Zagier was born in Heidelberg, West Germany. His mother was a psychiatrist, and his father was the dean of instruction at the American College of Switzerland. His father held five different citizenships, and he spent his youth living in many different countries. After finishing high school and attending Winchester College for a year, he studied for three years at MIT, completing his bachelor's and master's degrees and being named a Putnam Fellow in 1967 at the age of 16. He then wrote a doctoral dissertation on characteristic classes under Friedrich Hirzebruch at Bonn, receiving his PhD at 20. He received his Habilitation at the age of 23, and was named professor at the age of 24.
Work
Zagier collaborated with Hirzebruch in work on Hilbert modular surfaces. Hirzebruch and Zagier coauthored Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus, where they proved that intersection numbers of algebraic cycles on a Hilbert modular surface occur as Fourier coefficients of a modular form. Stephen Kudla, John Millson and others generalized this result to intersection numbers of algebraic cycles on arithmetic quotients of symmetric spaces. One of his results is a joint work with Benedict Gross. This formula relates the first derivative of the complex L-series of an elliptic curve evaluated at 1 to the height of a certain Heegner point. This theorem has some applications including implying cases of the Birch and Swinnerton-Dyer conjecture along with being an ingredient to Dorian Goldfeld's solution of the class number problem. As a part of their work, Gross and Zagier found a formula for norms of differences of singular moduli. Zagier later found a formula for traces of singular moduli as Fourier coefficients of a weight 3/2 modular form. Zagier collaborated with John Harer to calculate the orbifold Euler characteristics of moduli spaces of algebraic curves, relating them to special values of the Riemann zeta function. Zagier found a formula for the value of the Dedekind zeta function of an arbitrary number field at s = 2 in terms of the dilogarithm function, by studying arithmetic hyperbolic 3-manifolds. He later formulated a general conjecture giving formulas for special values of Dedekind zeta functions in terms of polylogarithm functions. He discovered a short and elementary proof of Fermat's theorem on sums of two squares. Zagier won the Cole Prize in Number Theory in 1987, the von Staudt Prize in 2001 and the Gauss Lectureship of the German Mathematical Society in 2007. He became a foreign member of the Royal Netherlands Academy of Arts and Sciences in 1997 and a member of the National Academy of Sciences in 2017.
Selected publications
. The First 50 Million Prime Numbers." Math. Intel. 0, 221–224, 1977.
"Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus" Invent. Math. 36 57-113
Hyperbolic manifolds and special values of Dedekind zeta functions Invent. Math. 83 285-302
Singular moduli J. reine Angew. Math. 355 191-220
Heegner points and derivative of L-series Invent. Math. 84 225-320
Polylogarithms, Dedekind zeta functions, and the algebraic K-theory of fields in Arithmetic Algebraic Geometry, Prog. in Math. 89, Birkhäuser, Boston 391-430
How often should you beat your kids?'' https://www.jstor.org/stable/2691064.
