============ 2009. 01. 08 ============ Liu, J. (Sandong) LinnikĄ¯s problem and analytic multiplicity one for automorphic representations Abstract: Recently, Linnik's famous theorem on the least prime on an arithmeticprogression has been generalized to the context of automorphic representations. Iwill report some new results in this direction, for example, sign changes of Dirichlet coefficients of automorphic L-functions, and the analytic multiplicity one for automorphic representations. Ghate, E. (TIIFR) Counting weight one forms Abstract: We count the number of octahedral forms of prime level. We show that on average the number of such forms is bounded by a constant. This is joint work with Manjul Bhargava. Ikeda, T. (Kyoto) On lifting of modular forms Abstract: Given a holomorphic Hecke eigenform of one variable, I will construct a lifting to a Siegel cusp form of degree 2n, which is also a Hecke eigenform.The Fourier coefficients and L-function of the Siegel cusp form can be given explicitly.It also possible to construct a lifting of automorphic representation of PGL(2) to a symplectic or metaplectic group over an arbitrary totally real field. Choi, D. (Korea Aerospace University) Weakly holomorphic modular forms of half-integral weight with non-vanishing constant terms modulo Abstract: In this talk, we study modular forms of half-integral weight whose coefficients are supported on finitely many square classes modulo primes l>3. We obtain that, as conjecture of Balog, Darmon and Ono, if the constant term of a modular form of half-integral weight is not zero modulo l and its coefficients are supported on finitely many square classes modulo l, then the weight is restricted. We also discuss some applications of this result. Sengupta, J. (TIFR) Sign changes of Hecke eigenvalues Abstract: It is well known that for a primitive form f there are infinitely many n such that the n-th Hecke eigenvalue is negative. In this talk we will give a bound for the least such n in terms of the parameters describing the form i.e. weight and level. We will also consider the same problem for Siegel cusp forms of genus two which are Hecke eigenforms. Choie, Y. (POSTECH) The first sign change of Fourier coefficients of cusp forms Abstract: Let f be a nonzero cusp form of even integral weight on the Hecke congruence subgroup with real Fourier coefficients a(n). Using analytic properties of the Hecke L-functions and Rankin-Selberg L-function attached to f it is not hard to see that a(n) changes sign infinitely often. Natural question arises when the first sign chagen occurs and one would hope for an upper bound for it depending only on k and N. This problem was taken up by several authors in the case when f is a normalized Hecke eigenform of square free level N that is a newform. In this talk we give the result for arbitrary cusp forms f of square free level N. This is a joint work with W. Kohnen. ============= 2009. 01. 09 ============= Coates, J. (Cambridge) Tate-Shafarevich groups of elliptic curves with complex multiplication Abstract: The lecture will discuss joint work with Z. Liang and R. Sujatha on the Tate-Shafarevich groups of elliptic curves E over Q with complex multiplication. It will show that, for all sufficently large good ordinary primes p for E, the number of copies of Qp/Zp occurring in the the Tate-Shafarevich group of E is at most 2p-g, where g is the rank of E(Q). No theoretical result of this kind seems to have been established before. It will also discuss the numerical determination of the p-primary subgroup of sha for certain E and all good ordinary primes p < 12, 500. Sujatha, R. (TIFR) On the fine Selmer group Abstract: We shall discuss some recent results, obtained jointly with J. Coates, related to the Fine Selmer group of elliptic curves Im, B.-H. (Chung-Ang University) On the rank of abelian varieties over large fields and the products of quadratic twists of elliptic curves over number fields Abstract: We will discuss large (infinite) extension field over which the Mordell-Weil groups of an abelian variety has infinite rank. And we will give examples of elliptic curves over number fields whose quadratic twists by d, d', and dd' are all of positive rank over the given number fields, which is a related result with the combination of Goldfeld's conjecture and Parity conjecture. Kakda, Mahesh (Princeton University or Cambridge?) Congruences in non-commutative Iwasawa theory Abstract: I will talk about the conjectural congruences between p-adic L functions implied by the conjectures in non-commutative Iwasawa theory. Then I will talk about the cases in which these congruences are known to be true. Fukaya, T. (Keio University) Local units in non-commutative Galois extensions Abstract: Iwasawa, Coates-Wiles, and Coleman obtained the `almost isomorphism' of the Iwasawa-modules between the tower of local units and Iwasawa algebra for the cyclotomic extensions of Qp. I discuss how to generalize this to a tower of non-commutative Galois extensions and give a partial result. Byeon, D. (Seoul National University) Rank-one quadratic twists of an infinite family of elliptic curves Abstract: A conjecture of Goldfeld implies that a positive proportion of quadratic twists of an elliptic curve E/Q has (analytic) rank 1. Here we confirm this assertion for infinitely many elliptic curves E/Q using the Heegner divisors, the 3-part of the class groups of quadratic fields, and a variant of the binary Goldbach problem for polynomials. This is joint work with Daeyeol Jeon and Chang Heon Kim. Han, L. (Inha University) semistable abelian varieties over Q having good reduction almost everywhere Abstract: There is no abelian varieties over Z. This is one of the interesting results proved by Fontaine in 1985 and relatively recently Schoof and other people made some progress in that direction by finding possible semistable abelian varieties over Q having good reduction almost everywhere. In this talk, starting from brief introduction to basic tools used in those arguments, I will discuss their meanings and applications, for instance, to the practical examples of Shafarevich conjecture, or simple examples of modularity of semistable abelian varieties. ============= 2009. 01. 10 ============= Kato, K. (Kyoto) On moduli of degenerations Abstract: I discuss about moduli of degenerating Hodge structures and the non-archimedean analogue. non-archimedean analogue Tian, Y. (Moningside Institute) Euler systems on Shiumura curves Abstract: In this talk, we will discuss above evidence of BSD conjecture for modular abelian varieties over totally real fields using Euler systems on Shimura curves. Kurihara, M. (Keio) Iwasawa theory and Stickelberger ideals Abstract: Beginning with the classical Stickelberger's theorem, I will talk on the Stickelberger ideals, and on their real meaning from the algebraic side. I will also discuss on a refinement of Iwasawa theory. Xu, Fei (Academy of Science,China) Integral points on homogenous spaces of algebraic tori Abstract: Recently Harari has shown that the Brauer-Manin obstruction is the only obstruction for existence of the integral points on homogenous spaces of algebraic tori. However the Brauer-Manin obstruction is infinite for algebraic tori. In this talk, we will explain that one can always find finite subgroups of Brauer-Manin obstruction depending the integral model of the homogenous spaces to test the existence of the integral points. Several examples will be provided. Kawada, Koichi (Iwate) On the Waring-Goldbach problem and related topics Abstract: The Waring-Goldbach problem is a research area that features representations of natural numbers as sums of powers of primes. Right after Vinogradov established his celebrated three prime theorem, Hua launched research in this area, and showed, for example, that every sufficiently large odd integer is the sum of nine cubes of primes. In the first half of this talk, brief history and several currently known results in this area will be mentioned together with rough outlines of the methods used. In the second half of the talk, some recent related results will be introduced mainly in the case of cubes. Amongst others, I will talk about a result obtained jointly with Joerg Bruedern which asserts that every sufficiently large integer can be written as the sum of eight cubes of $P_2 $-numbers (an integer is called $P_2$, when it is either a prime or the product of two primes). Liu, C. (Beijing Normal University) The generic Newton polygon for L-function of exponential sums associated to Laurent polynomials Abstract: Taguchi, Y. (Kyushu) Extensions of truncated discrete valuation rings Abstract: A truncated discrete valuation ring is a commutative ring which is isomorphic to a quotient of finite length of a discrete valuation ring. We give an equivalence between the category of at most $a$-ramified finite separable extensions of a complete discrete valuation field $K$ and the category of at most $a$-ramified finite extensions of the ``length-$a$ truncation" of the integer ring of $K$. This extends a theorem of Deligne, in which he proved this fact assuming the residue field is perfect. Our theory depends heavily on Abbes-Saito's ramification theory. ============= 2009. 01. 11 ============= Greenberg, R. (U. Washington) Galois representations with open image Abstract: We will describe a novel way of constructing continuous representations from the absolute Galois group $G_{\QQ}$ to $GL_n(\ZZ_p)$ which have an open image. Here $p$ will be an odd prime and $n$ will be a positive integer. The construction works for many choices of $n$ and $p$. For example, if $p$ is a regular prime and $p \ge 2n+1$, then such continuous representations exist. The construction depends on the notion of $p$-rationality and on certain properties of a Sylow pro-$p$ subgroup of $GL_n(\ZZ_p)$. Ngo, B-C. (IAS, Princeton) Hitchin fibration and fundamental lemma Abstract: Zheng, Weizhe (Paris) Integrality, Rationality, and Independence of $l$ in $l$-adic Cohomology over Local Fields Abstract: I will discuss two problems on traces in $l$-adic cohomology over local fields with finite residue field. In the first part, I will describe the behavior of integral complexes of L-adic sheaves under Grothendieck's six operations and the nearby cycle functor. In the second part, I will talk about rationality and independence of~$l$. More precisely, I will introduce a notion of compatibility for systems of $l$-adic complexes and explain the proof of its stability by the above operations, in a slightly more general context (equivariant under finite groups The main tool in this talk is a theorem of de Jong on alterations. Kim, M. (U.C. London) Selmer varieties Abstract: We report on the program to study the Diophantine geometry of hyperbolic curves using non-abelian cohomology varieties associated to non-abelian fundamental groups.