Ashay A. Burungale

activity

20132026

p

-adic Abel-Jacobi image of generalised Heegner cycles modulo

p

, II: Shimura curves

30 citations · 86 across the 26 of their papers we have counts for

collaborators

27 papers

math.NT2026

A proof of Sylvester's conjecture

Ashay Burungale, Ye Tian

We prove Sylvester's conjecture, originating in his 1879 study of ternary cubic equations, that every prime $p \equiv 4, 7, 8 (mod 9)$ is a sum of two rational cubes. Elkies announced a…

math.NT2026

Anticyclotomic Iwasawa theory of CM elliptic curves at ramified primes

Ashay Burungale, Shinichi Kobayashi, Kentaro Nakamura +1

We propose an integral framework for the anticyclotomic Iwasawa theory of CM elliptic curves $E$ at primes $p$ ramified in the CM field. The $ε$ -constants of the geometri…

math.NT2026

A refined non-vanishing of the $p$ -adic logarithm of a rational point on an abelian variety

Ashay Burungale, Christopher Skinner, Xin Wan

Inspired by a beautiful formula of Bertolini, Darmon, and Prasanna -- the oft-termed BDP formula -- we address questions about the non-vanishing of non-torsion points under $p$ -adi…

math.NT2025

A local sign decomposition for symplectic self-dual Galois representations of rank two

Ashay Burungale, Shinichi Kobayashi, Kentaro Nakamura +1

We prove the existence of a new structure on the first Galois cohomology of generic families of symplectic self-dual $p$ -adic representations of $G_{Q_{p}}$ of rank two (a l…

math.NT2025

Mod $ℓ$ non-vanishing of self-dual Hecke $L$ -values over CM fields and applications

Ashay Burungale, Wei He, Ye Tian +1

Let $λ$ be a self-dual Hecke character over a CM field $K$ . Let $p$ be a degree one prime of the maximal totally real subfield $F$ of $K$ and $Γ_{p}$ the Galo…

math.NT2025

The $p$ -adic valuation of local resolvents, generalized Gauss sums and anticyclotomic Hecke $L$ -values of imaginary quadratic fields at inert primes

Ashay Burungale, Shinichi Kobayashi, Kazuto Ota

We prove an asymptotic formula for the $p$ -adic valuation of Hecke $L$ -values of an imaginary quadratic field at an inert prime $p$ along the anticyclotomic $Z_{p}$ -tower. T…

math.NT2025

A rank zero $p$ -converse to a theorem of Gross--Zagier, Kolyvagin and Rubin

Ashay A. Burungale, Ye Tian

Let $E$ be a CM elliptic curve defined over $Q$ and $p$ a prime. We show that $${\mathrm corank}_{\mathbb{Z}_{p}} {\mathrm Sel}_{p^{\infty}}(E_{/\mathbb{Q}})=0 \implies {\…

math.NT2025

Generation of Hecke fields by squares of cyclotomic twists of modular $L$ -values

Valentin Blomer, Ashay Burungale, Philippe Michel +1

Let $f$ be a non-CM elliptic newform without a quadratic inner twist, $p$ an odd prime and $χ$ a Dirichlet character of $p$ -power order and sufficiently large $p$ -power conductor.…

math.NT2024★ 3 cited

Zeta elements for elliptic curves and applications

Ashay Burungale, Christopher Skinner, Ye Tian +1

Let $E$ be an elliptic curve defined over $Q$ with conductor $N$ and $p ∤ 2 N$ a prime. Let $L$ be an imaginary quadratic field with $p$ split. We prove the existence of…

math.NT2024★ 1 cited

Hecke $L$ -values, definite Shimura sets and Mod $ℓ$ non-vanishing

Ashay A. Burungale, Wei He, Shinichi Kobayashi +1

Let $λ$ be a self-dual Hecke character over an imaginary quadratic field $K$ of infinity type $(1, 0)$ . Let $ℓ$ and $p$ be primes which are coprime to $6N_{K/\mathbb{Q}}({\mathrm…

math.NT2024

Dimension of the deformation space of ordinary representations in the cyclotomic limit

Ashay A. Burungale, Laurent Clozel, Barry Mazur

The weight two ordinary deformations are unobstructed in the cyclotomic limit under certain assumptions. We show that such an ordinary deformation ring over the cyclotomic tower ca…

math.NT2024

Base change and Iwasawa Main Conjectures for $GL_{2}$

Ashay Burungale, Francesc Castella, Christopher Skinner

Let $E$ be an elliptic curve defined over $Q$ of conductor $N$ , $p$ an odd prime of good ordinary reduction such that $E [p]$ is an irreducible Galois module, and $K$ an im…

math.NT2024

On the Frobenius fields of abelian varieties over number fields

Ashay A. Burungale, Haruzo Hida, Shilin Lai

Let $A$ be a non-CM simple abelian variety over a number field $K$ . For a place $v$ of $K$ such that $A$ has good reduction at $v$ , let $F (A, v)$ denote the Frobenius field generate…

math.NT2023★ 1 cited

Non-vanishing of Kolyvagin systems and Iwasawa theory

Ashay Burungale, Francesc Castella, Giada Grossi +1

Let $E / Q$ be an elliptic curve and $p$ an odd prime. In 1991 Kolyvagin conjectured that the system of cohomology classes for torsion quotients of the $p$ -adic Tate module…

math.NT2023

Anticyclotomic Iwasawa theory of abelian varieties of $GL_{2}$ -type at non-ordinary primes II

Ashay Burungale, Kâzım Büyükboduk, Antonio Lei

Let $E / Q$ 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 $…

math.NT2022

Anticyclotomic Iwasawa theory of abelian varieties of $GL_{2}$ -type at non-ordinary primes

Ashay Burungale, Kâzım Büyükboduk, Antonio Lei

Let $p \geq 5$ be a prime number, $E / Q$ an elliptic curve with good supersingular reduction at $p$ and $K$ an imaginary quadratic field such that the root number of $E$ over…

math.NT2022

The conjecture of Birch and Swinnerton-Dyer for certain elliptic curves with complex multiplication

Ashay Burungale, Matthias Flach

Let $E / F$ be an elliptic curve over a number field $F$ with complex multiplication by the ring of integers in an imaginary quadratic field $K$ . We give a complete proof of the conj…

math.NT2021

The even parity Goldfeld conjecture: congruent number elliptic curves

Ashay Burungale, Ye Tian

In 1979 Goldfeld conjectured: 50\% of the quadratic twists of an elliptic curve defined over the rationals have analytic rank zero. In this expository article we present a few rece…

math.NT2019★ 28 cited

A proof of Perrin-Riou's Heegner point main conjecture

Ashay Burungale, Francesc Castella, Chan-Ho Kim

Let $E / Q$ be an elliptic curve of conductor $N$ , let $p > 3$ be a prime where $E$ has good ordinary reduction, and let $K$ be an imaginary quadratic field satisfying the Hee…

math.NT2019

Ordinary deformations are unobstructed in the cyclotomic limit

Ashay Burungale, Laurent Clozel

The deformation theory of ordinary representations of the absolute Galois groups of totally real number fields (over a finite field $k$ ) has been studied for a long time, starting…

math.NT2018★ 1 cited

Indivisibility of Heegner points and arithmetic applications

Ashay Burungale, Francesc Castella, Chan-Ho Kim

We upgrade Howard's divisibility towards Perrin-Riou's Heegner point main conjecture to the predicted equality. Contrary to previous works in this direction, our main result allows…

math.NT2018★ 8 cited

On the non-vanishing of $p$ -adic heights on CM abelian varieties, and the arithmetic of Katz $p$ -adic $L$ -functions

Ashay Burungale, Daniel Disegni

Let $B$ be a simple CM abelian variety over a CM field $E$ , $p$ a rational prime. Suppose that $B$ has potentially ordinary reduction above $p$ and is self-dual with root number $-…

math.NT2017★ 4 cited

Horizontal variation of Tate--Shafarevich groups

Ashay A. Burungale, Haruzo Hida, Ye Tian

Let $E$ be an elliptic curve over $Q$ . Let $p$ be an odd prime and $ι : Q ↪ C_{p}$ an embedding. Let $K$ be an imaginary quadratic f…

math.NT2017

Horizontal non-vanishing of Heegner points and toric periods

Ashay A. Burungale, Ye Tian

Let $F / Q$ be a totally real field and $A$ a modular $\GL_2$-type abelian variety over $F$ . Let $K / F$ be a CM quadratic extension. Let $χ$ be a class group character over $…

A proof of Sylvester's conjecture

Anticyclotomic Iwasawa theory of CM elliptic curves at ramified primes

A refined non-vanishing of the p-adic logarithm of a rational point on an abelian variety

A local sign decomposition for symplectic self-dual Galois representations of rank two

Mod ℓ non-vanishing of self-dual Hecke L-values over CM fields and applications

The p-adic valuation of local resolvents, generalized Gauss sums and anticyclotomic Hecke L-values of imaginary quadratic fields at inert primes

A rank zero p-converse to a theorem of Gross--Zagier, Kolyvagin and Rubin

Generation of Hecke fields by squares of cyclotomic twists of modular L-values

Zeta elements for elliptic curves and applications

Hecke L-values, definite Shimura sets and Mod ℓ non-vanishing

Dimension of the deformation space of ordinary representations in the cyclotomic limit

Base change and Iwasawa Main Conjectures for GL2​

On the Frobenius fields of abelian varieties over number fields

Non-vanishing of Kolyvagin systems and Iwasawa theory

Anticyclotomic Iwasawa theory of abelian varieties of GL2​-type at non-ordinary primes II

Anticyclotomic Iwasawa theory of abelian varieties of GL2​-type at non-ordinary primes

The conjecture of Birch and Swinnerton-Dyer for certain elliptic curves with complex multiplication

The even parity Goldfeld conjecture: congruent number elliptic curves

A proof of Perrin-Riou's Heegner point main conjecture

Ordinary deformations are unobstructed in the cyclotomic limit

Indivisibility of Heegner points and arithmetic applications

On the non-vanishing of p-adic heights on CM abelian varieties, and the arithmetic of Katz p-adic L-functions

Horizontal variation of Tate--Shafarevich groups

Horizontal non-vanishing of Heegner points and toric periods

A refined non-vanishing of the $p$ -adic logarithm of a rational point on an abelian variety

Mod $ℓ$ non-vanishing of self-dual Hecke $L$ -values over CM fields and applications

The $p$ -adic valuation of local resolvents, generalized Gauss sums and anticyclotomic Hecke $L$ -values of imaginary quadratic fields at inert primes

A rank zero $p$ -converse to a theorem of Gross--Zagier, Kolyvagin and Rubin

Generation of Hecke fields by squares of cyclotomic twists of modular $L$ -values

Hecke $L$ -values, definite Shimura sets and Mod $ℓ$ non-vanishing

Base change and Iwasawa Main Conjectures for $GL_{2}$

Anticyclotomic Iwasawa theory of abelian varieties of $GL_{2}$ -type at non-ordinary primes II

Anticyclotomic Iwasawa theory of abelian varieties of $GL_{2}$ -type at non-ordinary primes

On the non-vanishing of $p$ -adic heights on CM abelian varieties, and the arithmetic of Katz $p$ -adic $L$ -functions