Searcharxiv⌕ Search

arXiv subjects

Kâzım Büyükboduk

Publications and source records attributed to Kâzım Büyükboduk.

15 recordsLinked to original sources
arXiv2026

On the Artin formalism for triple product $p$-adic $L$-functions

Our main objective in the present article is to study the factorisation problem for triple-product $p$-adic $L$-functions, particularly in the scenarios when the defining properties of the $p$-adic $L$-functions involved have no bearing on this problem, although Artin formalism would suggest such a factorisation. Our analysis, which is guided by the ETNC philosophy, recasts this problem as a comparison of diagonal cycles, Beilinson--Kato elements, and Heegner cycles.

math.NT↗
arXiv2026

A proof of $p$-adic Gross--Zagier theorem via BDP formula

This paper provides a new proof of the $p$-adic Gross--Zagier formula for the $p$-adic $L$-function associated with the base change of a normalised cuspidal eigen-newform $f$ of weight $k \geq 2$ (and families of such) to an imaginary quadratic field $K$. Our results encompass both the classical $p$-ordinary cases and non-ordinary scenarios, including new cases where $k > 2$ and $\mathrm{ord}_p(a_p(f)) > 0$. Unlike the traditional approach of comparing geometric and analytic kernels, we employ a ``wall-crossing'' strategy centred on the BDP formula and the theory of Beilinson--Flach elements.

math.NT↗
arXiv2026

Anticyclotomic Iwasawa theory of abelian varieties of $\mathrm{GL}_2$-type at non-ordinary primes II

Let $E/\mathbb{Q}$ an elliptic curve with good supersingular reduction at a prime $p\geq 5$, and $K$ an imaginary quadratic field such that the root number of $E$ over $K$ equals $-1$. When $p$ splits in $K$, Castella and Wan formulated the plus/minus Heegner point main conjectures for $E$ along the anticyclotomic $\mathbb{Z}_p$-extension of $K$, and proved them for semistable curves. We generalize their results to two settings: 1. For $p$ split in $K$, we formulate Sprung-type main conjectures for $\mathrm{GL}_2$-type abelian varieties at non-ordinary primes and prove them under some conditions. 2. For $p$ inert in $K$, we formulate, relying on the work of the first-named author with Kobayashi and Ota, plus/minus Heegner point main conjectures for elliptic curves, and prove the minus main conjecture for semistable curves. The latter yields a $p$-converse to the Gross--Zagier and Kolyvagin theorem for semistable elliptic curves $E$ at supersingular primes $p\geq 5$, complementing the pioneering $p$-converse theorems of Skinner and Zhang. Our method relies on Howard's framework of bipartite Euler systems, Zhang's resolution of Kolyvagin's conjecture and the recent proof of cyclotomic main conjecture at non-ordinary primes.

math.NT↗
arXiv2025

Trito-non-ordinary Iwasawa theory of diagonal cycles

Our goal in this paper is to introduce and study the Euler system of signed diagonal cycles associated with a trito-non-ordinary triple product of the form $f^B \times g^B \times \mathbf{h}^B$, where $f^B$ (resp. $g^B$) is a $p$-ordinary (resp. non-ordinary) eigenform on an indefinite quaternion algebra $B_{/\mathbb{Q}}$ of weight $2$, and $\mathbf{h}^B$ is a primitive Hida ($p$-ordinary) family. When $B=\mathrm{M}_2(\mathbb{Q})$ is split and $\mathbf{h}=\mathbf{h}^B$ has CM by an imaginary quadratic field, this allows us to develop the signed anticyclotomic Iwasawa theory for the base change $\mathrm{BC}_{K/\mathbb{Q}}(π_f)\times \mathrm{BC}_{K/\mathbb{Q}}(π_g)\times ψ$, where $ψ$ is a Hecke character of $K$. We formulate a signed Perrin-Riou-style Iwasawa main conjecture in this setting, and obtain a result on one inclusion in this conjecture. Our methods also allow us to extend Hsieh's construction of the balanced triple-product $p$-adic $L$-function to the trito-non-ordinary scenario, and to define its signed counterparts.

math.NT↗
arXiv2025

Wall-crossing and $p$-adic Artin formalism for ${\rm GSp}_4 \times {\rm GL}_2 \times {\rm GL}_2$

The goal of this article is to develop a $p$-adic Artin formalism in the context of $p$-adic families of automorphic forms on ${\rm GSp}_4 \times {\rm GL}_2 \times {\rm GL}_2$. Our treatment is guided by the (double) wall-crossing principle, emphasising an interplay between arithmetic GGP and $p$-adic explicit GGP formulae. Although the picture we present remains largely conjectural, we provide evidence in favour of our conjectures (a) in terms of algebraic $p$-adic $L$-functions, and (b) in endoscopic scenarios.

math.NT↗
arXiv2024

Functional equations of algebraic Rankin-Selberg $p$-adic $L$-functions

This article presents an approach to the algebraic functional equation for Selmer complexes, which in turn have applications in the Iwasawa theoretic study of Rankin-Selberg products of the Hida and Coleman families. Our treatment establishes the functional equation for algebraic $p$-adic $L$-functions (which are given in terms of characteristic ideals of Selmer groups, which arise as the cohomology of appropriately defined Selmer complexes in degree $2$). This is achieved by recovering the characteristic ideal as the determinant of the said Selmer complex, once we prove (under suitable but rather mild) hypotheses that the Selmer complex in question is perfect with amplitude $[1,2]$, and its cohomology is concentrated in degree-2. The perfectness of these Selmer complexes turns out to be a delicate problem, and the required properties require a study of Tamagawa factors in families, which may be of independent interest.

math.NT↗
arXiv2024

On the Artin formalism for triple product $p$-adic $L$-functions: Chow--Heegner points vs. Heegner points

Our main objective in this paper (which is expository for the most part) is to study the necessary steps to prove a factorization formula for a certain triple product $p$-adic $L$-function guided by the Artin formalism. The key ingredients are: a) the explicit reciprocity laws governing the relationship of diagonal cycles and generalized Heegner cycles to $p$-adic $L$-functions; b) a careful comparison of Chow--Heegner points and twisted Heegner points in Hida families, via formulae of Gross--Zagier type.

math.NT↗
arXiv2024

Arithmetic of critical $p$-adic $L$-functions

Our objective in the present work is to develop a fairly complete arithmetic theory of critical $p$-adic $L$-functions on the eigencurve. To this end, we carry out the following tasks: a) We give an "étale" construction of Bellaïche's $p$-adic $L$-functions at a $θ$-critical point on the cuspidal eigencurve. b) We introduce the algebraic counterparts of these objects (which arise as appropriately defined Selmer complexes) and develop Iwasawa theory in this context, including a definition of an Iwasawa theoretic $\mathscr L$-invariant $\mathscr{L}^{\rm cr}_{\rm Iw}$. c) We formulate the (punctual) critical main conjecture and study its relationship with its slope-zero counterparts. Along the way, we also develop descent theory (paralleling Perrin-Riou's work). d) We introduce what we call thick (Iwasawa theoretic) fundamental line and the thick Selmer complex to counter Bellaïche's secondary $p$-adic $L$-functions. This allows us to formulate an infinitesimal thickening of the Iwasawa main conjecture, and we observe that it implies both slope-zero and punctual critical main conjectures, but it seems stronger than both. e) We establish an $\mathcal{O}_{\mathcal{X}}$-adic leading term formula for the two-variable $p$-adic $L$-function over the affinoid neighbourhood $\mathcal{X}={\rm Spm}(\mathcal{O}_{\mathcal{X}})$ in the eigencurve about a $θ$-critical point. Using this formula we prove, when the Hecke $L$-function of $f$ vanishes to order one at the central critical point, that the derivative of the secondary $p$-adic $L$-function can be computed in terms of the second order derivative of an $\mathcal{O}_{\mathcal{X}}$-adic regulator (rather than a regulator itself).

math.NT↗
arXiv2023

Anticyclotomic Iwasawa theory of abelian varieties of $\mathrm{GL}_2$-type at non-ordinary primes

Let $p\ge 5$ be a prime number, $E/\mathbb{Q}$ an elliptic curve with good supersingular reduction at $p$ and $K$ an imaginary quadratic field such that the root number of $E$ over $K$ is $+1$. When $p$ is split in $K$, Darmon and Iovita formulated the plus and minus Iwasawa main conjectures for $E$ over the anticyclotomic $\mathbb{Z}_p$-extension of $K$, and proved one-sided inclusion: an upper bound for plus and minus Selmer groups in terms of the associated $p$-adic $L$-functions. We generalize their results to two new settings: 1. Under the assumption that $p$ is split in $K$ but without assuming $a_p(E)=0$, we study Sprung-type Iwasawa main conjectures for abelian varieties of $\mathrm{GL}_2$-type, and prove an analogous inclusion. 2. We formulate, relying on the recent work of the first named author with Kobayashi and Ota, plus and minus Iwasawa main conjectures for elliptic curves when $p$ is inert in $K$, and prove an analogous inclusion.

math.NT↗
arXiv2023

Iwasawa theory of twists of elliptic modular forms over imaginary quadratic fields at inert primes

Our primary goal in this article is to study the Iwasawa theory for semi-ordinary families of automorphic forms on $\mathrm{GL}_2\times\mathrm{Res}_{K/\mathbb{Q}}\mathrm{GL}_1$, where $K$ is an imaginary quadratic field where the prime $p$ is inert. We prove divisibility results towards Iwasawa main conjectures in this context, utilizing the optimized signed factorization procedure for Perrin-Riou functionals and Beilinson--Flach elements for a family of Rankin--Selberg products of $p$-ordinary forms with a fixed $p$-non-ordinary modular form. The optimality enables an effective control on the $μ$-invariants of Selmer groups and $p$-adic $L$-functions as the modular forms vary in families, which is crucial for our patching argument to establish one divisibility in an Iwasawa main conjecture in three variables.

math.NT↗
arXiv2021

Iwasawa theory for $\mathrm{GL}_2\times\mathrm{Res}_{K/\mathbb{Q}}\mathrm{GL}_1$

Let $K$ be a imaginary quadratic field where the prime $p$ splits. Our goal in this article is to prove results towards the Iwasawa main conjectures for $p$-nearly-ordinary families associated to $\mathrm{GL}_2\times \mathrm{Res}_{K/\mathbb{Q}}\mathrm{GL}_1$ with a minimal set of assumptions. The main technical input is an improvement on the locally restricted Euler system machinery that allows the treatment of residually reducible cases, which we apply with the Beilinson--Flach Euler system.

math.NT↗
arXiv2021

Interpolation of Beilinson-Kato elements and $p$-adic $L$-functions

Our objective in this series of two articles, of which the present article is the first, is to give a Perrin-Riou-style construction of $p$-adic $L$-functions (of Bellaïche and Stevens) over the eigencurve. As the first ingredient, we interpolate the Beilinson-Kato elements over the eigencurve (including the neighborhoods of $θ$-critical points). Along the way, we prove étale variants of Bellaïche's results describing the local properties of the eigencurve. We also develop the local framework to construct and establish the interpolative properties of these $p$-adic $L$-functions away from $θ$-critical points.

math.NT↗
arXiv2021

Iwasawa theory for Symmetric Square of non-$p$-ordinary eigenforms

Let $f$ be a normalized cuspidal eigen-newform of level coprime to $p$ with $a_p(f)=0$. We formulate both integral signed Iwasawa main conjectures and analytic Iwasawa man conjectures attached to the symmetric square motive of $f$ twisted by an auxiliary Dirichlet character. We show that the Beilinson--Flach elements attached to the symmetric square motive factorize into integral signed Beilinson--Flach elements, giving evidence towards the existence a rank-two Euler system predicted by Perrin-Riou. We use these integral elements to prove one inclusion in the integral and analytic Iwasawa main conjectures.

math.NT↗
arXiv2019

Rank--two Euler systems for symmetric squares

Let $p\ge 7$ be a prime number and $f$ a normalized eigen-newform with good reduction at $p$ such that its $p$-th Fourier coefficient vanishes. We construct a rank-two Euler system attached to the $p$-adic realization of the symmetric square motive of $f$. Furthermore, we show that the non-triviality is guaranteed by the non-vanishing of the leading term of the relevant $L$-value and the non-vanishing of a certain $p$-adic period modulo $p$.

math.NT↗