David Zywina

Curriculum Vitae

Teaching: Math 6370 : Algebraic Number Theory (Spring 2025)

• Classification of modular curves with low gonality
• Rank one elliptic curves and rank stability
• There are infinitely many elliptic curves over the rationals of rank 2
• An elliptic surface with infinitely many fibers for which the rank does not jump
• Drinfeld modules with maximal Galois action
• Open image computations for elliptic curves over number fields
• Improved bounds for integral points on modular curves using Runge's method
• Torsion bounds for a fixed abelian variety and varying number field with Samuel Le Fourn and Davide Lombardo
• Explicit open images for elliptic curves over Q
• Determining monodromy groups of abelian varieties
• Computing actions on cusp forms
• Families of abelian varieties and large Galois images
• Galois groups arising from families with big orthogonal monodromy
• An effective open image theorem for abelian varieties
• Modular curves of prime-power level with infinitely many rational points with Andrew Sutherland
• The inverse Galois problem for orthogonal groups
• Possible indices for the Galois image of elliptic curves over Q
• Modular forms and some cases of the inverse Galois problem
• An explicit Jacobian of dimension 3 with maximal Galois action
• Bounds for the Lang-Trotter conjectures
• On the surjectivity of mod l representations associated to elliptic curves
• On the possible images of the mod l representations associated to elliptic curves over Q
• The inverse Galois problem for PSL2(Fp)
• The splitting of reductions of an abelian variety
• Explicit class field theory for global function fields
• The Sato-Tate Law for Drinfeld modules
• Bounds for Serre's open image theorem
• Abelian varieties over large algebraic fields with infinite torsion
• Hilbert's irreducibility theorem and the larger sieve
• Splitting fields of characteristic polynomials of random elements in arithmetic groups with F. Jouve and E. Kowalski
• The Chebotarev invariant of a finite group with E. Kowalski (short version)
• A refinement of Koblitz's conjecture
• Drinfeld modules with maximal Galois action on their torsion points
• Elliptic curves with maximal Galois action on their torsion points
• The Lang-Trotter conjecture and mixed representations (old preprint, main results are now part of "Bounds for the Lang-Trotter conjectures")
• The large sieve and Galois representations
• Arithmetic E8 lattices with maximal Galois action with Tony Várilly-Alvarado
• An explicit integral polynomial whose splitting field has Galois group W(E8) with F. Jouve and E. Kowalski

• Inverse Galois problem for small simple groups