TY - JOUR

T1 - Modular invariants for genus 3 hyperelliptic curves

AU - Ionica, Sorina

AU - Kilicer, Pinar

AU - Lauter, Kristin

AU - Garcia, Elisa Lorenzo

AU - Massierer, Maike

AU - Manzateanu, Adelina

AU - Vincent, Christelle

PY - 2019/1/2

Y1 - 2019/1/2

N2 - In this article we prove an analogue of a theorem of Lachaud, Ritzenthaler, and Zykin, which allows us to connect invariants of binary oc-tics 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 value 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.

AB - In this article we prove an analogue of a theorem of Lachaud, Ritzenthaler, and Zykin, which allows us to connect invariants of binary oc-tics 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 value 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.

KW - math.NT

U2 - 10.1007/s40993-018-0146-6

DO - 10.1007/s40993-018-0146-6

M3 - Article

SN - 2363-9555

VL - 5

JO - Research in Number Theory

JF - Research in Number Theory

IS - 9

ER -