TY - JOUR
T1 - Genus two curves with full √3 -level structure and Tate–Shafarevich groups
AU - Bruin, Nils
AU - Flynn, E. Victor
AU - Shnidman, Ari
N1 - Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
PY - 2023/7
Y1 - 2023/7
N2 - We give an explicit rational parameterization of the surface H3 over Q whose points parameterize genus 2 curves C with full 3 -level structure on their Jacobian J. We use this model to construct abelian surfaces A with the property that [InlineEquation not available: see fulltext.] for a positive proportion of quadratic twists Ad . In fact, for 100 % of x∈ H3(Q) , this holds for the surface A= Jac (Cx) / ⟨ P⟩ , where P is the marked point of order 3. Our methods also give an explicit bound on the average rank of Jd(Q) , as well as statistical results on the size of # Cd(Q) , as d varies through squarefree integers.
AB - We give an explicit rational parameterization of the surface H3 over Q whose points parameterize genus 2 curves C with full 3 -level structure on their Jacobian J. We use this model to construct abelian surfaces A with the property that [InlineEquation not available: see fulltext.] for a positive proportion of quadratic twists Ad . In fact, for 100 % of x∈ H3(Q) , this holds for the surface A= Jac (Cx) / ⟨ P⟩ , where P is the marked point of order 3. Our methods also give an explicit bound on the average rank of Jd(Q) , as well as statistical results on the size of # Cd(Q) , as d varies through squarefree integers.
KW - Higher genus curves
KW - Jacobians
KW - Tate–Shafarevich group
UR - http://www.scopus.com/inward/record.url?scp=85160033235&partnerID=8YFLogxK
U2 - 10.1007/s00029-023-00839-w
DO - 10.1007/s00029-023-00839-w
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AN - SCOPUS:85160033235
SN - 1022-1824
VL - 29
JO - Selecta Mathematica, New Series
JF - Selecta Mathematica, New Series
IS - 3
M1 - 42
ER -