Number Theory Seminars 2013

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January 10, 2013 Shin, Dong Hwa (Hankuk University of Foreign Studies)

Title: Applications of modular units

Abstract: Let $X(N)$ be the modular curve of level $N$ ($\geq1$), which is a compact Riemann surface. We denote its function field by $\mathbb{C}(X(N))$. As is well-known, $X(1)$ is of genus zero and $\mathbb{C}(X(1))=\mathbb{C}(j)$, where \begin{equation*} j=j(\tau)=q^{-1}+744+196884q+21493760q^2 +864299970q^3+\cdots\quad(q=e^{2\pi i\tau}) \end{equation*} is the elliptic modular function. Furthermore, $\mathbb{C}(X(N))$ is a Galois extension of $\mathbb{C}(X(1))$ whose Galois group is isomorphic to $\mathrm{SL}_2(\mathbb{Z}/N\mathbb{Z})/\{\pm I_2\}$. Let $\mathcal{O}_N$ be the integral closure of $\mathbb{C}[j]$ in $\mathbb{C}(X(N))$. We call the invertible elements in $\mathcal{O}_N$ \textit{modular units} of level $N$ (over $\mathbb{C}$), which are precisely those functions in $\mathbb{C}(X(N))$ having no zeros and poles on $\mathbb{H}$. In this talk we shall show that a weakly holomorphic modular function can be written as a sum of modular units of higher level. \par On the other hand, let $L/F$ be an extension of number fields and let $\mathcal{O}_L$ and $\mathcal{O}_F$ be the rings of integers of $L$ and $F$, respectively. We say that an element $\alpha$ of $L$ forms a \textit{relative power integral basis} for $L/K$ if $\mathcal{O}_L=\mathcal{O}_F[\alpha]$. For example, if $N$ is a positive integer, then $\zeta_N=e^{2\pi i/N}$ forms a (relative) power integral basis for the extension $\mathbb{Q}(\zeta_N)/\mathbb{Q}$. In general, not much has been known about relative power integral bases except for extensions of degree less than or equal to $9$. Now, let $K$ be an imaginary quadratic field other than $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$. Let $m$ and $n$ be positive integers such that $m$ has at least two prime factors and each prime factor of $mn$ splits in $K/\mathbb{Q}$. As an application of moudlar units we shall construct a relative power integral basis for the ray class field modulo $(mn)$ over the compositum of the ray class field modulo $(m)$ and the ring class field of the order of conductor $mn$ of $K$, in terms of a Weierstrass unit.

January 25, 2013 Noya Umezaki (University of Tokyo)

Title: On the degree of the field extension of which the action of local monodromy is unipotent

Abstract: On the $l$-adic representation, Grothendieck showed that the action of the inertia group of non-archimedian local field which we call local monodromy, become unipotent after some field extension. In this talk, we consider the degree of this field extension in the case of the $l$-adic etale cohomology of a smooth projective variety. I will show that it depends only on some invariants of the variety and does not depend on $l$. I will also present a similar result for the global field case.

January 25, 2013 Luc Illusie (Université Paris-Sud)

Title: Quotient stacks and mod l equivariant étale cohomology algebras :
Quillen's theory revisited (joint work with W. Zheng)

Abstract: In the early 70's Quillen constructed a theory of mod $l$ equivariant cohomology for actions of compact Lie groups on topological spaces. I will explain how questions of Serre on actions of finite groups and traces in $l$-adic cohomology (algebraic variants of Smith theory) led us to develop, in the setting of mod $l$ étale cohomology, an analogue of Quillen's theory for actions of algebraic groups on schemes separated and of finite type over an algebraically closed field.

January 25, 2013 Takeshi Saito (University of Tokyo)

Title: Wild Ramification and the Cotangent Bundle

Abstract: We define the characteristic cycle of a locally constant étale sheaf on a smooth variety in positive characteristic ramified along boundary as a cycle in the cotangent bundle of the variety, at least on a neighborhood of the generic point of the divisor on the boundary. We discuss a compatibility with pull-back and local acyclicity in non-characteristic situations. We also give a relation with the characteristic cohomology class and the Euler-Poincaré characteristic.

January 25, 2013 Kentaro Nakamura (Hokkaido University)

Title: Iwasawa theory of de Rham $(\phi,\Gamma)$-modules over the Robba ring

Abstract: In my talk, I will develop the theory of Bloch-Kato's exponential map and Perrin-Riou's exponential map in the framework of $(\phi, \Gamma)$-modules over the Robba ring. In particular, I will generalize Perrin-Riou's exponential map for all the de Rham $(\phi, \Gamma)$-modules and also generalize and prove her $\delta(V)$-conjecture.

February 14, 2013 Hyun, Jong Yoon (Ewha Womans University)

Title: Construction of strongly regular graphs using Gauss sums

Abstract: We construct strongly regular graphs and association schemes using the weakly regular $p$-ary bent functions (from $\mathbb{Z}^n_p$ to $\mathbb{Z}_p$), where $p$ is a prime number.

February 27, 2013 Kim, Min-Soo (Kyungnam University)

Title: On Dedekind-Rademacher sums generating the Dedekind Sums involving Euler functions and their reciprocity theorem

Abstract: We introduce Dedekind-Rademacher sums involving Euler functions $$ T_m(h,k\mid x,y)=\sum_{\mu=0}^{|k|-1}(-1)^\mu \overline{E}_{1}(\frac{\mu+y}{k})\overline{E}_m(\frac{h(\mu+y)}{k}+x). $$ The first aim of this talk is to state and prove that our Dedekind sums satisfies reciprocity theorem using analytic methods of Euler polynomials which is an analogue of the reciprocity theorem of the classical Dedekind sums dates back to Dedekind. From the derived reciprocity theorem, we obtain an explicit expression for these sums. Let us put $e(z)=e^{2\pi i z}.$ We give a systematic treatment of certain exponential sums $$T_{a,b}(x,y)=\sum_{\lambda\text{ (mod } c)}(-1)^{\lambda+\left[\frac{\lambda a}c\right] +\left[\frac{\lambda b}c\right]}e\left(\left\{\frac{\lambda a}{c}\right\}x+\left\{\frac{\lambda b}{c}\right\}y\right)$$ and $$\mathfrak{F}_{c}^{a,b}(x,y)=[e(x)+1]^{-1}[e(y)+1]^{-1}T_{a,b}(x,y)$$ generating $$ T_{m,n}\binom{a~~b}{c}=\sum_{\lambda=0}^{c-1}(-1)^\lambda\overline{E}_{m}(\frac{\lambda a}{c})\overline{E}_n(\frac{\lambda b}{c}). $$ The second aim of this talk is to state and prove a three-term relation of a new type $T_{m,n}\binom{a~~b}{c}$ which implies (in extended form) all classical reciprocity theorems. Finally, we will also introduce a similar result for the $p$-adic case.

March 5, 2013 Shin HATTORI(Kyushu University)

Title: On lower ramification subgroups and canonical subgroups

Abstract: In this talk, I will explain a description of lower ramification subgroups of finite flat group schemes killed by some p-power over the integer ring of p-adic fields via Breuil-Kisin modules, and its application to showing an overconvergence of higher canonical subgroups of abelian varieties.

March 14, 2013 Hiro-Fumi YAMADA (Okayama University)

Title: A peripheral combinatorics of partitions

Abstract: I will define 3 types of ¡±weight¡± for each partition, and show that the products of these weights over partitions coincide. This product turns out to be the Shapovalov determinant for the basic representation of the quantized universal enveloping algebra of affine type A. This is a joint work with M. Ando and T. Suzuki.

March 27, 2013 Yoshiyasu OZEKI (RIMS)

Title: Full faithfulness theorem for torsion crystalline representations

Abstract: M. Kisin proved that a certain ``restriction functor'' on crystalline p-adic representations is fully faithful. In this talk, I will explain the torsin analogue of this result with some restrictions for absolute ramification index e and (prescribed) Hodge-Tate weights. I will also explain that this restriction is the best possible for the full faithfulness at least the case where e=1. As an application, I will show the non-existence of crystalline lifts, with (prescribed) small Hodge-Tate weights, of certain torsion representations.

March 27, 2013 Atsushi SHIHO (University of Tokyo)

Title: On restriction of overconvergent isocrystals

Abstract: An overconvergent isocrystal is a p-adic object on a variety of characteristic p>0 which is the p-adic analogue of a smooth l-adic sheaf or a local system. In this talk, after explaining basic notions concerning overconvergent isocrystals, we explain some results on the relation between the differential Artin conductor of a given overconvergent isocrystal and that of the restriction of it to curves.

April 11, 2013 Choi, Dohoon (Korea Aerospace University)

Title: Density one property for a Hecke field in term of a Galois representation

Abstract: Let f be a newform of weight k and a(p,f) be the eigenvalue of f for the pth Hecke operator. We call as the Hecke field of f the field generated by all a(p,f). It was proved by Ribet and Koo, Stein and Wiese that this is the simple extension field of a(p,f) for primes with density one. These results can be restated in term of a Galois representation attached to a newform f. In this talk, we study a generalization of these results for general Galois representations over a local field. This is a joint work with Y. Taguchi.

April 25, 2013 Kim, Byoung Du (Victoria University)

Title: The congruences of congruent modular forms for non-ordinary primes

Abstract: This is joint work with Suh Hyun Choi. Let p be a prime number. Suppose we have two modular forms whose weights are congruent modulo p^r(p-1), and q-expansions are congruent modulo p^r. (For example, consider modular forms given by topologically close points on an eigencurve.) People who do Iwasawa Theory believe that their p-adic L-functions are also congruent modulo p^r. In fact, if we push this idea further, we can also imagine there is a big p-adic L-function over an eigencurve which is integral and smooth. This is known in the ordinary prime case (i.e. the case where the slope of modular form is a p-adic unit), and in this case, the big p-adic L-function over the eigencurve is called the Kitagawa-Mazur p-adic L-function. In the non-ordinary case, so far we know relatively little. In this presentation, we will prove that the (non-integral) p-adic L-functions that I constructed are congruent for the above-said congruent modular forms assuming that Hecke algebras are Gorenstein. (The same technique can be applied to different p-adic L-functions.) We believe that this is one step towards a big integral smooth p-adic L-function over an eigencurve for a non-ordinary prime.

May 9, 2013 Kim, Jaeseon(POSTECH)

Title: Genus field of a cyclic cubic field

Abstract: Let $K$ be a cyclic cubic field. The genus field $K^{*}$ of $K$ is, by definition of Leopoldt, the maximal absolute abelian number field containing $K$, which is unramified at all finite prime ideals of $K$. $K^{*}$ is explicitly determined by Ishida in terms of its conductor. Our goal is to determine the maximal 3-elementary abelian extension $E$ of $K$ which is the fixed field of the cube of the ideal class group of $K$ via class field theory. In this talk, we will study some properties of $E$ related to $K^{*}$.

May 23, 2013 Yang, Minsuk (Yonsei University)

Title: Asymptotic behaviour of a sampling of the Riemann zeta function on the critical line

Abstract: We investigate the distributional behaviour of the random sampling ¥æ(1/2 + iXt(¥ø)) of the Riemann zeta function on the critial line. Our main result states that if Xt is an increasing random sampling with gamma distribution, then for all sufficiently large t \[P(|\zeta(1/2+iX_t)|>\log t) \ll 1/\log t.\]

June 13, 2013 Yoo, Hwajong(UC Berkeley)

Title: Modularity of residually reducible Galois representations

Abstract: This lecture concerns reducible two dimensional mod ` Galois representations that arise from newforms. If a newform of weight two, trivial character, and a square-free level yields a reducible mod l representation, the semisimpli cation of this representation will be a direct sum of 1 and chi where chi is the mod l cyclotomic character. The natural question is to describe the set of square-free levels N for which there exists a newform of weight two for Gamma_0(N) that gives rise to a direct sum of 1 and chi. After providing some background, I will explain how level raising method sheds light on the question.

June 13, 2013 Kim, Yeansu(Purdue University)

Title: L-functions from Langlands-Shahidi Method and the generic Arthur L-packet conjecture

Abstract: L-functions are very interesting tools that number theorists have been using since 18th century. Those also appear in the local Langlands conjecture. Briefly, the local Langlands conjecture asserts that there exists a `natural' bijection between two di erent sets of objects: Arithmetic (Galois or Weil-Deligne) side and analytic (representation theoretic) side. In each side, we can define the L-functions of those objects. The L-functions from analytic side are defined by Shahidi (Langlands-Shahidi method) and the L-functions from arithmetic side are Artin L-functions. The natural question is whether two L-functions are equal through the local Langlands correspondence. If it is, we can use the properties of the L-functions from arithmetic side to study L-packet, the object in the analytic side, which is the set of irreducible admissible representations of quasi split group G over p-adic field. The equality of L-functions has an interesting application in proving the generic Arthur L-packet conjecture. The generic Arthur L-packet conjecture states that if the L-packet attached to Arthur parameter has a generic member, then it is tempered. (Remark that this conjecture relates to the generalized Ramanujan conjecture). In this talk, I will explain those in the case of split GSpin groups. Furthermore, I will explain the classification of strongly positive discrete series representations of GSpin groups over p-adic field which is one of the main tools in the proof of the equality of L-functions.

June 13, 2013 Lee, Chong Gyu(Soongsil University)

Title: Dynamics of small topological degree

Abstract: Algebraic dynamics is a field of studying preperiodic points of some algebraic maps on a projective variety. After Northcott and Weil provide the height function to measure arithmetic complexity of points, lots of mathematicians apply this useful tool to study algebraic dynamical systems. In this talk, we review recent progress on arithmetic dynamics, the arithmetic result for dynamical systems of various cases. And, we introduce new result on the dynamical system of polynomial maps of small topological degree, which is one of interesting topics in traditional complex. dynamics.

July 24, 2013 Choiy, Kwangho(Oklahoma State University)

Title: $p$-adic inner forms and invariants of Langlands parameters

Abstract: A new Galois action on a linear algebraic group over a $p$-adic field $F$ gives rise to a new structure of $F$-rational points as well as a new category of smooth complex representations. It appears that the old and new categories are related each other in the framework of Langlands functoriality. In this talk, we shall consider a quasi-split group $G$ over $F,$ and its $F$-inner form $G'$ which is induced from a new Galois action twisted by a Galois $1$-cocycle in the inner automorphisms of $G.$ We will discuss the behavior of representation-theoretic objects, such as Plancherel measures and formal degrees, between two categories of smooth complex representations of $G(F)$ and $G'(F)$ in terms of Langlands parameters. This work allows us to understand the structure of representations of $G'(F)$ using that of $G(F),$ and vice versa.

July 25, 2013 Baek, Sanghoon(KAIST)

Title: Codimension 2 cycles on products of projective homogeneous surfaces

Abstract: In this talk, we discuss the torsion in the codimension 2 Chow groups of the products of projective homogeneous surfaces: we provide general bounds for the torsion subgroup. In particular, we show that these bounds are sharp for the product of three Severi-Brauer surfaces and the product of four Pfister quadric surfaces. We also find an upper bound for the torsion in the codimension 2 Chow groups of the product of three quadric surfaces with the same discriminant.

August 8, 2013 Yuuki Takai(Keio University)

Title: A determining condition of Hilbert modular forms

Abstract: The elliptic holomorphic modular forms of weight $k$ and level $\Gamma_1(N)$ are determined by the first $(k/12)[\Gamma_1(1): \Gamma_1(N)]$ Fourier coefficients. The mod $\ell$ analogue of the fact is called Sturm's theorem. In this talk, I will give a generalization of the first fact and Sturm's theorem for Hilbert modular forms. To prove them, I use geometric properties of some compacitifications of Hilbert modular varieties.

August 16, 2013 Lee, Jung-Jo(Seoul National University)

Title: An application of Mumford's gap principle

Abstract: We study a Dirichlet series attached to a polynomial first defined by Rubin and Silverberg in their study of ranks of quadratic twists of a fixed elliptic curve. We apply Mumford¡¯s gap principle to show that the series converges if the associated polynomial has distinct roots and degree at least 5. This is a joint work with Ram Murty.

August 16, 2013 M. Ram Murty(Queen's University)

Title: Li's criterion and transcendence

Abstract: In 1997, Li discovered a very simple criterion for the truth of the Riemann hypothesis. We will discuss its relation to transcendental number theory and report on some recent work with S. Gun and P. Rath.

August 22, 2013 John Binder(MIT)

Title: An introduction to the Local Langlands Correspondence for GL(n)

Abstract: We'll introduce the 'players' in the Local Langlands Correspondence over $GL_n$: we'll introduce Weil-Deligne representations on the Galois side and irreducible admissible representations of GL_n on the Automorphic side. We'll discuss the properties that a good Local Langlands correspondence should satisfy: it should respect contragredients and it should match $L$-functions and $\epsilon$ factors (which we will define on both the Galois and Automorphic Side). We'll talk about some consequences of the Local Langlands Correspondence on the Automorphic side. Time permitting, we'll talk about some extensions (to the global case, or to other reductive groups).

August 22, 2013 Jo, Sihun(KIAS)

Title: On Maass waveforms of half integral weight with eta multiplier system

Abstract: Let $L^2(\Gamma_0(N)\backslash \mathfrak{H})$ be the space of $\Gamma_0(N)$-invariant functions such that $L^2$-norm is finite, and $M(\Gamma_0(M),v_{\eta},\frac{1}{2})$ be the space of Maass waveform of $\frac{1}{2}$ weight on $\Gamma_0(M)$ with $\eta$ multiplier system. We investigate the relation between $L^2(\Gamma_0(N)\backslash \mathfrak{H})$ and $M(\Gamma_0(M),v_{\eta},\frac{1}{2})$ using spectral decomposition and the residue theory of complex analysis.

August 23, 2013 John Binder(MIT)

Title: The Bernstein-Zelevinski Reduction

Abstract: We'll discuss the structure of Weil-Deligne representations: in particular, every indecomposable representation corresponds to an irreducible representation and an integer, and any representation is a direct sum of indecomposables. We'll then discuss the results of Bernstein and Zelevinski in their 1976, 1977, and 1980 papers that describe a similar structure on the set of irreducible automorphic representations. We'll then show how this reduces the correspondence to simply finding a correspondence between irreducible representations on the Galois side, and the "supercuspidal" representations on the Automorphic side. Finally, we'll show that this reduction 'makes sense' in the context of talk 1: that the extension to a full correspondence will respect the match of $L$-functions, $\epsilon$-factors, and contragredients, so long as the sub-correspondence does.

August 23, 2013 John Binder(MIT)

Title: Some major ideas in BZ

Abstract: We'll discuss some major ideas in the proof of the Bernstein-Zelevinski reduction. We'll discuss the 'geometric lemma' that shows the uniqueness of the cuspidal support. We'll discuss the representation ring, the mirabolic subgroup, and the 'derivative' of a representation. Finally, we'll discuss the properties of derivatives that make them so miraculous, and try to give some examples of these properties in action.

August 23, 2013 Kentaro Ihara(University of Osaka)

Title: The harmonic product of multiple zeta values and several maps

Abstract: This is a joint work with M. Hoffman.

September 10, 2013 Park, Jeehoon(POSTECH)

Title: Lie algebra representations attached to hypersurfaces

Abstract: I will explain how to attach a certain Lie algebra representation to a hypersurface X via the Schrodinger representation of the Heisenberg Lie algebra, whose 0-th Lie algebra homology is canonically isomorphic to the middle dimensional primitive cohomology group of X. Then we derive a homotopy Lie algebra from this and present a method to analyze the period integrals of X based on, so called, homotopy probability theory. This is a joint work with Jae-Suk Park.

September 10, 2013 Yoo, Hwajong(UC Berkeley)

Title: Serre conjectures

Abstract: 1. Serre conjectures : Definition of optimal weight, 2. Serre conjectures : Some proofs.

September 12, 2013 Kim, Dohyeong(POSTECH)

Title: Non-commutative p-adic Hecke L-functions

Abstract: It is a report on a recent progress on construction of non-commutative p-adic L-functions. While we have satisfactory results due to Kakde and Ritter-Weiss for totally even Artin L-functions for which Iwasawa mu invariants are zero, it is widely open for the remaining cases. The goal of my project is to construct non-commutative p-adic L-functions for Hecke L-functions. I will quickly review the p-adic Hecke L-function of Katz and then describe the congruence between different Katz measures predicted by the non-commutative Iwasawa theory. As a consequence, we obtain the non-commutative p-adic Hecke L-function for false Tate curve extensions over imaginary quadratic field.

September 26, 2013 Ambrus Pal(Imperial College London)

Title: Strong Weil curves over function fields

Abstract: The analogy between number fields and global function fields is very deep. One interesting example of such an analogy is between elliptic modular curves over Q and Drinfeld modular curves. For example the analogue of the Taniyama-Shimura-Weil conjecture for function fields has been a theorem since the early seventies. Continuing this analogy it is possible to define strong Weil curves. This way we can single out a special elliptic curve in any isogeny class. I will talk about whether it is possible to characterise these elliptic curves without reference to modular parametrisations.

October 10, 2013 Jin, Seokho(KIAS)

Title: Periods of Jacobi forms and Hecke operators

Abstract: A Hecke action on the space of periods of cusp forms, that is compatible with that on the space of cusp forms, was first computed by Manin using continued fraction and an explicit algebraic formula of Hecke operators acting on the space of period functions of modular forms was derived by Choie-Zagier by studying the rational period functions. As an application an elementary proof of the Eichler-Selberg trace formula was derived by Zagier. A similar modification has been applied to the space of period functions of Maass cusp forms with spectral parameter s. In this talk we study the space of period functions of Jacobi forms by means of Jacobi integral and give an explicit description of the action of Hecke operators on this space. A Jacobi Eisenstein series $E_{2,1}(\tau,z)$ of weight 2 and index 1 is discussed as an example.

October 11, 2013 Yoon, Dong Sung (NIMS)

Title: Construction of class fields over imaginary biquadratic fields

Abstract: In 1900 Hilbert asked at the Paris ICM, as his 12-th problem, that what kind of analytic functions and algebraic numbers are necessary to construct all abelian extensions of given number fields. For any number field F and a modulus $\mathfrak{m},$ it is well known that there is a unique maximal abelian extension of $F$ unramified outside $\mathfrak{m}$ with certain ramification condition, which we call the $\textit{ray class field}$ of $F$ modulo $\mathfrak{m}$. Hence, as a first step toward the problem we need to construct ray class fields for given number fields. In this talk, we will focus on the case of imaginary biquadratic fields K. There are two imaginary quadratic subfields $K_1$, $K_2$ and one real quadratic subfield $K_3$ in $K.$ For a nonnegative integer $\mu$ and an odd prime $p$, let $K_{(p^\mu)}$ and $(K_i)_{(p^\mu)}$ for $i=1,2,3$ be the ray class fields of $K$ and $K_i,$ respectively, modulo $p^\mu.$ We first present certain class fields $\widetilde{K_{p,\mu}^{1,2}}$ of $K$ which are generated by ray class invariants of $(K_i)_{(p^{\mu+1})}$ for $i=1,2$ over $K_{(p^\mu)}$ and find the exact extension degree $[K_{(p^{\mu+1})}:\widetilde{K_{p,\mu}^{1,2}}].$ And we shall further construct a primitive generator of the composite field $K_{(p^\mu)}(K_3)_{(p^{\mu+1})}$ over $K_{(p^\mu)}$ by means of norms of the above ray class invariants, which is a real algebraic integer. Using these values, we also generate a primitive generator of $(K_3)_{(p)}$ over the Hilbert class field of the real quadratic field $K_3$, and further find its normal basis.

October 24, 2013 Kim, Kwang-Seob(POSTECH)

Title: Construction of unramified extensions with a prescribed Galois group

Abstract: We shall prove that for any finite solvable group $G$, there exist infinitely many abelian extensions $K/\mathbb{Q}$ and Galois extensions $M/\mathbb{Q}$ such that the Galois group $\textrm{Gal}(M/K)$ is isomorphic to $G$ and $M/K$ is unramified.

November 15, 2013 Sun, Hae-Sang(Chungbuk National University)

Title: Distribution of modular symbols and a skew product from Gauss map

Abstract: This is a research in progress. We begin the talk by introducing the problem of distribution of modular symbols in the homology group of modular curves $X_0(N)$, which can be regarded as an infinite length version of the famous theorem on the distribution of closed geodesics on a hyperbolic manifold. After the introduction, we present how to relate the problem to a study on the transfer operator of skew product of two maps; One is the Gauss map on the unit interval and the other one is a map on the set $SL_2(\mathbb{Z})/\Gamma_0(N)$ of left cosets of $\Gamma_0(N)$, induced by the Gauss map.

November 20, 2013 Anatoly Libgober(University of Illinois at Chicago)

Title: Mordell Weil ranks of isotrivial abelian varieties and singularities

Abstract: I will discuss relation between ranks of Mordell-Weil groups of abelian varieties over field of rational functions in two variables and the fundamental groups of the complements to plane curves with arbitrary singularities. Key tools which will be described in the talk are use of abelian varieties of CM type associated with singularities and the structure of Albanese varieties of cyclic covers of the plane.

November 28, 2013 Im, Bo-Hae(Chung-Ang University)

Title: Positive rank quadratic twists of elliptic curves and $\theta$-congruent numbers

Abstract: We discuss positive rank quadratic twists of four elliptic curves and the relation between positive rank quadratic twists of a certain family of elliptic curves and $\theta$-congruent numbers.

December 12, 2013 Lim, Jongryul(POSTECH)

Title: On the Fourier coefficients of certain Jacobi forms and its application

Abstract: In this talk, we would like to review the basic theory of Jacobi forms in terms of Jacobi Eisenstein series and its Fourier coefficients. In the theory of modular forms, it is well-known that cusp forms satisfy the Hecke bound. As a counterpart, we will introduce the Hecke bound for Jacobi cusp forms and we will prove that if a Jacobi form satisfies the Hecke bound, it must be a Jacobi cusp form, using the exact information of Fourier coefficients of Jacobi Eisenstein series.

December 19, 2013 Woo, Youngho(KAIST)

Title: Introduction to the moduli stack of Curves I

Abstract: We will survey basic concepts of algebraic stacks. The main goal of this talk is to introduce basic languages like 2-functors, category fibred in groupoids, Grothentieck topologies, algebraic spaces which are necessary to define and understand the moduli space of curves as a stack.

December 20, 2013 Woo, Youngho(KAIST)

Title: Introduction to the moduli stack of Curves II

Abstract: We will survey basic concepts of algebraic stacks. The main goal of this talk is to introduce basic languages like 2-functors, category fibred in groupoids, Grothentieck topologies, algebraic spaces which are necessary to define and understand the moduli space of curves as a stack.

December 20, 2013 Woo, Youngho(KAIST)

Title: Introduction to the moduli stack of Curves III

Abstract: We will survey basic concepts of algebraic stacks. The main goal of this talk is to introduce basic languages like 2-functors, category fibred in groupoids, Grothentieck topologies, algebraic spaces which are necessary to define and understand the moduli space of curves as a stack.

December 26, 2013 Cho, Jaehyun(SUNY at Buffalo)

Title: Two-level densities of Artin $L$-functions

Abstract: We study two-level density of a family of Artin $L$-function twisted by a cuspidal representation, $L(s,\pi\times\rho)$, where $\pi$ is a fixed self-dual cuspidal representation of $GL_m$, and $\rho$ is given by $L(s,\rho)=\zeta_K(s)/\zeta(s)$ attached to cubic number fields $K$. A new phenomenon occurs: Namely, if $L(s,\pi, Sym^2)$ has a pole at $s=1$, and the root number $\epsilon(\pi\times\rho)=-1$, the one-level density shows that its symmetry type should be symplectic, but for the two-level density, we need to add the Dirac distribution to the usual symplectic kernel function due to the trivial zero at the central point.