing on general principles. Specific numerical choices are surely computable, if not by hand, in Sage or similar. Likewise, with quadratic spaces of dimension $2n$ and harmonic polynomials of degrees $d,d,\ldots,d$ (with $d>0$), you can make cuspforms of weight $(n+d,n+d,\ldots,n+d)$, with non-vanishing assured by a sufficiently strong congruence condition.

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created Mar 17, 2017 at 17:38

paul garrett

You can construct some by (pluri-) harmonic theta series on even-dimensional quadratic spaces, imposing congruence conditions to assure not-identical-vanishing. For example, $$ \theta(z_1,z_2) \;=\; \sum_{m,n\in \mathfrak o,\,m=m_o,\,n=n_o\!\!\!\mod\!\! N} P_1(m_1,n_1)\,P_2(m_2,n_2)\,e^{2\pi i ((m_1^2+n_1^2)z_1+(m_2^2+n_2^2)z_2)} $$ where $P_1,P_2$ are homogeneous, harmonic, both of degree $d$, $m_1,m_2$ are the two real imbeddings of $m\in\mathfrak o$, is a cuspform of weight $(1+d,1+d)$. Sufficiently increasing $N$ will guarantee non-vanishing on general principles. Specific numerical choices are surely computable, if not by hand, in Sage or similar. Likewise, with quadratic spaces of dimension $2n$ and harmonic polynomials of degrees $d,d,\ldots,d$ (with $d>0$), you can make cuspforms of weight (n+d,n+d,\ldots,n+d)$, with non-vanishing assured by a sufficiently strong congruence condition.