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I have heard about the Schottky problem and the related Novikov's conjecture about the characterization of matrices in the Siegel upper half-space which are indeed the Riemann matrix of a compact Riemann surface.

Instead of such a global statement, I was wondering about the following: given $\Omega$ a Riemann matrix of an existing compact Riemann surface, what kind of operations on $\Omega$ could guarantee me that the result is again the Riemann matrix of a surface?

For example, is it true that $2\Omega$ is the Riemann matrix of a surface if $\Omega$ is?

The last specific question comes from the desire for a geometric interpretation with surfaces of the theta functions appearing in the formula expressing the product of two theta functions with a matrix $\Omega$ as a sum of product of theta functions with a matrix $2\Omega$.

Thank you in advance for your insights

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No, this is not true. If $C$ is a general curve of genus $\geq 4$ with period matrix $\Omega$ and $p$ is a prime, $p\Omega$ is not the period matrix of a curve. This is proved (in an equivalent, more geometric, form) by Donagi and Livné, Ann. Sc. Norm. Sup. Pisa 28, no. 2 (1999), p. 323-339 — see §7.

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  • $\begingroup$ Thank you very much for this quick and precise answer, and the specific reference! $\endgroup$
    – azbuk8
    Commented Dec 8, 2021 at 16:04

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