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PreprintCC BY 4.0arXiv

Type of homomorphisms of complex tori

  • Juliana Coelho
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Abstract

Using determinantal divisors of integral matrices, that is, greatest common divisors of minors of fixed order, we introduce the notion of type of a homomorphism f of complex tori, which is similar to the type of a polarization. We show that the type is invariant under composition with isomorphisms, and that it completely describes the kernel of f as a group. More precisely, the quotient of the kernel by its connected component containing 0 is a product of cyclic groups whose orders are determined by the type of f. Since the type can be computed from any rational representation of f, this gives an effective way to determine the kernel of a homomorphism. As a consequence, we compute the classic invariants degree and exponent of f, when f has finite kernel. When f is an isogeny, we also compute the type of its inverse isogeny from that of f. Finally, we compare the type of a polarization to the type of its associated isogeny.

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Cite

BibTeXRIS
BibTeX
@misc{coelho2026type,
  title = {{Type of homomorphisms of complex tori}},
  author = {Juliana Coelho},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.28655},
  eprint = {2608.28655},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.28655},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Type of homomorphisms of complex tori
AU  - Juliana Coelho
PY  - 2026
DA  - 2026/08/20
PB  - arXiv
DO  - 10.48550/arxiv.2608.28655
UR  - https://arxiv.org/abs/2608.28655
AB  - Using determinantal divisors of integral matrices, that is, greatest common divisors of minors of fixed order, we introduce the notion of type of a homomorphism f of complex tori, which is similar to the type of a polarization. We show that the type is invariant under composition with isomorphisms, and that it completely describes the kernel of f as a group. More precisely, the quotient of the kernel by its connected component containing 0 is a product of cyclic groups whose orders are determined by the type of f. Since the type can be computed from any rational representation of f, this gives an effective way to determine the kernel of a homomorphism. As a consequence, we compute the classic invariants degree and exponent of f, when f has finite kernel. When f is an isogeny, we also compute the type of its inverse isogeny from that of f. Finally, we compare the type of a polarization to the type of its associated isogeny.
ER  -