Ionica, Sorina; Kılıçer, Pınar; Lauter, Kristin; Lorenzo García, Elisa; Manzateanu, Maria-AdelinaISTA; Massierer, Maike; Vincent, Christelle
In this article we prove an analogue of a theorem of Lachaud, Ritzenthaler, and Zykin, which allows us to connect invariants of binary octics to Siegel modular forms of genus 3. We use this connection to show that certain modular functions, when restricted to the hyperelliptic locus, assume values whose denominators are products of powers of primes of bad reduction for the associated hyperelliptic curves. We illustrate our theorem with explicit computations. This work is motivated by the study of the values of these modular functions at CM points of the Siegel upper half-space, which, if their denominators are known, can be used to effectively compute models of (hyperelliptic, in our case) curves with CM.
Research in Number Theory
The authors would like to thank the Lorentz Center in Leiden for hosting the Women in Numbers Europe 2 workshop and providing a productive and enjoyable environment for our initial work on this project. We are grateful to the organizers of WIN-E2, Irene Bouw, Rachel Newton and Ekin Ozman, for making this conference and this collaboration possible. We thank Irene Bouw and Christophe Ritzenhaler for helpful discussions. Ionica acknowledges support from the Thomas Jefferson Fund of the Embassy of France in the United States and the FACE Foundation. Most of Kılıçer’s work was carried out during her stay in Universiteit Leiden and Carl von Ossietzky Universität Oldenburg. Massierer was supported by the Australian Research Council (DP150101689). Vincent is supported by the National Science Foundation under Grant No. DMS-1802323 and by the Thomas Jefferson Fund of the Embassy of France in the United States and the FACE Foundation.
Ionica S, Kılıçer P, Lauter K, et al. Modular invariants for genus 3 hyperelliptic curves. Research in Number Theory. 2019;5. doi:10.1007/s40993-018-0146-6
Ionica, S., Kılıçer, P., Lauter, K., Lorenzo García, E., Manzateanu, M.-A., Massierer, M., & Vincent, C. (2019). Modular invariants for genus 3 hyperelliptic curves. Research in Number Theory. Springer Nature. https://doi.org/10.1007/s40993-018-0146-6
Ionica, Sorina, Pınar Kılıçer, Kristin Lauter, Elisa Lorenzo García, Maria-Adelina Manzateanu, Maike Massierer, and Christelle Vincent. “Modular Invariants for Genus 3 Hyperelliptic Curves.” Research in Number Theory. Springer Nature, 2019. https://doi.org/10.1007/s40993-018-0146-6.
S. Ionica et al., “Modular invariants for genus 3 hyperelliptic curves,” Research in Number Theory, vol. 5. Springer Nature, 2019.
Ionica S, Kılıçer P, Lauter K, Lorenzo García E, Manzateanu M-A, Massierer M, Vincent C. 2019. Modular invariants for genus 3 hyperelliptic curves. Research in Number Theory. 5, 9.
Ionica, Sorina, et al. “Modular Invariants for Genus 3 Hyperelliptic Curves.” Research in Number Theory, vol. 5, 9, Springer Nature, 2019, doi:10.1007/s40993-018-0146-6.