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Tate provided an explicit way to kill a nontrivial homology class of a commutative differential graded algebra over a commutative noetherian ring R in Tate (Ill J Math 1:14–27, 1957). The goal of this article is to generalize his result to the case of GBV (Gerstenhaber–Batalin–Vilkovisky) algebras and, more generally, the descendant L ∞ -algebras. More precisely, for a given GBV algebra (A = ⊕ m ≥ 0 A m , δ , ℓ 2 δ) , we provide another explicit GBV algebra (A ~ = ⊕ m ≥ 0 A ~ m , δ ~ , ℓ 2 δ ~) such that its total homology is the same as the degree zero part of the homology H 0 (A , δ) of the given GBV algebra (A , δ , ℓ 2 δ) . [ABSTRACT FROM AUTHOR]

Abstract: When an elliptic curve of square-free conductor N has a rational point of odd prime order , Dummigan (2005) in explicitly constructed a rational point of order l on the optimal curve E, isogenous over to , under some conditions. In this paper, we show that his construction also works unconditionally. And applying it to Heegner points of elliptic curves, we find a family of elliptic curves such that a positive proportion of quadratic twists of has (analytic) rank 1. This family includes the infinite family of elliptic curves of the same property in Byeon, Jeon, and Kim (2009) . [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 / 2 ℤ ⊕ ℤ / 2 ℤ , ℤ / 2 ℤ ⊕ ℤ / 4 ℤ or ℤ / 2 ℤ ⊕ ℤ / 6 ℤ. [ABSTRACT FROM AUTHOR]

In 2017, we decided to start the annual series of MathCrypt Workshops in order to encourage more mathematicians and computational number theorists to propose and work on hard problems in cryptography. In fact, a Program Officer from NSF spoke at the first MathCrypt workshop in August and encouraged participants to apply to the NSF SaTC program for potential support for their research. The founders of MathCrypt, Jung Hee Cheon and Kristin Lauter, acted as Program Chairs, and were joined by Jintai Ding to form the Organizing Committee. [Extracted from the article]

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]