1. 김찬호, On the Iwasawa main conjecture for modular forms

• Give me an elliptic curve E over the rationals and a (sufficiently large) good reduction prime p. Then let me give you back the equality of the Iwasawa main conjecture for E over the cyclotomic Z_p-extension of Q (if I can access SAGE). This is joint work with Myoungil Kim and Hae-Sang Sun.

2. Joachim König , Grunwald problems and specialization of function field extensions

• Grunwald problem (for a finite group G and a number field k) is the following question: Given a finite set S of primes of k, and for each p ∈ S a Galois extension L_p/k_p with Galois group embedding into G, does there exist a G-extension L/k whose completion at p equals L_p/k_p for all p ∈ S ? Famously, the Grunwald-Wang theorem gives a positive answer (with exceptions at primes extending 2) in the case that G is abelian. We use specializations of function field extensions E/k(T) to find solutions to Grunwald problems. In the case where all L_p/k_p are unramified, this approach has been used successfully by Dèbes and Ghazi. I will present partial positive as well as negative results on the solvability of ramified Grunwald problems via specialization of a function field extension. As a sample application, I will give new results on the classical Q-admissibility conjecture, asking about the existence of G-crossed product division algebras. (Based on joint work with Francois Legrand and Danny Neftin)

3. 김완수, Equivariant BSD conjecture over global function fields

• Under a certain finiteness assumption of Tate-Shafarevich groups, Kato and Trihan showed the BSD conjecture for abelian varieties over global function fields of positive characteristic. We explain how to obtain the equivariant generalisation of the work of Kato and Trihan for semi-stable abelian varieties with respect to (not necessarily abelian) Galois extensions L/K of global function fields that are at most tamely ramified everywhere. This is a joint work in progress with David Burns and Mahesh Kakde.

4. 김병두, Construction of Anti-Cyclotomic Euler Systems of Abelian Varieties Associated to X₁(N)

• (Join work with Daeyeol Jeon and Chang Heon Kim) Let K be an imaginary quadratic field, N be a positive integer, f(z) be a newform of Γ₁ (N), and A_f be the abelian variety associated to f. I will demonstrate certain points that D. Jeon, C. Kim, and I have constructed on A_f defined over an extended ring class field of K of level N. Our construction generalizes Birch’s construction of the Heegner points to the abelian varieties associated to modular forms of level Γ₁ (N) and nontrivial character. Then, I will show that these points satisfy the distribution and congruence relations of an Euler system, thus I will argue that they form an Euler system, and can be applied to find the ranks of A_f over K.

5. 이종규, Multiplicative functions, additive on polygonal numbers

• Spiro showed that multiplicative function which is additive on prime numbers should be the identity function. After Spiro's work, there are many variations. One of such result is work of Chung and Phong; multiplicative function which is additive on triangular numbers should be the identity. But there is non-identity functions which is multiplicative and is additive on square numbers. In this talk, we check the difference of triangular and square numbers and expand it the other polygonal numbers, pentagonal and hexagonal numbers.

6. 이정원, On the recent conjecture of Mazur-Rubin

• Mazur and Rubin have formulated some conjectures on the distribution of modular symbols. We discuss recent results and our strategy mostly based on the transfer operator approach (joint with Hae-Sang Sun).

7. 최도훈, Values of harmonic Maass forms

• In this talk, I will present some properties for values of harmonic Maass forms.

8. 이정연, Number fields generated by cyclotomic Hecke L-values of totally real fields.

• (This is a joint work with Byungheup Jun and Hae-Sang Sun.) In this talk, we study a relation between a filed generated by Hecke L-values of totally real field twisted by cyclotomic characters and a field generated by cyclotomic character values.

9, 김태경, Elliptic curves having everywhere good reduction

• Given an elliptic curve E defined over number field k, lots of important global arithmetic information are encoded in reductions of E modulo various places of k. Somewhat conversely, we may ask about elliptic curves over $k$ having prescribed reduction types at certain primes of k. In particular, in this talk, we focus on elliptic curves having everywhere good reduction (EGR). For k quadratic fields, I present some methods to count the number of isomorphism classes of elliptic curves over k having EGR. Amongst them, if we add some constraints on the torsion subgroups of elliptic curves, we have some non-existence results on curves having EGR.

10. 이현석, Conjectures for N-Exponentials.

• In this talk, I will present some conjectures on the Four Exponentials Conjecture and Six Exponentials with Related Statements.