A.C. COJOCARU Professor of Mathematics
University of Illinois
Chicago, IL, USA
Scientific Researcher
Institute of Mathematics of the Romanian Academy
Bucharest, Romania


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Research



A.C. Cojocaru's research lies in number theory, a branch of pure mathematics devoted to understanding the integers and their generalizations. The principal objects of study in number theory are the primes, that is, positive integers, not 1, whose only divisors are 1 and the integer itself. For example, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, and 197 are primes. Since every positive integer different from 1 can be written uniquely as a product of primes, the primes are the building blocks of the integers.

It has been known, at least since Euclid (around 300 BCE), that there are infinitely many primes. It has also been observed that there are many primes in patterned sequences. For example, 5 = 3 + 2, 7 = 5 + 2, 13 = 11 + 2, 19 = 17 + 2, 31 = 29 + 2 are primes of the form p + 2, with p another prime, while 5 = 2^2 + 1, 17 = 4^2 + 1, 37 = 6^2 + 1, 101 = 10^2 + 1, 197 = 14^2 + 1 are primes of the form n^2 + 1, with n an integer.

Are there infinitely many primes satisfying an observed pattern? In a given interval, how many primes satisfying an observed pattern are there?

For the above explicit patterns, precise conjectural answers were famously formulated by Hardy and Littlewood in the 1920s. Cojocaru's research centers on investigations of similar questions, with the prime pattern arising in a geometric setting such as the one defined by an elliptic curve.

An elliptic curve, call it E, is the geometric locus of an equation y^2 = f(x), where f(x) is a polynomial of degree 3. For example, y^2 = x^3 + 1 is the equation of an elliptic curve. For the points (x, y) on E there is an analogue of the addition of integers, which gives rise to a group structure associated to E. If f(x) has complex coefficients, this group may be visualized as a torus. If f(x) has integral coefficients, each prime p defines a new curve, call it E_p, whose points give rise to another group structure. Of major interest in the field is to understand the group structures coming from E and from E_p, as the prime p varies.

During the 1950s-1980s, reputed number theorists formulated now-celebrated problems, most still open, about the reductions E_p of an elliptic curve E, including: the Birch and Swinnerton-Dyer Conjecture, the Koblitz Conjecture, the Lang-Trotter Conjectures, and the Sato-Tate Conjecture. While motivated by algebraic and cryptographic objectives, these geometric problems echo classical ones about prime patterns, such as the Hardy-Littlewood Conjectures alluded to above.

Thanks to generous support provided by agencies such as NSERC, NSF, and the Simons Foundation, Cojocaru has investigated and proved several results about the Koblitz Conjecture and the Lang-Trotter Conjectures in the setting of elliptic curves and that of higher dimensional abelian varieties (e.g., related to equations y^2 = f(x) with f(x) a polynomial of degree higher than 3), as well as the setting of Drinfeld modules (e.g., related to polynomial generalizations of tori). For details on this research, see Cojocaru's papers.




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A.C. COJOCARU Professor of Mathematics
University of Illinois
Chicago, IL, USA
Scientific Researcher
Institute of Mathematics of the Romanian Academy
Bucharest, Romania