Arithmetic statistics of Prym surfaces.

Area of Science:

• Algebraic Geometry
• Number Theory

Background:

• Prym varieties are central to understanding algebraic cycles and arithmetic properties of families of abelian varieties.
• Abelian surfaces with polarization of type (1, 2) are a key area of study in arithmetic geometry.

Purpose of the Study:

• To investigate the average size of Selmer groups for a specific family of abelian surfaces.
• To establish bounds on the average rank of these Prym varieties.
• To provide evidence for heuristics concerning non-principally polarized abelian varieties.

Main Methods:

• Analysis of Lie algebra embeddings and invariant theory.
• Utilizing a geometric construction by Pantazis.
• Studying Néron component groups of Prym surfaces.
• Applying Bhargava's orbit-counting techniques.

Main Results:

• The average size of the Selmer group for the (1, 2)-polarization is 3.
• The average size of the 2-Selmer group is bounded above by 5.
• An upper bound on the average rank of these Prym varieties is established.

Conclusions:

• The results support Poonen and Rains' heuristics for families of non-principally polarized abelian varieties.
• This work offers significant insights into the arithmetic of Prym varieties.