POTENTIAL PROBLEM DESCRIPTIONS Number Theory: The distribution of primes has long been a central theme in number theory. One important conjecture in this area is the Lang-Trotter conjecture. Let E : y 2 = x3 +Ax+B be an elliptic curve defined over Q and let aE (p) denote the number of points on the reduction of E modulo the prime p. Let r ∈ Z and put πE,r (X) = #{p < X | p is prime and aE (p) = r}. Lang and Trotter [6] conjectured that if E does not have complex multiplication (this condition is satisfied by almost all curves) or if r 6= 0, then √ X (1) πE,r (X) ∼ CE,r , log X where the constant depends only on E and r. To appreciate the importance of this conjecture one should note that this is a refinement of the recently proved Sato-Tate conjecture and is much more precise than what can be deduced from the Chebotarev density theorem in this setting. I. Champion primes: An interesting related topic is the distribution of champion primes for an elliptic curve, that is a prime p for which aE (p) achieves the maximal/minimal value allowed √ by Hasse’s theorem ±[|2 p|]. Put √ πEmax (X) = {p < X : aE (p) = −b2 pc} √ π min (X) = {p < X : a (p) = b2 pc} E

E

πE± (X) = πEmax (X) + πEmin (X). Participants in our 2012 REU were able to prove upon assumption of the extended Riemann Hypothesis and additional mild assumptions that if E has complex mulX 3/4 tiplication, then πE± (X) ∼ CE log where CE is an explicit constant (see [5]). This X result was made unconditional in [2] and in fact asymptotics for all of the above prime counting functions were obtained for E an elliptic curve with complex multiplication. There is some current work towards characterizing the average behavior of these functions in the non-CM case. It would be interesting to extend this work and the work of [2] to the setting of number fields as in [1, 3, 4]. It will also be interesting to consider related prime counting functions. Let f (t) be √ a positive valued function which grows more slowly than p (-e.g. f (t) = log t). √ πEf (X) = {p < X : aE (p) = [|2 p − f (p)|]} 1

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POTENTIAL PROBLEM DESCRIPTIONS

Participants could attempt to employ the techniques of [2] to obtain similar asymptotics for πEf (X) for CM elliptic curves. They could also formulate conjectures concerning these functions in the non-CM case. Computations yielding data to support the various conjectures about the behaviors of these prime counting functions in the non-CM case would also be desirable. II. Lang-Trotter Constant Computation: The constant in the Lang-Trotter conjecture is given explicitly in terms of the Galois representations arising from E. However, it is difficult to explicitly compute the constant for a given curve E and integer r. Participants in our 2003 REU investigated the Lang-Trotter conjecture computationally. In particular, we computed the ratio √ of primes p < 107 with ap (E) = r to X/ log X for various curves E, r ∈ Z. In light of the Lang-Trotter conjecture, we expect that this ratio should tend to the constant CE,r of equation (1). In our 2008 REU, participants made some progress in developing algorithms to explicitly compute the constant CE,r . Participants in future REUs will continue this work allowing one to extend the minimal computational evidence for the Lang-Trotter conjecture. III. Lang-Trotter For Modular Forms: The Lang-Trotter conjecture can be generalized to the setting of modular forms [7, 8]. More generally, it can be formulated to the case of Galois representations where the Galois representations are required to satisfy some hypotheses. As a first step in this project we would produce more computational evidence for this form of the conjecture in the case of elliptic modular forms by using SAGE. Once the students are comfortable using SAGE, we would move on to studying the conjecture for Siegel modular forms. These are modular forms that live on Siegel upper half-space (a generalization of the complex upper half-plane) and transform under subgroups of Sp2n (Z). In this setting there has been no computational evidence, so we will produce the appropriate SAGE code to compute with such forms and then test the conjecture. In the case that n = 2 it is known these modular forms have an associated 4-dimensional Galois representation so we can also check whether these Galois representations fit into the class of representations the conjectures apply to.

References [1] B. Faulkner, K. James, M. King, and D. Penniston. Average Frobenius distributions for elliptic curves over abelian extensions. Acta Arith. to appear. [2] Kevin James and Paul Pollack. Extremal primes for elliptic curves with complex multiplication. J. Number Theory, 172:383–391, 2017. [3] Kevin James and Ethan Smith. Average Frobenius distribution for elliptic curves defined over finite Galois extensions of the rationals. Math. Proc. Cambridge Philos. Soc., 150(3):439–458, 2011.

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[4] Kevin James and Ethan Smith. Average Frobenius distribution for the degree two primes of a number field. Math. Proc. Cambridge Philos. Soc., 154(3):499–525, 2013. [5] Kevin James, Brandon Tran, Minh-Tam Trinh, Phil Wertheimer, and Dania Zantout. Extremal primes for elliptic curves. J. Number Theory, 164:282–298, 2016. [6] S. Lang and H. Trotter. Frobenius distributions in GL2 extensions, volume 504 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1976. [7] M.R. Murty, V.K. Murty, and N. Saradha. Modular forms and the Chebotarev density theorem. Amer. J. Math., 110(2):253–281, 1988. [8] V.K. Murty. Frobenius distributions and Galois representations. In Automorphic Forms, Automorphic Representations, and Arithmetic, volume 66 of Proc. Sympos. Pure. Math, pages 193–211, Providence, RI, 1999. Amer. Math. Soc.