Number Theory Seminars 2015

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January 8, 2015 Jang, Junmyeong (University of Ulsan)

Title: Non-symplectic automorphism and the Frobenius invariant of a $K3$ surface.

Abstract: In this talk, we will see some relation between the representations of the automorphism group of a $K3$ surface over field of odd characteristic on the two forms, on the transcendental cycles (etale or crystalline) and the discriminant group of the Neron-Severi group (when the given $K3$ surface is supersingular). Using these results we show that the Frobenius invariant of a $K3$ surface with a non-symplectic automorphism of sufficiently large order is determined by the congruence class of the base characteristic modulo the order.

January 22, 2015 Shin, Dong Hwa (Hankuk University of Foreign Studies)

Title: Determination of Fricke families.

Abstract: For a positive integer $N$ divisible by $4$ we shall determine explicit generators of the ring of weakly holomorphic functions for the congruence subgroup $\Gamma^1(N)$ with rational Fourier coefficients, from which we can classify all Fricke families of level $N$. To this end we use various modular units, especially Siegel functions and Weierstrass units. This result is a joint work with Dr. Eum and Prof. Koo.

January 29, 2015 Hann, Jaeho (Seoul National University)

Title: The Gross-Prasad conjecture for some non-tempered Arthur parameter.

Abstract: The Gross-Prasad conjecture concerns the restriction problem in the representation theory and it has generated much interests in recent years. In this talk, we introduce the conjecture for the unitary group and explain our recent result concerning the non-tempered aspects of the conjecture. More precisely, for a pair of tempered $L$-parameters of $(U(n),U(n-1))$, it is known that there is a unique pair of representations in their associateed Vogan $L$-packets which produces the unique Bessel model of these $L$-parameters. We showed that this is ture for some pair of $L$-parameters involving a non-tempered one and precisely suggested it in the framework of the local Langlands correspondence for unitary group. As an applicaiton of these results, we prove an analog of Ichino-Ikeda conejcture for some non-tempered case. For more information, refer to arXiv:1501.00885/

February 4, 2015 Lee, Chul-hee (The University of Queensland)

Title: Introduction to Mahler Measure.

Abstract: For a (Laurent) polynomial with complex coefficients, we can define a quantity called the Mahler measure. It was originally studied in attempts to find large primes. And later its multivariate version was introduced as a tool in transcendental number theory. In more recent times it has become an active area of research due to its mysterious connections with many other subjects. For example, there are many conjectural formulas relating Mahler measures to special values of L-functions of elliptic curves and they also show up in hyperbolic geometry. In this talk, I will give an introductory survey on this topic.

February 10, 2015 Sun, Hae-Sang (UNIST)

Title: Special values of Hecke L-values of totally real field.

Abstract: In the talk, first I will introduce how to represent the special values of Dirichlet L-function as a period integral over a special cycle in the first homology group of a punctured cylinder. Next I will discuss how to extend this homological setting to the case of totally real field.

February 26, 2015 Kim, Byoung Du (Victoria University of Wellington)

Title: Signed-Selmer groups and p-adic L-functions for elliptic curves at non-ordinary/supersingular primes

Abstract: Finding new Selmer groups and $p$-adic $L$-functions which are suitable for studying the arithmetic of elliptic curves for primes with bad or non-ordinary reduction is one of the major themes of Iwasawa Theory. Of particular interest lately is good supersingular reduction. At the turn of the century, the idea of plus/minus Selmer groups and $p$-adic $L$-functions were introduced by S. Kobayashi and R. Pollack, and has become standard. These groups and functions were constructed mostly for certain $\mathbb{Z}_p$-extensions over $\mathbb{Q}$, and people wondered whether one could construct something similar for $\mathbb{Z}^{2}_{p}$-extensions over an imaginary quadratic field (especially because Iwasawa Theory for $\mathbb{Z}^{2}_{p}$-extensions over an imaginary quadratic field is often key to understanding Iwasawa Theory for the cyclotomic $\mathbb{Z}_{p}$-extensions over $\mathbb{Q}$). In this presentation, I will present my construction of $\pm / \pm$-Selmer groups, and D. Loeer's construction of $\pm / \pm$-$p$-adic $L$-functions. Together, they may provide a key to solving some conjectures of Iwasawa Theory in some cases.

March 19, 2015 Kim, Byungchan (Seoul National University of Science and Technology)

Title: On the odd-balanced unimodal sequences

Abstract: We define odd-balanced unimodal sequences and show that their generating function $\mathcal{V}(x,q)$ has the same remarkable features as the generating function for strongly unimodal sequences $U(x,q)$. In particular, we discuss (mixed) mock modularity, quantum modularity, and congruence properties modulo $2$ and $4$. This is a joint work with Subong Lim and Jeremy Lovejoy.

April 9, 2015 Kim, Taekyung (Seoul National University)

Title: A conjecture of Gross and Zagier for elliptic curves having rational torsion group of a 2-power order

Abstract: One of the most prominent results toward the Birch and Swinnerton-Dyer conjecture is the paper \emph{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 $\mathrm{ord}_{s=1} L(E/K, s) = \mathrm{rank} E(K) = 1$, then $L'(E/K,s)$ can be written as the 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 $E(\mathbb{Q})$ 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.

May 11, 2015 Haan, Jaeho (Korea Advanced Institute of Science and Technology)

Title: On the uniqueness of functions in the Selberg class.

Abstract: The Selberg class is an axiomatically defined class of $L$-functions which are of arithmetic interests. We prove a uniqueness theorem for functions in the Extended Selberg Class which states that for every $c\ne 0$, the functions $L(s)=\sum_{n=1}^{\infty}\frac{a(n)}{n^k}$ in the class having the positive degree are completely determined by $L^{-1}(c)$.

May 12, 2015 Satoshi Wakatsuki (Kanazawa University)

Title: Congruences modulo 2 for dimensions of spaces of cusp forms.

Abstract: In this talk, we give some congruences modulo $2$ for dimensions of spaces of Siegel cusp forms of degree one or two. First, we review some known results for congruences between dimensions of spaces of cusp forms and class numbers of imaginary quadratic fields. Next, we give our main result, which is a congruence relation for the degree two case. It is related to Arthur's conjecture for $PGS_p(2)$.

May 14, 2015 Kim, Henry (University of Toronto)

Title: The average of the smallest prime in a conjugacy class.

Abstract: Let $C$ be a conjugacy class of $S_n$, and $K$ be an $S_n$-field. Let $N_{K,C}$ be the smallest prime $p$ such that the Frobenius automorphism $Frob_p$ belongs to $C$. Under GRH (generalized Riemann hypothesis), it is known that $N_{K,C}<< (log |d_K|)^2$. An unconditional result is very poor. We look at the average of $N_{K,C}$ over a family of number fields without GRH. This is a joint work with Jaehyun Cho.

May 28, 2015 Lee, Junguk (Yonsei University)

Title: Uniform bound of elliptic curves over $\mathbb{Q}$ and nonstandard rational number fields.

Abstract: A nonstandard rational number field is a quotient field of a ring of a infinite product of rational number field $\mathbb{Q}$ by a nonprincipal maximal ideal. This nonstandard field satisfies same first order logical properties of $\mathbb{Q}$. In this talk, we review the first order logic fast, and we see which properties of elliptic curves can be expressed in the first order logic. Using nonstandard rational number fields, our main aim is to show the ranks of elliptic curves over $\mathbb{Q}$ is uniformly bounded if and only if the weak Mordell-Weil theorem holds for some(and every) nonstandard rational number field.

July 3, 2015 Dousse, Jehanne (LIAFA)

Title: Generalisation of two partition identities of Andrews

Abstract: A partition of a positive integer n is a nonincreasing sequence of positive integers (called parts) whose sum is n. The Rogers-Ramanujan identities state that for every n, the number of partitions of n into parts congruent to 1 or 4 modulo 5 is equal to the number of partitions of n such that the difference between two consecutive parts is at least 2. More generally, Rogers-Ramanujan type identities establish equalities between certain types of partitions with congruence conditions and partitions with difference conditions. In 1968 and 1969, Andrews proved two very general Rogers-Ramanujan type identities which generalise Schur's theorem. Those identitites went on to become important theorems in the theory of partitions, with applications in combinatorics, representation theory and quantum algebra. In this talk, we will show that we can generalise Andrews' identities to overpartitions (partitions in which the last occurence of a part may be overlined) using a new technique which consists in going back and forth between q-difference equations on generating functions and recurrence equations on their coefficients.

July 6, 2015 Matthias Flach (Caltech)

Title: Weil etale cohomology and Zeta functions of arithmetic schemes

Abstract: We report on joint work with Baptiste Morin in which we give a description of the vanishing order and leading Taylor coefficient of the Zeta function of an arithmetic scheme at integer arguments in terms of Weil-etale cohomology complexes. This extends work of Lichtenbaum for the Dedekind Zeta function at s=0 and of Milne, Lichtenbaum and Geisser for varieties over finite fields. We show how our description is compatible with the Tamagawa number conjecture of Bloch, Kato, Fontaine, Perrin-Riou, at least in certain cases. We discuss in some d etail the example of the Dedekind Zeta function at any integer n and elliptic curves over Q at s=1.

July 6,7,8, 2015 Karl Mahlburg (Louisiana State University)

Title: I. Introduction to classical partition identities. Discovery, proofs, and historical significance. II. Automorphic properties. The role of modular forms, Jacobi forms, mock modular forms, etc., and applications. III. Additional discussion. Other applications in combinatorics, probability, and algebra.

Abstract: In this series of talks I will discuss the deep connections between automorphic forms and partition identities. These include classical results such as the famous "sum-product" identities of Euler, Rogers-Ramanujan, Andrews-Gordon, and Schur, as well as more recent identities arising from affine Lie algebras and Lepowsky-Wilson's program of vertex operator algebras. Although these identities are of interest due to their intrinsic combinatorics and algebraic applications, they also often display automorphic properties, with examples of theta functions, modular functions, mock modular forms, and false theta functions. Some of these connections were only discovered recently, and have led to applications including asymptotic formulas, algebraicity, congruences, and probabilistic interpretations.

July 9, 2015 John Miller (Rutgers University)

Title: Class number of large degree number fields

Abstract: The calculation of the class number of number fields of large degree and discriminant is known to be a difficult problem. The Minkowski bound is too large to be useful, and the root discriminant of the field can be too large to be treated by Odlyzko's discriminant bounds. This talk describes a new approach. By finding nontrivial lower bounds for sums over prime ideals of the Hilbert class field, we establish upper bounds for class numbers of fields of larger discriminant. This analytic upper bound, together with algebraic arguments concerning the divisibility properties of class numbers, allows us to determine the class numbers of many number fields that have previously been untreatable by any known method.

July 17, 2015 Hwajong Yoo (IBS-CGP)

Title: On Eisenstein ideals and the cuspidal group of J_0(N)

Abstract: In this talk, we discuss Eisenstein ideals of the Hecke ring of level N with N square-free. The main theorem of this talk is the following. 'Every Eisenstein maximal ideal has support at the cuspidal group.'

July 23, 2015 Arno Kret (Max Planck Institute)

Title: Galois representations for the general symplectic group

Abstract: We explain how to construct Galois representations attached to automorphic representations of the general symplectic group over a totally real number field, under local simplifying hypotheses. This is joint work with Sug Woo Shin.

August 6, 2015 Seungki Kim (Stanford)

Title: The behavior of lattice reduction algorithms

Abstract: The celebrated LLL algorithm takes as an input a basis of a lattice in $R^n$, and outputs a 'nice' basis of that lattice in polynomial time. It is a well-known curious phenomenon that, most of the time, LLL gives a much nicer basis than theoretically guaranteed. We use some theory of automorphic forms on $SL(n)$ to suggest that this happens because the algorithm prefers certain bases over others. This is a joint work in progress with Akshay Venkatesh.

August 11, 2015 Lukasz Pankowski (Nagoya University)

Title: On large values of $L$-functions from the Selberg class

Abstract: During the talk we discuss some Omega results for $L$-functions from the Selberg. Using the so-called resonance method due to Soundararajan, we prove estimates for large values of $L$-functions from the Selberg class with polynomial Euler product in the strip $\sigma_m. Moreover, we show that under the Riemann Hypothesis, Montgomerys approach gives slightly weaker estimates but in the entire right open half of the critical strip.

August 20, 2015 Myungjun Yu (UC Irvine)

Title: Selmer ranks of Jacobians of hyperelliptic curves in families of quadratic twists

Abstract: Mazur and Rubin found sufficient conditions for elliptic curves to have infinitely many quadratic twists that have $2$-Selmer ranks $r$, for any given non-negative integer $r$. We generalize this result to hyperelliptic curves.

September 10, 2015 Cristian Virdol (Yonsei University)

Title: Artins conjecture for abelian varieties

Abstract: Artin's primitive root conjecture (1927) states that, for any integer $a\neq\pm1$ or a perfect square, there are infinitely many primes $p$ for which $a$ is a primitive root (mod $p$). This conjecture is not known for any specific $a$. In my talk I will prove the equivalent of this conjecture unconditionally for general abelian varieties for all $a$.

September 24, 2015 Choi, Suh Hyun (KAIST)

Title: Potential modularity and finiteness of certain Galois representations

Abstract: This is a joint work with Prof. Dohoon Choi. I will introduce several results in modularity, potential modularity and finiteness of Galois representations and show our result on finiteness of some potentially 2-dimensional crystalline Galois representations of totally real fields.

October 8, 2015 Robert Osburn (University College Dublin)

Title: $q$-series and quantum knot invariants

Abstract: There has been recent interest in the appearance of classical partition theoretic identities in knot theory. In this talk, we prove conjectures due to Garoufalidis, Le and Zagier concerning Rogers-Ramanujan type identities associated to tails of colored Jones polynomials. We also discuss a recent extension to all alternating knots up to 10 crossings. This is joint work with Adam Keilthy (TCD) and Paul Beirne (UCD).

October 8, 2015 Jeremy Lovejoy (Paris 7 and CNRS)

Title: Torus knots and quantum modular forms

Abstract: We compute a $q$-hypergeometric expression for the coefficients in the cyclotomic expansion of the colored Jones polynomial for the left-handed torus knot $(2,2t+1)$. We use this to define a family of $q$-series, the simplest case of which is the generating function for strongly unimodal sequences. Special cases of these $q$-series are quantum modular forms, and at roots of unity these are dual to the generalized Kontsevich--Zagier series. This duality generalizes a result of Bryson, Pitman, Ono, and Rhoades. This is joint work with Kazuhiro Hikami.

October 22, 2015 Jungwon Lee (Postech)

Title: On $p$-adic $L$-functions and elliptic units

Abstract: The conjecture of Birch and Swinnerton-Dyer asserts an extraordinary precise link between the behaviour of special $L$-values and the arithmetic of an elliptic curve. In the mid-1970s, Coates and Wiles observed that there is a generalization of the classical Iwasawa theory that can be used to attack this conjecture. I will discuss their Iwasawa theory of CM elliptic curves focusing on the role of elliptic units in the construction of $p$-adic $L$-functions, and give a numerical example for the certain family of quadratic twists by analyzing the structure of Iwasawa module.

October 29, 2015 Jeehoon Park (Postech)

Title: Deformation theory of period integrals of smooth projective complete intersections

Abstract: Period integrals of algebraic varieties are important invariants which are defined as the integrals of their de-Rham cohomology classes over singular homology classes. In this talk, we will present an explicit algorithmic formula between the period integrals of two algebraically different smooth projective complete intersection varieties with the same degree and same dimension (but topologically same). Our method is based on the theory of algebraic Dwork complexes and homotopy Lie theory (so called, $L_\infty$ -algebras and $L_\infty$-morphisms). This is a joint work with Yesule Kim (Postech).

November 6, 2015 Giuseppe Molteni (Universitá statale di Milano)

Title: Recent results about the prime ideal theorem

Abstract: Let $\psi_{\mathbb{K}}$ be Chebyshev's function of a number field $\mathbb{K}$. Let also $\psi^{(1)}_\mathbb{K}(x) := \int_{0}^{x}\psi_\mathbb{K}(t)\,\mathrm{d} t$ and $\psi^{(2)}_\mathbb{K}(x) := 2\int_{0}^{x}\psi^{(1)}_\mathbb{K}(t)\,\mathrm{d} t$, which are smoothed versions of $\psi_{\mathbb{K}}$. Assuming the Generalized Riemann Hypothesis we prove explicit inequalities for $|\psi_\mathbb{K}(x) - x|$, $|\psi^{(1)}_\mathbb{K}(x) - \tfrac{x^2}{2}|$ and $|\psi^{(2)}_\mathbb{K}(x) - \tfrac{x^3}{3}|$. Two results about the existence of ideals having small norms and about the computation of the residue of Dedekind's zeta function are also given. This is an account of a joint work with Loïc Grenié.

November 25-26, 2015 Seunghwan Chang (Ewha Womans University)

Title: The field of norms and ($\varphi$, $\Gamma$)-modules I, II, III

Abstract: Fontaine's theory of ($\varphi$, $\Gamma$)-modules provides means to do explicit computations regarding $p$-adic local Galois representations. There have been efforts to generalize the theory, which is contingent upon lifting the field of norms of more general extensions than cyclotomic. In this seminar, I will talk about field of norms construction and Berger's results on liftability of the field of norms.

December 10, 2015 Peter Jaehyun Cho (University of Waterloo)

Title: Extreme residues of Dedekind zeta functions

Abstract: In a family of $S_{d+1}$-fields ($d=2,3,4$), we obtain the true upper and lower bound of the residues of Dedekind zeta functions except for a density zero set. For $S_5$-fields, we need to assume the strong Artin conjecture. We also show that there exists an infinite family of number fields with the upper and lower bound, resp. This is a joint work with Henry Kim.

December 22-23, 2015 Taekyung Kim (Seoul National University)

Title: Integral Models of Algebraic Curves

Abstract: In this series of expository lectures, I will introduce fundamental notions of integral models of algebraic curves. With arithmetic in mind, the main motivation for the study is antique but also cutting-edge techniques of reduction. So after rapid set-up of some basic notions from geometry, I will move on to more arithmetic-flavoured topics such as reductions of elliptic curves, Néron models and Kodaira-Néron classifications. I will also discuss the famous theorem of Deligne-Mumford for curves of higher genera. The main reference of this series is Chapter X in Qing Liu's text.

December 29, 2015 Junghwan Lim (Oxford University)

Title: Covering spaces and Galois sections

Abstract: I will introduce about some usages of covering spaces in airthmetic geometry and its connection with Galois sections arising in theory of arithmetic fundamental groups.