4th Meeting of Young Number Theorists |
| Program | Home > Program |
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6월24일(수) |
6월25일(목) |
6월26일(금) |
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09:30~10:30 |
하준수 |
이석형 |
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11:00~12:00 |
조시훈 |
이정욱 |
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중 식 |
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14:00~15:00 |
김태경 |
권영욱 |
이완 |
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15:30~16:30 |
한겨울 |
윤동성 |
최서현 |
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17:00~18:00 |
한재호 |
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18:00~ |
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저녁만찬 |
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[초록]
1. 권영욱
제목: Congruence for traces of singular moduli on genus zero groups
초록: In 2005, Ono and Ahlgren found some mod p congruences for the traces of CM values of certain modular functions of level 1 by using Hecke operators. After their work, many authors obtained generalizations modulo prime powers. But most of the results were restricted to level 1 case. In this talk, we discuss a generalization of these results to some modular functions on certain genus zero groups. We also discuss similar properties for the Hurwitz-Kronecker class numbers.
2. 김태경
제목 : A conjecture of Gross and Zagier for elliptic curves having rational torsion group of a 2-power order
초록 : One of the most prominent results toward the Birch and Swinnerton-Dyer conjecture is the paper Heegner points and derivatives of L-series by B. Gross and D. Zagier. In the paper, they showed that if K is an imaginary quadratic field with ‘Heegner hypothesis’, and E is an elliptic curve such that ord s=1L(E=K; s) = rankE(K) = 1, then L0(E=K; s) can be written asthe product of various arithmetic invariants. This formula looks very alike to the conjectural formula of Birch and Swinnerton-Dyer. Equating these two formulae, Gross and Zagier was able to formulate the conjecture that the order of the rational torsion subgroup of E(Q) divides the product of
Tamagawa number, Manin constant, and the square root of the order of Tate–Shafarevich group of E over K. In this talk, after giving some preliminary stuffs needed to formulate the conjecture, I will give the proof of the conjecture, when the elliptic curve does not have points of odd order.
3. 윤동성
제목: Applications of modular units
초록: A modular unit is defined to be a modular function which has neither zeros nor poles on the complex upper half-plane. It has many advantages to construct generators of modular function fields and rings of the weakly holomorphic modular functions. The special values of modular functions at imaginary quadratic arguments belong to abelian extensions of corresponding imaginary quadratic fields. Hasse proved by using the theory of complex multiplication that abelian extensions of imaginary quadratic fields can be generated by special values of the elliptic modular function and the Weber function. Moreover, Ramachandra explained this result in terms of modular units. In this talk, I will introduce these classical results and recent applications of modular units.
4. 이석형
제목: Counting Arithmetic Objects: A Brief Overview
초록: 수체(number field), 대수적 곡선, 곡선의 유리점 등의 정수론적 대상을 세는 문제는 Manjul Bhargava의 Higher Composition Laws 논문들을 시작으로 많은 발전을 거두었습니다. 이런 문제들에 대해서 그는 대상을 group representation의 integral orbit들과 대응시키고, 그 개수를 geometry of numbers를 이용해 세는 방법을 제공했습니다. 이 발표에서는 "Arithmetic invariant theory"라 불리기도 하는 이 방법을 간단히 소개하고, 수체의 경우를 다룬 Higher Composition Laws 논문의 결과들을 더욱 자세히 살펴보고자 합니다.
5. 이완
제목: Cohomological dimension of Galois groups with restricted ramification
초록: We discuss about cohomological p-dimension of maximal S-ramified Galois group of number fields for set of primes S and related phenomenon.
6. 이정욱
제목: 모델론적으로 수론문제 접근하기(Number theory : Model theoretic approach)
초록: 모델론은 수리논리학의 한 분야로, 여러 수학적 대상들을 1차 논리를 사용하여 표현하고 이해하려고 한다. 예를 들어서, 유리수체를 이해하기 위해, 유리수체 위에서 1차 논리로 표현할 수 있는 대상들을 '직접적'으로 살펴 볼 수가 있고, 또는 같은 성질이 논스텐다드 유리수체, 예를 들어 유리수체의 울트라 파워(ultrapower of rational number field), 에 대해서는 어떻게 되는지 살펴봄으로서 원하는 성질을 이해할 수 가 있다. 이 발표에서는 1차 논리에 대해서 간단히 살펴보고, 구체적인 수론의 문제의 예로. 힐버트 체들(Hilbertian fields)을 어떻게 모델론적으로 이해할 수 있는지와 최근 발표자가 연구하고 있는 유리수 체 위에서 정의된 타원곡선들의 랭크가 균등하게 상계(uniformly bounded)되기 위한 조건에 대한 내용을 보도록 할 것이다.
7. 조시훈
제목: Large gaps between prime numbers
초록: The research about consecutive prime gaps is one of the most important subjects in Number theory. Recently, there are many progresses about the maximum gap between consecutive primes less than X. In this talk, we introduce recent results and ideas about large gaps between consecutive prime numbers.
8. 최서현
1. 권영욱
제목: Congruence for traces of singular moduli on genus zero groups
초록: In 2005, Ono and Ahlgren found some mod p congruences for the traces of CM values of certain modular functions of level 1 by using Hecke operators. After their work, many authors obtained generalizations modulo prime powers. But most of the results were restricted to level 1 case. In this talk, we discuss a generalization of these results to some modular functions on certain genus zero groups. We also discuss similar properties for the Hurwitz-Kronecker class numbers.
2. 김태경
제목 : A conjecture of Gross and Zagier for elliptic curves having rational torsion group of a 2-power order
초록 : One of the most prominent results toward the Birch and Swinnerton-Dyer conjecture is the paper Heegner points and derivatives of L-series by B. Gross and D. Zagier. In the paper, they showed that if K is an imaginary quadratic field with ‘Heegner hypothesis’, and E is an elliptic curve such that ord s=1L(E=K; s) = rankE(K) = 1, then L0(E=K; s) can be written asthe product of various arithmetic invariants. This formula looks very alike to the conjectural formula of Birch and Swinnerton-Dyer. Equating these two formulae, Gross and Zagier was able to formulate the conjecture that the order of the rational torsion subgroup of E(Q) divides the product of
Tamagawa number, Manin constant, and the square root of the order of Tate–Shafarevich group of E over K. In this talk, after giving some preliminary stuffs needed to formulate the conjecture, I will give the proof of the conjecture, when the elliptic curve does not have points of odd order.
3. 윤동성
제목: Applications of modular units
초록: A modular unit is defined to be a modular function which has neither zeros nor poles on the complex upper half-plane. It has many advantages to construct generators of modular function fields and rings of the weakly holomorphic modular functions. The special values of modular functions at imaginary quadratic arguments belong to abelian extensions of corresponding imaginary quadratic fields. Hasse proved by using the theory of complex multiplication that abelian extensions of imaginary quadratic fields can be generated by special values of the elliptic modular function and the Weber function. Moreover, Ramachandra explained this result in terms of modular units. In this talk, I will introduce these classical results and recent applications of modular units.
4. 이석형
제목: Counting Arithmetic Objects: A Brief Overview
초록: 수체(number field), 대수적 곡선, 곡선의 유리점 등의 정수론적 대상을 세는 문제는 Manjul Bhargava의 Higher Composition Laws 논문들을 시작으로 많은 발전을 거두었습니다. 이런 문제들에 대해서 그는 대상을 group representation의 integral orbit들과 대응시키고, 그 개수를 geometry of numbers를 이용해 세는 방법을 제공했습니다. 이 발표에서는 "Arithmetic invariant theory"라 불리기도 하는 이 방법을 간단히 소개하고, 수체의 경우를 다룬 Higher Composition Laws 논문의 결과들을 더욱 자세히 살펴보고자 합니다.
5. 이완
제목: Cohomological dimension of Galois groups with restricted ramification
초록: We discuss about cohomological p-dimension of maximal S-ramified Galois group of number fields for set of primes S and related phenomenon.
6. 이정욱
제목: 모델론적으로 수론문제 접근하기(Number theory : Model theoretic approach)
초록: 모델론은 수리논리학의 한 분야로, 여러 수학적 대상들을 1차 논리를 사용하여 표현하고 이해하려고 한다. 예를 들어서, 유리수체를 이해하기 위해, 유리수체 위에서 1차 논리로 표현할 수 있는 대상들을 '직접적'으로 살펴 볼 수가 있고, 또는 같은 성질이 논스텐다드 유리수체, 예를 들어 유리수체의 울트라 파워(ultrapower of rational number field), 에 대해서는 어떻게 되는지 살펴봄으로서 원하는 성질을 이해할 수 가 있다. 이 발표에서는 1차 논리에 대해서 간단히 살펴보고, 구체적인 수론의 문제의 예로. 힐버트 체들(Hilbertian fields)을 어떻게 모델론적으로 이해할 수 있는지와 최근 발표자가 연구하고 있는 유리수 체 위에서 정의된 타원곡선들의 랭크가 균등하게 상계(uniformly bounded)되기 위한 조건에 대한 내용을 보도록 할 것이다.
7. 조시훈
제목: Large gaps between prime numbers
초록: The research about consecutive prime gaps is one of the most important subjects in Number theory. Recently, there are many progresses about the maximum gap between consecutive primes less than X. In this talk, we introduce recent results and ideas about large gaps between consecutive prime numbers.
8. 최서현
제목: finiteness of certain 2-dimensional Galois representations
초록:This is a joint work with professor Dohoon Choi. In 1993, Fontaine and Mazur suggested several conjectures on properties of Geometric Galois representations, and one of the conjecture is about the finiteness of Geometric Galois representations when the Hodge-Tate weight and level of the representation is fixed. I will introduce some related works of other mathematicians and state our result for dimension 2 crystalline Galois representations for absolute Galois group of totally real fields.
9. 하준수
제목: Bounded Gaps between Primes : Historic overview of Yitang Zhang's result
초록: The gap of two consecutive primes is one of the most classic topics in number theory. The twin prime conjecture, which states there are infinitely many primes p such that p+2 is also a prime, is a famous example of the easy-to-state but hard-to-solve problem. A recent breakthrough of Yitang Zhang brought current mathematicians into the era of very small gaps of primes – they are less than 70,000,000 infinitely often. A year later, Maynard proved the bounded gaps of prime in a clever new way using sieve-theoretic methods. In this talk, we breifly survey the history of previous efforts and discuss a few key tools and ideas behind their proof.
10. 한겨울
제목: Elliptic curves with all quartic twists of the same root number
초록: Let $E/K: y^2 = x^3 + ax$ be an elliptic curve with $j$-invariant $1728$ defined over a number field $K$. The root number of elliptic curves is the one of the most important notion in the study of the elliptic curves. In this talk, I will give a formula of the root number of $E/K$ and using this formula, I will give a simple condition on $K$ which determines whether all quartic twists of $E/K$ have the same root number or not. This completes a series of the works on the same root number of twists; quadratic twists by T. Dokchitser and V. Dokchitser and cubic(or sextic) twists by D. Byeon and N. Kim. This is a joint work with Prof. D. Byeon.
11. 한재호
제목: On the Fourier-Jacobi model for some endoscopic Arthur packet of $U(3) times U(3)$ : the non-tempered case
초록: For a tempered $L$-parameter of $U(n)times U(n)$, it is known that there is a unique representation in their associated relevant Vogan $L$-packet which produces the unique Fourier-Jacobi model. We showed that this is ture for some non-tempered Arthur parameter of $U(3)times U(3)$. Furthermore, we specified such representation under the local Langlands correspondence for unitary group.
9. 하준수
제목: Bounded Gaps between Primes : Historic overview of Yitang Zhang's result
초록: The gap of two consecutive primes is one of the most classic topics in number theory. The twin prime conjecture, which states there are infinitely many primes p such that p+2 is also a prime, is a famous example of the easy-to-state but hard-to-solve problem. A recent breakthrough of Yitang Zhang brought current mathematicians into the era of very small gaps of primes – they are less than 70,000,000 infinitely often. A year later, Maynard proved the bounded gaps of prime in a clever new way using sieve-theoretic methods. In this talk, we breifly survey the history of previous efforts and discuss a few key tools and ideas behind their proof.
10. 한겨울
제목: Elliptic curves with all quartic twists of the same root number
초록: Let $E/K: y^2 = x^3 + ax$ be an elliptic curve with $j$-invariant $1728$ defined over a number field $K$. The root number of elliptic curves is the one of the most important notion in the study of the elliptic curves. In this talk, I will give a formula of the root number of $E/K$ and using this formula, I will give a simple condition on $K$ which determines whether all quartic twists of $E/K$ have the same root number or not. This completes a series of the works on the same root number of twists; quadratic twists by T. Dokchitser and V. Dokchitser and cubic(or sextic) twists by D. Byeon and N. Kim. This is a joint work with Prof. D. Byeon.
11. 한재호
제목: On the Fourier-Jacobi model for some endoscopic Arthur packet of $U(3) times U(3)$ : the non-tempered case
초록: For a tempered $L$-parameter of $U(n)times U(n)$, it is known that there is a unique representation in their associated relevant Vogan $L$-packet which produces the unique Fourier-Jacobi model. We showed that this is ture for some non-tempered Arthur parameter of $U(3)times U(3)$. Furthermore, we specified such representation under the local Langlands correspondence for unitary group.