Number Theory Seminars 2018

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December 21 (Fri), 2018, 4:00pm-5:00pm, Room 1424

Title: Distribution of integral division points on the algebraic torus

Speaker: Su-Ion Ih (University of Colorado Boulder)

Abstract: I will begin by giving a brief introduction to integral points on an algebraic variety defined over a number field and paying attention to a finiteness result about integral points on the algebraic torus. Then I will discuss how this property fits in well with some prior known results and how it may lead to a general conjecture. (This is joint work with P. Habegger.)

December 19 (Wed), 2018, 4:00pm-5:00pm, Room 1424

Title: On the conjectures of Mazur-Rubin: a dynamical approach

Speaker: Jungwon Lee (UNIST)

Abstract: We present a transfer operator approach to study of distribution of modular symbols, motivated by the work of Baladi-Vall?e on dynamics of continued fractions. The approach leads to a few applications. We show an average version of Mazur-Rubin conjectures about statistical properties of modular symbols and further obtain the quantitative non-vanishing mod p of modular L-values with Dirichlet twists. (joint with Hae-Sang Sun)

December 6 (Thu), 2018, 1:30pm-2:30pm, Room 1424

Title: Class number problem for certain real quadratic fields

Speaker: Jigu Kim (SNU)

Abstract: Gauss calculated class numbers of definite quadratic forms which can be reinterpreted as class numbers of imaginary quadratic fields and observed that the class number goes to infinity as the discriminant goes to negative infinity. In 1976 D. Goldfeld proved a result which connected the conjecture of Birch and Swinnerton-Dyer with the conjecture of Gauss and gave a positive resolution of the effective version of Gauss's class number conjecture. But it is far from optimal, because it simultaneously covers the cases of both real and imaginary quadratic fields. In 1985 considering only imaginary quadratic fields, J. Oesterl\'{e} made Goldfeld's result explicit and solved the class number 3 problem. In this talk, I extend Oesterl\'{e}'s result to real quadratic fields by Hecke's argument in [Hecke1917]. This is joint work with Dongho Byeon.

November 15 (Thu), 2018, 2:00pm-3:30pm, Room 1424

Title: A classical introduction to Gross-Keating invariants

Speaker: Chul-hee Lee (KIAS)

Abstract: The Gross-Keating invariant of a quadratic form over p-adic integers is a relatively recent but fundamental concept in the study of quadratic forms. I will explain how it is related to some classical topics of number theory, such as the representation of integers by quadratic forms and the classical modular polynomials for the j-invariant. The rest of the talk will be devoted to a computer program that computes the Gross-Keating invariant of a quadratic form over Zp, and other related quantities.

November 9 (Thu), 2018, 5:00pm-6:00pm, Room 8309

Title: Fourier-Jacobi period and Central $L$-value

Speaker: Jaeho Haan (CMC)

Abstract: The Gan-Gross-Prasad conjecture predicts that the non-vanishing of certain periods is equivalent to the non-vanishing of certain central value of some $L$-function. There are two types of periods: Bessel period, Fourier-Jacobi(FJ) period. Bessel period is period of automorphic forms on orthogonal group or hermitian unitary group and FJ period is period on symplectic and metaplectic group or skew hermitian unitary group. In a seminal paper of Wei Zhang (2014. Ann. of Math), he proved Bessel case for unitary group and thereafter Hang Xue proved FJ-case for unitary group in a similar way. But both results are under some local restrictions to apply relative trace formula. In this talk, we prove one direction of the FJ case for unitary group without local restriction. This is a joint work with Hiraku Atobe.

October 31 (Wed), 2018, 4:00pm-6:00pm, Room 1424

Title: The a-values of Riemann zeta function near the critical line

Speaker: Junsoo Ha

Abstract: Let zeta(s) be the Riemann zeta function. We call the solutions to zeta(s)=a the a-values of Riemann zeta function. Selberg has studied the vertical value distribution of zeta function near the critical line and shown that log zeta(1/2+it) for t in [T,2T} behaves like Gaussian random variable of the mean 0 and the variance log log T. From this, he obtained the number of a-values on the box sigma>=1/2 and t in [T,2T]. Recently, Lamzouri Lester and Radziwill used Jensen's formula carefully to obtain the discrepancy between the Riemann zeta function and its randomized model, and achieved the a-value result for sigma > sigma_0 >1/2 for some fixed sigma_0. In this talk, we survey the key ideas behind their works and present the explicit range for which their result remains valid. Substantial portion of this talk is dedicated to basic tools and concepts of Riemann zeta function so (hopefully) no prior knowledge on the analytic number theory and the theory of zeta functions are necessary. This is a joint work with Yoonbok Lee.

October 25 (Thu), 2018, 4:00pm-6:00pm, Room 1423
October 26 (Fri), 2018, 2:00pm-4:00pm, Room 1423

Title: Level-raising problems for GL_n

Speaker: Aditya Karnataki (Beijing International Center for Mathematical Research)

Abstract: I will first explain the formalism of level-raising problems for the classical case of elliptic modular forms and later proceed to explain some new results for automorphic forms on GL_n over CM number fields.

October 2 (Tue), 2018, 4:00pm-5:00pm, Room 1423

Title: Eisenstein ideals and cuspidal subgroups of Jacobians of modular curves

Speaker: Ken Ribet (UC Berkeley)

Abstract: In my lecture, I will try to explain the main points of the proof of a theorem that I proved in 2015 with B. Jordan and A. Scholl. Although the theme of this seminar talk is closely related to that of the plenary lecture that I will give later in the week at the KMS-DMV joint meeting, that lecture will be for a general audience and will have few proofs, if any. The statement of the theorem requires some notation: we take a classical modular curve X (for definiteness, we'll take X = X0(N) for some N >= 1)) and consider the set of cusps on X. The cuspidal group attached to X is the image C in the Jacobian of X of the group of degree-0 divisors on X that are supported on the set of cusps of X. A theorem of Manin-Drinfeld asserts that C is finite. There is a great deal of literature to the effect that C is ``large'' in some qualitative sense. For example, view C as the quotient of the group of degree-0 cuspidal divisors on X by the group of divisors of the ``modular units'' on X. Theorems of Kubert and Kubert-Lang identify the group of modular units completely in terms of units that were constructed earlier by Siegel. Because all modular units may be described in terms of previously known units, it is natural to say that there are not too many of them. Qualitatively, to say that the group of modular units is small is to say that C is large. Our theorem states that C is large in the sense that its annihilator in a ring of Hecke operators is as small as possible: namely, the annihilator is the Eisenstein ideal of the ring. We don't prove this statement completely, but do prove it locally at all prime numbers bigger than 3 that are prime to N.

September 6 (Thu), 2018, 4:00pm-5:30pm, Room 1424

Title: p-local theory for elliptic curves of supersingular reduction

Speaker: Jaehoon Lee (UCLA)

Abstract: This talk will be half-expository and half-new (Joint with Chan-Ho Kim). We will start by reviewing the (relative) Lubin-Tate theory of de Shalit and prove the cyclicity of the maximal ideals in the Lubin-Tate tower as Galois modules. Then we will see how this result can be used to generate the "Norm-Compatible" points on (the formal group of) elliptic curve, which gives the description of the structure of the Mordell-Weil group over Z_p-extensions.

August 9 (Thu), 2018, 4:00pm-6:00pm, Room 8101

Title: The structure of local Galois deformation rings

Speaker: Stefano Morra (University of Montpellier)

Abstract: Via the Taylor-Wiles method, problems in modularity lifting can be reduced to questions on local Galois deformation spaces. In a limited number of cases (e.g. for elliptic curves over Q) their geometry is understood using moduli of finite flat group schemes. But in general it is related to affine flag varieties over Zp and determined by transcendental conditions via p-adic Hodge theory. In this talk we show that such conditions can be algebrized in most cases, obtaining properties as their connectedness, with applications to modularity lifting theorems, Serre weight, and Breuil-M\'{e}zard conjectures. This is joint work with Dan Le, Bao Le Hung and Brandon Levin.

August 7 (Tue), 2018, 2:00pm-3:30pm, Room 1424

Title: Structure of Mordell-Weil groups over \mathbb{Z}_p extensions

Speaker: Jaehoon Lee (UCLA)

Abstract: We will talk about the structure of Mordell-Weil, Selmer, and Tate-Shafarevich groups over \mathbb{Z}_p extensions as \Lambda-modules. If time permits, we will talk about the similar results for towers of modular curves in the setting of Hida's recent paper "Analytic variation of Tate-Shafarevich groups".

August 2 (Thu), 2018, 4:00pm-5:30pm, Room 1424

Title: Relative crystalline representations and p-divisible groups

Speaker: Yong Suk Moon (Purdue)

Abstract: Let k be a perfect field of characteristic p > 2, and let R be a relative base ring over W (k) which is unramified. Examples of R include R = W(k)[X1, . . . , Xd] and R = W(k)(X \pm 1,...,X \pm 1). We define relative B-pairs and study their relations to 1d weakly admissible R[1]-modules and Qp-representations. As an application, we show that when R = W(k)[X] with k = k, every horizontal crystalline representation of rank 2 with Hodge-Tate weights in [0, 1] arises from a p-divisible group over SpecR. Furthermore, we characterize all admissible R[1]-modules of rank 2 which are generated by parallel elements, and give an example of a B-pair which arises from a weakly admissible R[1]-module but does not arise from a Qp-representation. fields.

August 1 (Wed), 2018, 4:00pm-5:30pm, Room 1424

Title: Fitting ideals of Selmer groups

Speaker: Chan-Ho Kim (KIAS)

Abstract: We discuss the Fitting ideals of Selmer groups of elliptic curves in the context of the Mazur-Tate conjecture and the Kurihara conjecture. This talk is based on joint work with Masato Kurihara.

July 20 (Fri), 2018, 4:00pm-5:00pm, Room 1424

Title: Specialization and pullback of Galois covers

Speaker: Joachim Konig (KAIST)

Abstract: Given a Galois cover f : X -> P^1(k) with group G over a number field k, a very general question in inverse Galois theory is: what does the set Sf of all specializations (i.e., residue extensions of f at k-points) look like? Hilbert's irreducibility theorem yields that this set contains infinitely many G-extensions of k. On the other hand, recent results show that ``usually'' this set also misses infinitely many G-extensions. The precise structure of Sf is a great mystery even in seemingly very simple cases such as hyperelliptic covers. To gain some evidence, we consider a function field analog over k = C, replacing the notion of specialization by rational pullback. In joint work with Debes, Legrand and Neftin, we show that the finite subgroups of PGL2(C) are the only finite groups G such that all Galois covers X ! P1(C) of group G can be obtained from those with a bounded number of branch points by pullback along rational maps P1(C) ! P1(C). In fact, apart from those few exceptions, the set of all pullbacks is very small in a precise geometric sense. A worthwhile consequence for inverse Galois theory is that letting the branch point number grow always provides many truly new Galois realizations F = C(T) of group G. I will also discuss analogs over more general fields such as ample fields and number fields.

July 12 (Thu), 2018, 4:00pm-5:30pm, Room 1424

Title: Geometric perspective for ring parametrizations

Speaker: Seokhyeong Lee (Princeton)

Abstract: Since Manjul Bhargava's phenomenal 'Higher Compositions Laws' (HCL) works which gave parametrizations of number fields of degree 2, 3, 4, 5 using certain integral forms, various generalizations and improvements of HCL came out so far. One of them was Melanie Wood's works where she generalized some part of HCL to arbitrary bases (degree 2 algebra over arbitrary schemes, etc.) and added some additional correspondences in the list. In this explanatory talk, I will go over how Wood's hypercohomology construction provides the alternative scheme-theoretic approach for HCL and gives explicit procedure for computations. I will especially focus on its computational aspects and show explicit calculations of global functions in some cases.

September 13 (Thu), 2018, 5:00pm-6:00pm, Room 1424

Title: A bijective proof of Amdeberhan's conjecture on the number of $(s,s+2)$-core partitions with distinct parts

Speaker: Hayan Nam (UC Irvine)

Abstract: Amdeberhan conjectured that the number of $(s, s + 2)$-core partitions with distinct parts for an odd integer $s$ is $2^{s-1}$. This conjecture was first proved by Yan, Qin, Jin, and Zhou, then subsequently by Zaleski and Zeilberger. Since the formula for the number of such core partitions is so simple one can hope for a bijective proof. We give the first direct bijective proof of this fact by establishing a bijection between the set of $(s, s + 2)$-core partitions with distinct parts and a set of lattice paths.

May 29 (Tue), 2018, 1:00pm-2:00pm, Room 1423

Title: Computing isogenies and endomorphism rings of supersingular elliptic curves

Speaker: Travis Morrison (Waterloo)

Abstract: A quantum computer would break currently deployed public key cryptosystems. In light of this, NIST is currently organizing a public evaluation of ``quantum-secure'' cryptographic protocols. One such protocol bases its security on the hardness of computing isogenies between supersingular elliptic curves. This problem is deeply related to computing the endomorphism ring of a given supersingular elliptic curve. In this talk, I will give some examples of how isogenies are used to build protocols. I will also discuss how the problem of computing isogenies reduces to the problem of computing endomorphism rings. This is joint work with Eisentraeger, Hallgren, Lauter, and Petit. If time permits, I will also discuss a necessary condition on when two cycles in a supersingular isogeny graph generate an endomorphism ring (joint work with Bank, Camacho, Eisentraeger, Park).

May 24 (Thu), 2018, 4:00pm-5:00pm, Room 1423

Title: Number theoretic results in a family of number fields

Speaker: Henry Kim (Toronto)

Abstract: Unconditional results without an unproved hypothesis such as the generalized Riemann hypothesis (GRH) are very weak for an individual number field. But if we consider a family of number fields, one can prove just as strong results as we would assume GRH, in the form: (1) average result in the family; (2) the result is valid for almost all members except for a density zero set. We will explain this philosophy using examples of logarithmic derivatives of L-functions, residues of Dedekind zeta functions, and least primes in a conjugacy class.

April 30 (Mon), 2018, 1:30pm-2:30pm, Room 1423

Title: Motivic cohomology of fat points in Milnor range

Speaker: Jinhyun Park (KAIST)

Abstract: We give an algebraic cycle complex that describes the Milnor K-groups of truncated polynomials over a field of characteristic 0. We discuss how deformation theory and certain non-archimedean topology could be useful in studies of algebraic cycles and K-groups. Joint work with Sinan Unver.

April 20 (Fri), 2018, 5:00pm-6:00pm, Room 1424

Title: The asymptotic formulas for coefficients and algebraicity of Jacobi forms expressed by infinite product

Speaker: Seok-Ho Jin (CAU)

Abstract: We determine asymptotic formulas for the Fourier coefficients of Jacobi forms expressed by infinite products with Jacobi theta functions and the Dedekind eta function. These are generalizations of results about the growth of the Fourier coefficients of Jacobi forms given by an inverse of Jacobi theta function to derive the asymptotic behavior of the Betti numbers of the Hilbert scheme of points on an algebraic surface by Bringmann- Manshot and about the asymptotic behavior of the \chi_y -genera of Hilbert schemes of points on K3 surfaces by Msnshot-Rolon. And we get the algebraicity of the generating functions given by G\"{o}ttsche for the Hilbert schemes associated to general algebraic surfaces.

March 30 (Fri), 2018, 4:00pm-5:00pm, Room 1424

Title: Galois actions on finite groups and some applications

Speaker: Junghwan Lim (Oxford)

Abstract: I will discuss about the Galois actions on finite groups arising as characteristic quotients of the geometric fundamental groups of once-punctured elliptic curves and related topics.

February 20 (Tue), 2018, 10:30am-noon, Room 1423
February 20 (Tue), 2018, 2:00pm-3:30pm, Room 1424(!)
February 22 (Thu), 2018, 10:30am-noon, Room 1423
February 22 (Thu), 2018, 2:00pm-3:30pm, Room 1423

Title: Hasse-Weil zeta functions of modular curves: introduction to Langlands-Kottwitz-Scholze method

Speaker: Dong Uk Lee

Abstract: We explain P. Scholze's proof (IMRN 2010), extending the Langlands-Kottwitz method to bad reductions, of the (already estalbished) conjecture that the Hasse-Weil zeta functions of modular curves $X(N)$ is a product of automorphic L-functions on $GL_2$. A new contribution is certain explicit computatons concerning the semi-simple trace of Frobenius on nearby cycle sheaves.

January 3 (Wed), 2018, 5:00pm-6:00pm, Room 1423

Title: Algebraic functional equation of Selmer groups

Speaker: Jaehoon Lee (UCLA)

Abstract: We will start from introducing various "exotic" modular curves and Jacobians arose from those curves. Assuming control results of Selmer groups studied by Hida in his recent paper, we will show algebraic functional equation between two \Lambda-adic Selmer groups. If time permits, we will also talk about structure of cohomology groups arose from \Lambda-adic Barsotti-Tate groups.

January 3 (Wed), 2018, 4:00pm-5:00pm, Room 1423

Title: Fontaine-Messing Theory in the Relative setting

Speaker: Yong Suk Moon (Purdue)

Abstract: Let k be a perfect field of characteristic p > 2, and let R be a formal power series ring over W(k). For any proper smooth scheme X over R, we extend Fontaine-Messing theory in this setting to compare torsion crystalline cohomology of X with its torsion etale cohomology.