TY - JOUR

T1 - Modular invariants for genus 3 hyperelliptic curves

AU - Ionica, Sorina

AU - Kılıçer, Pınar

AU - Lauter, Kristin

AU - Lorenzo García, Elisa

AU - Mânzăţeanu, Adelina

AU - Massierer, Maike

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 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.

AB - 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.

KW - Bad reduction

KW - Complex multiplication

KW - Hyperelliptic curve

KW - Invariant of curve

KW - Siegel modular form

KW - Theta constant

UR - http://www.scopus.com/inward/record.url?scp=85059473259&partnerID=8YFLogxK

U2 - 10.1007/s40993-018-0146-6

DO - 10.1007/s40993-018-0146-6

M3 - Article (Academic Journal)

AN - SCOPUS:85059473259

SN - 2363-9555

VL - 5

JO - Research in Number Theory

JF - Research in Number Theory

IS - 1

M1 - 9

ER -