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Let E be an elliptic curve defined over ℚ of conductor N , c the Manin constant of E , and m the product of Tamagawa numbers of E at prime divisors of N. Let K be an imaginary quadratic field where all prime divisors of N split in K , P K the Heegner point in E (K) , and III (E / K) the Shafarevich–Tate group of E over K. Let 2 u K be the number of roots of unity contained in K. Gross and Zagier conjectured that if P K has infinite order in E (K) , then the integer c ⋅ m ⋅ u K ⋅ | III (E / K) | 1 2 is divisible by | E (ℚ) tor |. In this paper, we show that this conjecture is true if E (ℚ) tor ≅ ℤ / 2 ℤ ⊕ ℤ / 2 ℤ , ℤ / 2 ℤ ⊕ ℤ / 4 ℤ or ℤ / 2 ℤ ⊕ ℤ / 6 ℤ. [ABSTRACT FROM AUTHOR]

Let E be an elliptic curve defined over ℚ of conductor N , c the Manin constant of E , and m the product of Tamagawa numbers of E at prime divisors of N. Let K be an imaginary quadratic field where all prime divisors of N split in K , P K the Heegner point in E (K) , and III (E / K) the Shafarevich–Tate group of E over K. Let 2 u K be the number of roots of unity contained in K. Gross and Zagier conjectured that if P K has infinite order in E (K) , then the integer c ⋅ m ⋅ u K ⋅ | III (E / K) | 1 2 is divisible by | E (ℚ) tor |. In this paper, we show that this conjecture is true if E (ℚ) tor ≅ ℤ / 3 ℤ. [ABSTRACT FROM AUTHOR]

Kakao is a fast growing software company focused on providing innovative smartphone applications. The company is rooted in the Korean mobile communication market where customer demands for high speed connectivity have been growing sharply. Kakao maintains the top position in the category of mobile communication utility tool for the last three years and competes successfully with other new comers. However, the company faces tremendous external threat from traditional smartphone carriers, SK Telecom, KT and LG U+. Those telecommunication conglomerates view the rapid growth of Kakao as a dangerous possibility that may deprive them of revenue streams from their traditional cash cows of texting and voice calling. Moreover, traditional software champions, Naver and Daum, steadily expand their market shares by differentiating themselves. Those competitors have strong brand assets, innovation capabilities and market sensitivity. After three years in the mobile software market, Kakao faces constraints to its growth potential. Kakao tries to overcome the situation by positioning itself as a new mobile platform leader that connects potential customers and game publishers. In addition, the company wants to speed up innovation for sustaining its reputation through a financial transfusion from external investors. Instead of choosing an easier option of being merged or acquired, the company clearly sets its course to fight against external challenges. How can Kakao overcome the difficulty and sustain growth? What kind of strategic choices can be executable? [ABSTRACT FROM AUTHOR]