## Webinar in Number Theory

YoungJu Choie (POSTECH), Bo-Hae Im (KAIST),
Laurent Berger (ENS Lyon)

Forthcoming speakers
• 2022/02/07: Gautier Ponsinet (Post doctoral at the Università degli Studi di Genova)
Title: ??

Abstract: T??

• 2022/02/21: Dr Jun-Yong Park (MPIM, Bonn)
Title: Arithmetic Moduli of Elliptic Surfaces

Abstract: By considering the arithmetic geometry of rational orbi-curves on modular curve $\overline{\mathcal{M}}_{1,1}$ where $\overline{\mathcal{M}}_{1,1}$ is the Deligne--Mumford stack of stable elliptic curves, we formulate the moduli stack of minimal elliptic fibrations over $\mathbb{P}^{1}$, also known as minimal elliptic surfaces with section over any base field $K$ with $\mathrm{char}(K) \neq 2,3$. Inspired by the classical work of [Tate] which allows us to determine the Kodaira--N\'eron type of fibers over global fields, we establish Tate's correspondence between the moduli stacks $\mathrm{Rat}_{n}^{\gamma}(\mathbb{P}^1, \overline{\mathcal{M}}_{1,1})$ of quasimaps with vanishing constraints $\gamma$ and $\mathrm{Hom}^{\Gamma}_n(\mathcal{C}, \overline{\mathcal{M}}_{1,1})$ of twisted maps with cyclotomic twistings $\Gamma$. Afterward, we acquire the exact arithmetic invariants of the moduli for each Kodaira--N\'eron types which naturally renders new sharp enumerations with a main leading term of order $\mathcal{B}^{\frac{5}{6}}$ and secondary & tertiary order terms $\mathcal{B}^{\frac{1}{2}} ~\&~ \mathcal{B}^{\frac{1}{3}}$ on $\mathcal{Z}_{\mathbb{F}_q(t)}(\mathcal{B})$ for counting elliptic curves over $\mathbb{P}_{\mathbb{F}_q}^{1}$ with additive reductions ordered by bounded height of discriminant $\Delta$. The emergence of non-constant lower order terms are in stark contrast with counting the semistable (i.e., strictly multiplicative reductions) elliptic curves. In the end, we formulate an analogous heuristic on $\mathcal{Z}_{\mathbb{Q}}(\mathcal{B})$ for counting elliptic curves over $\mathbb{Q}$ through the global fields analogy. This is a joint work with Dori Bejleri (Harvard) and Matthew Satriano (Waterloo).

Past speakers
• 2021/12/20: Professor Kwangho Choiy ( Southern Illinois University)
Title: Invariants in restriction of admissible representations of $p$-adic groups

Abstract: The local Langlands correspondence, LLC, of a $p$-adic group over complex vector spaces has been proved for several cases over decades. One of interesting approaches to them is the restriction method which was initiated for $SL(2)$ and its inner form. It proposes in line with the functoriality principle that the LLC of one group can be achieved from the LLC of the other group sharing the same derived group. In this context, we shall explain how the method is extended to some other cases of LLC's, the multiplicity formula in restriction, and the transfer of the reducibility of parabolic induction.

• 2021/11/22: Lucile Devin (Université du Littoral)
PDF Title: Chebyshev's bias and sums of two squares

Abstract: Studying the secondary terms of the Prime Number Theorem in Arithmetic Progressions, Chebyshev claimed that there are more prime numbers congruent to 3 modulo 4 than to 1 modulo 4. We will explain and qualify this claim following the framework of Rubinstein and Sarnak. Then we will see how this framework can be adapted to other questions on the distribution of prime numbers. This will be illustrated by a new Chebyshev-like claim : "for more than half of the prime numbers that can be written as a sum of two squares, the odd square is the square sof a positive integer congruent to 1 mod 4.

• 2021/11/08: Wansu Kim (KAIST)
PDF Title: Equivariant BSD conjecture over global function fields

Abstract: 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 generalise this to semi-stable abelian varieties twisted by Artin character'' over global function field (under some additional technical assumptions), and discuss further speculations for generalisations if time permits. This is a joint work with David Burns and Mahesh Kakde.

• 2021/10/18: Richard Griffon (University Clermont-Auvergne)
PDF Title: Isogenies of elliptic curves over function fields

Abstract: I will report on a recent work, joint with Fabien Pazuki, in which we study elliptic curves over function fields and the isogenies between them. More specifically, we prove analogues in the function field setting of two famous theorems about isogenous elliptic curves over number fields. The first of these describes the variation of the Weil height of the j-invariant of elliptic curves within an isogeny class. Our second main result is an ''isogeny estimate'' in the spirit of theorems by Masser-Wüstholz and by Gaudron-Rémond. After stating our results and giving quick sketches of their proof, I will, time permitting, mention a few Diophantine applications.

• 2021/10/04: Dr. Seoyoung Kim (Queen's University)
PDF Title: On the generalized Diophantine m-tuples

Abstract: For non-zero integers $n$ and $k\geq2$, a generalized Diophantine $m$-tuple with property $D_k(n)$ is a set of $m$ positive integers $\{a_1,a_2,\ldots, a_m\}$ such that $a_ia_j + n$ is a $k$-th power for any distinct i and j. Define by $M_k(n)$ the supremum of the size of the set which has property $D_k(n)$. In this paper, we study upper bounds on $M_k(n)$, as we vary $n$ over positive integers. In particular, we show that for $k\geq 3$, $M_k(n)$ is $O(\log n)$ and further assuming the Paley graph conjecture, $M_k(n)$ is $O((\log n)^{\epsilon})$. The problem for $k=2$ was studied by a long list of authors that goes back to Diophantus who studied the quadruple $\{1,33,68,105\}$ with property $D(256)$. This is a joint work with A. Dixit and M. R. Murty.

• 2021/06/21: Vlerë Mehmeti (Université Paris-Saclay, France)
PDF Title: Non-Archimedean analytic curves and the local-global principle

Abstract: In 2009, a new technique, called algebraic patching, was introduced in the study of local-global principles. Under different forms, patching had in the past been used for the study of the inverse Galois problem. In this talk, I will present an extension of this technique to non-Archimedean analytic curves. As an application, we will see various local-global principles for function fields of curves, ranging from geometric to more classical forms. These results generalize those of the previous literature and are applicable to quadratic forms. We will start by a brief introduction of the framework of non-Archimedean analytic curves and will conclude by a presentation of a first step towards such results in higher dimensions.

• 2021/06/07: Hae-Sang Sun (UNIST, Korea)
PDF Title: Cyclotomic Hecke L-values of a totally real field

Abstract: It is known that any Fourier coefficient of a newform of weight 2 can be expressed as a polynomial with rational coefficients, of a single algebraic critical value of the corresponding L-function twisted by a Dirichlet character of $p$-power conductor for a rational prime $p$. In the talk, I will discuss a version of this result in terms of Hecke L-function over a totally real field, twisted by Hecke characters of $p$-power conductors. The discussion involves new technical challenges that arise from the presence of the unit group, which are (1) counting lattice points in a cone that $p$-adically close to units and (2) estimating an exponential sum over the unit group. This is joint work with Byungheup Jun and Jungyun Lee.

• 2021/05/17: Riccardo Pengo (ENS Lyon, France)
PDF Title: Entanglement in the family of division fields of a CM elliptic curve

Abstract: Division fields associated to an algebraic group defined over a number field, which are the extensions generated by its torsion points, have been the subject of a great amount of research, at least since the times of Kronecker and Weber. For elliptic curves without complex multiplication, Serre's open image theorem shows that the division fields associated to torsion points whose order is a prime power are "as big as possible" and pairwise linearly disjoint, if one removes a finite set of primes. Explicit versions of this result have recently been featured in the work of Campagna-Stevenhagen and Lombardo-Tronto. In this talk, based on joint work with Francesco Campagna (arXiv:2006.00883), I will present an analogue of these results for elliptic curves with complex multiplication. Moreover, I will present a necessary condition to have entanglement in the family of division fields, which is always satisfied for elliptic curves defined over the rationals. In this last case, I will describe in detail the entanglement in the family of division fields.

• 2021/05/03: Chan-Ho Kim (KIAS, Korea)
PDF Title: On the Fitting ideals of Selmer groups of modular forms

Abstract: In 1980's, Mazur and Tate studied Iwasawa theory for elliptic curves over finite abelian extensions" and formulated various related conjectures. One of their conjectures says that the analytically defined Mazur-Tate element lies in the Fitting ideal of the dual Selmer group of an elliptic curve. We discuss some cases of the conjecture for modular forms of higher weight.