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Felix, Adam Tyler
- On the Asymptotics for Invariants of Elliptic Curves Modulo p
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Authors
Affiliations
1 Department of Mathematics & Computer Science, University of Lethbridge, Lethbridge, Alberta, CA
2 Department of Mathematics & Statistics, Queen’s University, Kingston, Ontario, CA
1 Department of Mathematics & Computer Science, University of Lethbridge, Lethbridge, Alberta, CA
2 Department of Mathematics & Statistics, Queen’s University, Kingston, Ontario, CA
Source
Journal of the Ramanujan Mathematical Society, Vol 28, No 3 (2013), Pagination: 271–298Abstract
Let E be an elliptic curve defined over Q. Let E(Fp) denote the elliptic curve modulo p. It is known that there exist integers i p and f p such that E(Fp) ∼= Z/i pZ × Z/i p fp Z. We study questions related to i p and f p. In particular, for any α > 0 and k ∈ N, we prove there exist positive constants cα and ck such that for any A > 0
Σ(log ip)α = cα li(x) + O(x/(logx)A)
and
Σ Tk(ip) = ck li(x) + O(x/(log x)A)
unconditionally for CMelliptic curves, where τk (n) is the number of ways of writing n as a product of k positive integers. For a CM curve E and 0 < α < 1, we prove that there exists a constant c'α > 0 such that Σ iαp = c'α li(x) + O(x3+α/4 (log x)1-α/2)