Article ID Journal Published Year Pages File Type
4595133 Journal of Number Theory 2008 17 Pages PDF
Abstract

Let H(N)H(N) denote the Hurwitz class number. It is known that if p is a prime, then∑|r|<2pH(4p−r2)=2p. In this paper, we investigate the behavior of this sum with the additional condition r≡c(modm). Three different methods will be explored for determining the values of such sums. First, we will count isomorphism classes of elliptic curves over finite fields. Second, we will express the sums as coefficients of modular forms. Third, we will manipulate the Eichler–Selberg trace formula for Hecke operators to obtain Hurwitz class number relations. The cases m=2,3m=2,3 and 4 are treated in full. Partial results, as well as several conjectures, are given for m=5m=5 and 7.

Related Topics
Physical Sciences and Engineering Mathematics Algebra and Number Theory
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