Murilo Corato Zanarella

I am a fifth year PhD candidate at MIT working under Wei Zhang. I am applying to jobs this year. I am interested in arithmetic aspects of the (relative) Langlands program, particularly in the role of Shimura varieties, and applications to the Beilinson--Bloch--Kato conjectures, Euler systems and Iwasawa theory.

Research projects

Consider a unitary group $G(\mathbb{A}_{F^+})=U_{2r}(\mathbb{A}_{F^+})$ over a CM extension $F/F^+$ with $G(\mathbb{A}_\infty)$ compact. In this article, we study the Beilinson--Bloch--Kato conjecture for motives associated to irreducible cuspidal automorphic representations $\pi$ of $G(\mathbb{A}_{F^+}).$ We prove that if $\pi$ is distinguished by the unitary Friedberg--Jacquet period, then the Bloch--Kato Selmer group (with coefficients in a favorable field) of the motive of $\Pi=\mathrm{BC}(\pi)$ vanishes.

We give explicit models for spherical functions on $p$-adic symmetric spaces $X=H\backslash G$ for pairs of $p$-adic groups $(G,H)$ of the form $(\mathrm{U}(2r),\mathrm{U}(r)\times \mathrm{U}(r)),$ $(\mathrm{O}(2r),\mathrm{O}(r)\times \mathrm{O}(r)),$ $(\mathrm{Sp}(4r),\mathrm{Sp}(2r)\times\mathrm{Sp}(2r))),$ $(\mathrm{U}(2r+1),\mathrm{U}(r+1)\times \mathrm{U}(r)),$ and $ (\mathrm{O}(2r+1),\mathrm{O}(r+1)\times \mathrm{O}(r)).$ As an application, we completely describe their Hecke module structure.

arXiv →

For all $r\ge1,$ we verify the following conjecture of Hironaka: for a $p$-adic field $F$ with $p$ odd, the space of spherical functions of $\mathrm{Sym}_{r\times r}(F)\cap\mathrm{GL}_r(F)$ is free of rank $4^r$ over the Hecke algebra.

arXiv →

We upgrade Howard's divisibility toward Perrin-Riou's Heegner point Main Conjecture to an equality under some mild conditions. We do this by exploiting Wei Zhang's proof of the Kolyvagin conjecture. The main ingredient is an improvement of Howard's Kolyvagin system formalism. As another consequence of it, we establish the equivalence between this main conjecture and the primitivity of the Kolyvagin system in certain cases, by also exploiting a explicit reciprocity law for Heegner points.

arXiv →

We adapt Wei Zhang's proof of Kolyvagin's conjecture for modular abelian varieties over $\mathbb{Q}$ to rely on the BDP main conjecture instead of on the cyclotomic main conjecture. The main ingredient is a reduction to a case that is tractable by the BDP main conjecture, in a similar spirit to Zhang's reduction to the rank one case. By using the BDP main conjecture instead of the cyclotomic main conjecture, our approach is more suitable than Zhang's to extend to modular abelian varieties over totally real fields.

arXiv →

Teaching & Mentorship

High School Enrichment Program Teacher 2021-2023

Virtual classes with students from my former high school on undergraduate-level topics in number theory.

Graduate TA at MIT

Fall 2023: 18.701 Algebra I
Fall 2023: 18.950 Differential Geometry
Fall 2022: 18.02 Multivariable Calculus
Spring 2021: 18.065 Matrix Methods in Data Analysis, Signal Processing & Machine Learning
Fall 2021: 18.701 Algebra I
Fall 2021: 18.700 Linear Algebra
Spring 2020: 18.702 Algebra II
Fall 2020: 18.100A Real Analysis

MIT directed reading program mentor

Winter 2022: Analytic Number Theory
Winter 2020: Modular Forms and Elliptic Curves


Fall 2022: Learning seminar on Euler systems


Graduate student seminar: An introduction to compactifications of Shimura varieties, with examples

Lecture series about Euler systems: Arithmetic level raising and reciprocity laws

Learning seminar on Euler systems: Iwasawa theory of elliptic curves

Learning seminar on Euler systems: Introduction to Iwasawa theory

Learning seminar on Euler systems: Euler system of cyclotomic units

Learning seminar on Rapoport–Zink spaces: Examples

Learning seminar on Rapoport–Zink spaces: Formulation of RZ data

STAGE: $p$-adic modular forms à la Katz

On Howard’s main conjecture and the Heegner point Kolyvagin system

Pure Math Graduate Student Seminar (PUMAGRASS): Regular primes and Bernoulli numbers

Notes & Expositories

18.748 Topics in Lie theory - final project: Fundamental lemma (Fall 2020)

18.906 Algebraic topology II (Spring 2020, taught by Haynes Miller)

18.726 Algebraic geometry II (Spring 2020, taught by Chenyang Xu)

18.705 Commutative algebra (Fall 2019, taught by Bjorn Poonen)

18.725 Algebraic geometry I (Fall 2019, taught by Chenyang Xu)

MAT 511 Class field theory - final project: Class field tower problem (Spring 2018)

Reading course on algebraic geometry - final project: Kuga--Satake construction (Spring 2018)

Junior seminar on modular symbols - talks and final project (Spring 2018)