About

Ashay Burungale is an Assistant Professor in the Department of Mathematics at the University of Texas at Austin. His research primarily concerns number theory and automorphic forms, with a particular focus on the arithmetic of elliptic curves, special values of L-functions, and zeta elements. Over recent years, he has been engaged with the Birch and Swinnerton-Dyer conjecture and non-ordinary Iwasawa theory. Burungale received his B. Math. degree from the Indian Statistical Institute Bangalore and his Ph.D. from the University of California, Los Angeles. Prior to his current position, he spent several years at the California Institute of Technology as a research assistant professor.

Research topics

• Combinatorics
• Mathematics
• Discrete mathematics
• Pure mathematics

Selected publications

• On the Tate-Shafarevich Groups of CM Elliptic Curves Over Anticyclotomic $$\mathbb {Z}_p$$-Extensions at Inert Primes

  Springer proceedings in mathematics & statistics · 2026-01-01

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• On the Frobenius fields of abelian varieties over number fields

  Algebra & Number Theory · 2026-03-24

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  Let A be a non-CM simple abelian variety over a number field K.For a place v of K where A has good reduction, let F.A; v/ denote the Frobenius field generated by the corresponding Frobenius eigenvalues.If A has connected monodromy groups, we show that the set of places v such that F.A; v/ is isomorphic to a fixed number field has upper Dirichlet density zero.Moreover, assuming the GRH, we give a power saving upper bound for the number of such places.

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

  ArXiv.org · 2025-03-19

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  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. We show that the compositum $\mathbb{Q}_{f}(χ)$ of the Hecke fields associated to $f$ and $χ$ is generated by the square of the absolute value of the corresponding central $L$-value $L^{\rm alg}(1/2, f \otimes χ)$ over $\mathbb{Q}(μ_p)$. The proof is based among other things on techniques used for the recent resolution of unipotent mixing conjecture by the first and third named authors.

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

  ArXiv.org · 2025-06-04

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  Let $E$ be a CM elliptic curve defined over $\mathbb{Q}$ and $p$ a prime. We show that $${\mathrm corank}_{\mathbb{Z}_{p}} {\mathrm Sel}_{p^{\infty}}(E_{/\mathbb{Q}})=0 \implies {\mathrm ord}_{s=1}L(s,E_{/\mathbb{Q}})=0 $$ for the $p^{\infty}$-Selmer group ${\mathrm Sel}_{p^{\infty}}(E_{/\mathbb{Q}})$ and the complex $L$-function $L(s,E_{/\mathbb{Q}})$. Along with Smith's work on the distribution of $2^\infty$-Selmer groups, this leads to the first instance of the even parity Goldfeld conjecture: For $50\%$ of the positive square-free integers $n$, we have $ {\mathrm ord}_{s=1}L(s,E^{(n)}_{/\mathbb{Q}})=0, $ where $E^{(n)}: ny^{2}=x^{3}-x $ is a quadratic twist of the congruent number elliptic curve $E: y^{2}=x^{3}-x$.

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• Base Change and Iwasawa Main Conjectures for GL2

  International Mathematics Research Notices · 2025-04-01 · 2 citations

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  Abstract Let $E$ be an elliptic curve defined over ${\mathbb{Q}}$ of conductor $N$, $p$ an odd prime of good ordinary reduction such that $E[p]$ is an irreducible Galois module, and $K$ an imaginary quadratic field with all primes dividing $Np$ split. We prove Iwasawa main conjectures for the ${\mathbb{Z}}_{p}$-cyclotomic and ${\mathbb{Z}}_{p}$-anticyclotomic deformations of $E$ over ${\mathbb{Q}}$ and $K,$ respectively, dispensing with any of the ramification hypotheses on $E[p]$ in previous works. The strategy employs base change and the two-variable zeta element associated to $E$ over $K$, via which the sought after main conjectures are deduced from Wan’s divisibility towards a three-variable main conjecture for $E$ over a quartic CM field containing $K$ and certain Euler system divisibilities. As an application, we prove cases of the two-variable main conjecture for $E$ over $K$. The aforementioned one-variable main conjectures imply the $p$-part of the conjectural Birch and Swinnerton-Dyer formula for $E$ if $\operatorname{ord}_{s=1}L(E,s)\leq 1$. They are also an ingredient in the proof of Kolyvagin’s conjecture and its cyclotomic variant in our joint work with Grossi [1].

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

  Pure and Applied Mathematics Quarterly · 2025-01-01

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

  Annals of Mathematics · 2025-12-31 · 1 citations

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  Let $E$ be a CM elliptic curve defined over $\mathbb{Q}$ and $p$ a prime. We show that $\mathrm{corank}_{\mathbb{Z}_{p}} \mathrm{Sel}_{p^\infty}(E_{/\mathbb{Q}})=0 \implies \mathrm{ord}_{s=1} L(s,E_{/\mathbb{Q}})=0$ for the $p^\infty$-Selmer group $\mathrm{Sel}_{p^\infty}(E_{/\mathbb{Q}})$ and the complex $L$-function $L(s,E_{/\mathbb{Q}})$. Along with Smith's work on the distribution of 2$^\infty$-Selmer groups, this leads to the first instance of the even parity Goldfeld conjecture: For $50\%$ of the positive square-free integers $n$, we have $\mathrm{ord}_{s=1} L(s,E^{(n)}_{/\mathbb{Q}})=0$, where $E^{(n)}: ny^2=x^3-x$ is a quadratic twist of the congruent number elliptic curve $E: y^2=x^3-x$.

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• Hecke $L$-values, definite Shimura sets and Mod $\ell$ non-vanishing

  arXiv (Cornell University) · 2024-08-25 · 1 citations

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  Let $λ$ be a self-dual Hecke character over an imaginary quadratic field $K$ of infinity type $(1,0)$. Let $\ell$ and $p$ be primes which are coprime to $6N_{K/\mathbb{Q}}({\mathrm cond}(λ))$. We determine the $\ell$-adic valuation of Hecke $L$-values $L(1,λχ)/Ω_K$ as $χ$ varies over $p$-power order anticyclotomic characters over $K$. As an application, for $p$ inert in $K$, we prove the vanishing of the $μ$-invariant of Rubin's $p$-adic $L$-function, leading to the first results on the $μ$-invariant of imaginary quadratic fields at non-split primes. Our approach and results complement the work of Hida and Finis. The approach is rooted in the arithmetic of a CM form on a definite Shimura set.The application to Rubin's $p$-adic $L$-function also relies on the proof of his conjecture. Along the way, we present an automorphic view on Rubin's theory.

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• Zeta elements for elliptic curves and applications

  arXiv (Cornell University) · 2024-09-02 · 3 citations

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  Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with conductor $N$ and $p\nmid 2N$ a prime. Let $L$ be an imaginary quadratic field with $p$ split. We prove the existence of $p$-adic zeta element for $E$ over $L$, encoding two different $p$-adic $L$-functions associated to $E$ over $L$ via explicit reciprocity laws at the primes above $p$. We formulate a main conjecture for $E$ over $L$ in terms of the zeta element, mediating different main conjectures in which the $p$-adic $L$-functions appear, and prove some results toward them. The zeta element has various applications to the arithmetic of elliptic curves. This includes a proof of main conjecture for semistable elliptic curves $E$ over $\mathbb{Q}$ at supersingular primes $p$, as conjectured by Kobayashi in 2002. It leads to the $p$-part of the conjectural Birch and Swinnerton-Dyer (BSD) formula for such curves of analytic rank zero or one, and enables us to present the first infinite families of non-CM elliptic curves for which the BSD conjecture is true. We provide further evidence towards the BSD conjecture: new cases of $p$-converse to the Gross--Zagier and Kolyvagin theorem, and $p$-part of the BSD formula for ordinary primes $p$. Along the way, we give a proof of a conjecture of Perrin-Riou connecting Beilinson--Kato elements with rational points.

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• Kato's epsilon conjecture for anticyclotomic CM deformations at inert primes

  Journal of Number Theory · 2024-08-10 · 3 citations

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Recent grants

• Arithmetic Aspects of Special Values of L-Functions

  NSF · $110k · 2020–2022

• Arithmetic Aspects of Special Values of L-Functions

  NSF · $53k · 2022–2024

• Iwasawa theory of elliptic curves and the Birch--Swinnerton-Dyer conjecture

  NSF · $152k · 2023–2027

Frequent coauthors

• Ye Tian

  Tianjin University

  13 shared

• Kazuto Ota

  Osaka University

  6 shared

• Shinichi Kobayashi

  6 shared

• Haruzo Hida

  6 shared

• Christopher Skinner

  5 shared

• Francesc Castella

  5 shared

• Laurent Clozel

  Université Paris-Saclay

  4 shared

• Antonio Lei

  University of Ottawa

  3 shared