Analytic Number Theory
- Non-vanishing of L-functions.
- Average values of twists of L-functions.
- Sato-Tate conjecture.
- Distribution of Fourier coefficients of modular forms.
- Lang-Trotter conjecture and generalizations.
- Families of L-functions.
Algebraic Number Theory
- Class numbers of solvable extensions.
- Effective versions of the Chebotarev density theorem.
- Growth of Selmer ranks of Abelian varieties in towers of number fields.
- Euclidean Algorithm.
- Bounded generation of Algebraic groups.
Arithmetic Algebraic Geometry
- Conjectures on algebraic cycles (Hodge, Tate).
- Tannakian groups.
- Algebraic cycles on Shimura varieties.
- Fundamental group of Satake compactifications.
Applications to Information Technology
- Algebraic curves and cryptography.
- Explicit arithmetic on Abelian varieties over finite fields.
- Counting points on Abelian varieties over finite fields.
- Data Integrity algorithms.
- Data Compression algorithms.
- Storage Security.