MA thesis available - University of Hawaii

More documents

Recommendations

Info

$Math 215 Worksheet 1 (1) Is it true that between any two distinct real ...$

$Applied Calculus Math 215 - University of Hawaii$

$Math 215 Derivative Practice Problems & Solutions 2009 July 11 ...$

$Linear Algebra (Math 311) – Final$

2 Acknowledgments I would like to express the deepest appreciation to my advisor, Professor Pavel Guerzhoy. You have been incredibly patient through the years to a student whose head is too often in the clouds. Without your guidance and persistent help, this project would remain in my dream. My gratitude also goes to Professor J.B. Nation. You have enriched my time here at the Mathmatics Department in a way that words cannot express. One would be hard-pressed to find a more enthusiastic and energentic graduate chair. There are many others to whom I am grateful in the mathematics department at the University of Hawaii, including all my professors, fellow graduate students. Also, thank you to Susan, Shirley, and Kevin for your love and attentions over the years. Finally, I will be forever grateful to all my friends and family. Each of you is in this accomplishment, for without your love and support it would not have been possible.
ON KANEKO CONGRUENCES MINGJING. CHI Abstract. We present a proof of certain congruences modulo powers of an odd prime for the coefficients of a series produced by repeated application of U-operator to a certain weakly holomorphic modular form. This kind of congruences were first observed by Kaneko as a result of numerical experiments, and later proved in a different (but similar) case by Guerzhoy [6]. It is interesting to note that, in our case, the congruences become different, both experimentally and theoretically, depending on whether the prime is congruent to 1 or 3 modulo 4. 1. Introduction In this paper we prove congruences of q-series coefficients modulo powers of primes. These congruences may be described in an elementary way, and we begin with this description. Consider the following formal power series in variable q e4(q) = 1+240 ⎛ ⎝ d 3 ⎞ ⎠ q n = 1+240q+2160q 2 +6720q 3 +17520q 4 +30240q 5 +60480q 6 +. . . and n>0 d|n η ∗ (q) = (1 − q n ) = 1 − q − q 2 + q 5 + q 7 − q 12 − q 15 + . . . n≥1 By construction, both series have integer coefficients, i.e. e4, η ∗ ∈ Z[[q]]. Moreover, it is easy to see that the series e4(q 4 ) η ∗ (q 4 ) 2 η ∗ (q 8 ) 2 = 1 + 242q4 + 2647q 8 + 12734q 12 + 49475q 16 + 151026q 20 + 429260q 24 + . . . also has integer coefficients. In order to observe the properties in question, we divide the series by q in order to shift the indexing by one, and introduce the notation c(n) for the series coefficients c(n)q n = n≥−1 e4(q 4 ) qη ∗ (q 4 ) 2 η ∗ (q 8 ) 2 = q−1 + 242q 3 + 2647q 7 + 12734q 11 + 49475q 15 + . . . Note that c(n) = 0 unless n ≡ 3 mod 4. The following division properties of the coefficients c(n) were first observed by M. Kaneko on the basis of numerical experiments. 3
Page 1: On Kaneko Congruences A thesis subm
Page 5 and 6: In this paper we prove the followin
Page 7 and 8: ON KANEKO CONGRUENCES 7 principle
Page 9 and 10: ON KANEKO CONGRUENCES 9 holomorphic
Page 11 and 12: ON KANEKO CONGRUENCES 11 T (mn) :=
Page 13 and 14: with a fourth root of unity uγ (wh
Page 15 and 16: ON KANEKO CONGRUENCES 15 Propositio
Page 17 and 18: Let β, β ′ be the roots of equa
Page 19 and 20: as β − β′ Ap 0 ≡ p ON KANEK

MA thesis available - University of Hawaii

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?