Daniel Kriz

MIT

I am a postdoctoral fellow at MIT. I graduated in 2018 from Princeton University with a Ph.D.

My advisers were Christopher Skinner and Shouwu Zhang. Previously, I was also an undergraduate at Princeton, and graduated in 2014.

In the Fall of 2016, I was a visitor to the I.H.E.S..

My thesis title was ``A New p-adic Maass-Shimura operator and supersingular Rankin-Selberg p-adic L-functions".

Mathematical Interests

I am interested in algebraic number theory, particularly in arithmetic geometry and Iwasawa theory. My work concerns mainly the study of p-adic objects such as Shimura varieties over p-adic rings, p-adic modular forms, p-adic L-functions and Iwasawa theory. Recently, I have developed the Iwasawa theory of imaginary quadratic fields at height 2 primes, formulating and proving Rubin-type main conjectures at ramified primes. For this, I also constructed new p-adic L-functions using techniques from relative p-adic Hodge theory developed in my book and thesis. These p-adic L-functions have a new type of interpolation property relating certain "twisted Fourier transforms" to twists of L-values by finite-order anticyclotomic characters.

These results have applications to elliptic curves, including the cubic twists and congruent number families. As a corollary, I prove the 1879 conjecture of Sylvester, showing that for any prime p = 4,7,8 mod 9, x^3 + y^3 = p has a solution with x,y rational numbers. Moreover, combined with Smith's Selmer distribution results, I show that 100% of square free d = 5,6,7 mod 8 are congruent numbers, and hence establish Goldfeld's conjecture for the congruent number family y^2 = x^3 -d^2x and solve the congruent number problem in 100% of cases.

Previously, I have constructed supersingular Rankin-Selberg p-adic L-functions for imaginary quadratic fields in the style of Katz, Bertolini-Darmon-Prasanna and Liu-Zhang-Zhang, resolving questions about the existence of such p-adic L-functions dating back to the 70s. I also established special value formulas for these p-adic L-functions and explored their arithmetic applications. Before that, I did work with Chao Li on establishing the rank and p-parts of the Birch and Swinnerton-Dyer conjecture for quadratic twist families of elliptic curves over Q, in particular showing that a positive proportion of quadratic twists satisfy rank BSD whenever the curve has a rational 3-isogeny (thus verifying a weak version of Goldfeld's conjecture for such curves). We also establish similar results for the Mordell sextic twists family y^2 = x^3 + k, showing a positive proportion have rank 0 (resp. 1).

In the past, I also did work in geometric topology regarding the Heegaard-Floer and Khovanov homologies of knot theory. In particular, I did work with I. Kriz showing that the E3 term of Ozsvath-Szabo's spectral sequence from Khovanov homology to Heegaard-Floer homology is a link invariant. I also did work with P. Hu and I. Kriz on refinements of Khovanov homology into a module over the connective K-theory spectrum.

Research

Here are some of my recent papers in reverse chronological order. Preprints posted here may be more up to date than arXiv versions.

Supersingular p-adic L-functions, Maass-Shimura Operators, and Waldspurger Formulas, Annals of Mathematics Studies, vol (405), 2021. ISBN: 9780691216461

Supersingular main conjectures, Sylvester's conjecture and Goldfeld's conjecture. arxiv.org/abs/2002.04767

Prime twists of elliptic curves (with Chao Li), Mathematical Research Letters, vol. 26 no. 4, pp. 1187-1195, 2019.

Goldfeld's conjecture and congruences between Heegner points (with Chao Li), Forum of Mathematics, Sigma, Volume 7, 2019, e15.

A New p-adic Maass-Shimura operator and supersingular Rankin-Selberg p-adic L-functions, Ph.D. thesis, 2018.

Generalized Heegner cycles at Eisenstein primes and the Katz p-adic L-function, Algebra and Number Theory vol. 10, no 2, 2016, pp. 309-374. arxiv.org/abs/1512.05032 (minor differences from the published version)

Field theories, stable homotopy theory, and Khovanov homology (with Po Hu, Igor Kriz), Topology Proceedings, vol 48, 2016, pp. 327-360. arxiv.org/abs/1203.4773

A Galois cohomological proof of Gross's factorization theorem, 2015.

A spanning tree cohomology theory for links (with Igor Kriz), Advances in Mathematics, vol 255, 1 (April 2014), 414-454. arxiv.org/abs/1109.0064

On a conjecture concerning the maximal cross number of unique factorization indexed sequences, appeared in J. Number Theory, vol 133, 9 (September 2013), 3033-3056. This paper is a result of the research I did as a participant in the Duluth REU, run by Professor Joe Gallian at the University of Minnesota Duluth.

Expository

Here is an introductory article to Étale cohomology, written for the final project of my algebraic geometry class with Nick Katz in Spring 2012.

An Excursion into Étale Cohomology.

Here are the notes for my Fall 2020 STAGE talk on Dwork's p-adic proof of rationality of the zeta function.

Dwork's p-adic proof of rationality of the zeta-function.

Seminars

In Fall 2020 and Spring 2021, I organized an Iwasawa theory seminar at MIT in which I gave talks mainly on my recent work on Iwasawa theory for imaginary quadratic fields at height 2 primes.

In Spring 2021, I co-organized the MIT STAGE seminar. The topic this semester is crystalline and prismatic cohomology.

In Spring 2020, I co-organized the MIT STAGE seminar. The topic of the semester was p-adic modular forms.

Music

I have played piano since I was 5 years old and violin since I was 10. I also compose music, usually for solo piano, and sometimes for ensembles. I find great inspiration in playing, performing and composing. Below you will find pdfs (and perhaps recordings) of some of my compositions.

Snowfall, for solo piano.

Pan's March, for piano and voice (and chorus).

The Grind, for voice and rock band.

Lullaby, a simple piece for solo piano.

Email

dkriz@mit.edu

This page last modified June 7, 2022.