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 two explicit patterns, precise conjectural answers were formulated by Hardy and Littlewood in the 1920s. Cojocaru's research centers on investigating such questions when the prime pattern arises in the geometric setting defined by an elliptic curve (or by a related, more general object).

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 gives rise to a torus. If f(x) has integer coefficients, each prime p defines a new curve, call it E_p, whose points give rise to yet another group structure. A theme of major research interest in number theory is that of unraveling properties of the groups defined by E and E_p.

During the 1950s-1980s, reputed number theorists formulated now-celebrated problems, most still open, about the reductions E_p of an elliptic curve E defined over the rational numbers. Some of these problems are known as the Birch and Swinnerton-Dyer Conjecture, the Koblitz Conjecture, the Lang-Trotter Conjectures, and the Sato-Tate Conjecture. While set in higher arithmetic-geometric settings, these problems echo classical ones about primes, such as the Hardy and Littlewood Conjectures mentioned 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 settings of elliptic curves and of higher dimensional abelian varieties (objects related to equations y^2 = f(x) with f(x) a polynomial of degree higher than 3), as well as in the setting of Drinfeld modules (objects related to polynomial generalizations of tori). For details, see Cojocaru's papers.

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