Research Statement — Debanjana Kundu
Introduction
My research lies at the intersection of number theory, algebra, arithmetic algebraic geometry and complex analysis;
more specifically I study the arithmetic of elliptic curves. These are precisely the genus 1 algebraic curves withrational points, and hence land in a sweet spot for mathematical enquiry. The arithmetic of simpler, genus 0,algebraic curves has been studied for centuries. On the other hand, higher genus curves are less accessible. A major
application was seen in Wiles’ proof of Fermat’s Last Theorem.
My primary focus is on Iwasawa theory. It is an area of number theory that emerged from the foundational workof Kenkichi Iwasawa in the 1950s. In Mazur (1972), the Iwasawa theory of Selmer groups of elliptic curves wasintroduced. The main goal is to study growth of arithmetic objects, like Galois modules over infinite towers ofnumber fields. The central theme is that it is often hard to study objects like class groups or Selmer groups inisolation; but in ‘nice’ families, they are more amenable since their properties stabilize.
A key observation is that a part of this growth exhibits regularity which can be described in terms of values ofmeromorphic functions known as L-functions, such as the Riemann zeta function. Through such descriptions,
Iwasawa theory unveiled intricate links between algebraic, geometric, and analytic objects of arithmetic nature.
The existence of such links is a common theme in many areas within arithmetic geometry. So Iwasawa theory has
found itself to be a subject of continued great interest. The literature on Iwasawa theory is often technical, but theunderlying ideas possess an undeniable beauty!
Iwasawa theory is heavily used in work on the Birch and Swinnerton-Dyer (BSD) conjecture, a Clay Math Millennium
Problem. The Iwasawa invariants associated to elliptic curves epitomize their arithmetic and Iwasawa theoretic
properties. Moreover, there is a deep relationship between the behaviour of Iwasawa invariants and the (p-adic) Birchand Swinnerton-Dyer formula. The first positive result to be proved in this direction, the Coates-Wiles theorem thatanalytic rank 0 implies algebraic rank 0 for elliptic curves over Q with complex multiplication (CM), was shownusing Iwasawa theory. More generally, almost all the results on BSD that we now have (thanks to Kolyvagin, Rubin,Kato, Perrin-Riou, Kobayashi, etc.) use the machinery of Iwasawa theory.
Iwasawa Theory of Fine Selmer Groups
In Coates and Sujatha (2005), the fine Selmer group of elliptic curves was defined. It is a subgroup of the Selmer
group with stronger finiteness properties, and it ‘interpolates’ between the class group and the Selmer group. Theyformulated a conjecture (Conjecture A) parallel to the Classical Iwasawa µ = 0 Conjecture: the conjecture predicts
that the fine Selmer group over the cyclotomic extension is a finitely generated Zp -module and it mimics precisely
the growth of the class group. On the contrary, the classical Selmer groups are often known to not satisfy thisproperty. Given a rank 0 elliptic curve, I have shown that for density 1 primes, the fine Selmer group is trivial in
cyclotomic extensions; thereby verifying Conjecture A. Lim and Murty (2016) showed that growth of p-ranks of fineSelmer group and ideal class groups in a such a tower is of the same order. Using these ideas, I have studied growthquestions in other infinite Abelian and non-Abelian towers (see Kundu (2020b), Kundu (2020c), and Kundu andRay (2021b)). Using analytic techniques, I improved results of Lim-Murty and Cesnavicius (2017) on growth of fineSelmer groups in degree p-extensions (see Kundu (2020a)). Iwasawa (1981) showed that class groups of number fields
are a good analogue of the p-power division points of the Jacobian variety of an algebraic curve and its λ-invariant is
an analogue of the genus of the curve. Combining this with the recently understood similarity in the growth patternof class groups and fine Selmer groups, I obtained a Riemann-Hurwitz type formula for λ-invariants of fine Selmergroups (see Kundu (2021)). In joint work with A. Lei, I have studied the q-part of the fine Selmer group in Zp -towers(where p ̸= q) and again made explicit the close relationship between fine Selmer groups and class groups.
Control Theorems play a key role in Iwasawa theory: they allow us to extract information about the Selmer groupand provide an invaluable approach towards studying the Birch and Swinnerton-Dyer Conjecture. In Kundu andLim (2021), we study Control Theorems for fine Selmer groups in arbitrary p-adic Lie extensions. This allows us todeduce properties of the fine Selmer group over an infinite tower from those at the finite layers (and vice versa).
Iwasawa Theory and Arithmetic Statistics
More recently I have started studying questions at the intersection of arithmetic statistics and Iwasawa theory. The
area of arithmetic statistics concerns the behaviour of number theoretic objects in families, and offers a probabilistic

model that seeks to explain numerous phenomena in the statistical behaviour of Selmer groups. In Kundu and Ray(2021a), we show that there is promise in the analysis of the average behaviour of Iwasawa invariants. In Kundu
et al. (2022a), have also explored related but more intricate questions in non-commutative extensions.
This subject area is relatively new and there are many unanswered questions. I am continuing to study questions
that have arisen from my previous projects. Two of the questions I am thinking about recently include:
1. To determine whether the λ-invariants for elliptic curves are bounded as p varies.
2. To determine when the λ-invariants are sharp bounds for the rank of the elliptic curve along the corresponding
cyclotomic extension.
Given a Diophantine equation, such as an elliptic curve E : y 2 = x3 + Ax + B where A and B are integers and
4A3 + 27B 2 ̸= 0, a natural question to ask is whether the equation has any integer solutions. If yes, then is thesolution-set finite or infinite? Another question of interest is whether the solution-set changes if we look for solutions
in more general rings (such as integers of number fields). Such questions fall under the umbrella of Diophantine
Stability. Using tools from Iwasawa theory and analytic number theory, in Beneish et al. (2021) we show that given
an elliptic curve with a finite number of solutions over Z, there are infinitely many number fields where the elliptic
curve does not gain any new solutions. Questions on Diophantine stability are known to have applications to Hilbert’stenth problem; an area I have recently started to explore, see for example Kundu et al. (2022b).
Other Research Interests
Questions in number theory can often be elementary to state, but challenging to solve. Success often relies on
establishing relationships with ostensibly unconnected areas of mathematics. One such connection is provided by theLanglands Program, which conceptualizes a deep structural relationship between number theory and representation
theory. There have been major advances made in recent years, however the Langlands Program remains largely
conjectural. Though only proven in special cases, the Langlands functoriality conjectures have become a cornerstoneof modern mathematics. The origins of these conjectures were in the area of automorphic forms, which lies at theintersection of number theory, representation theory, and harmonic analysis on Lie groups. The theory developed tounderstand and prove cases of the conjectures has become crucial to understanding these subjects, and has reachedbeyond them, leading to important applications in areas such as topology, algebraic geometry and mathematicalphysics. The trace formula is a powerful tool to study the analytic behaviour of automorphic L-functions. It is
one of the main techniques in attacking Langlands’ Functoriality Conjecture and related problems like the “BeyondEndoscopy” idea introduced in Langlands (2004).
The strategy of beyond endoscopy is a two-step process. The first step is to isolate via the trace formula the (packetsof) cuspidal automorphic representations whose L-functions (for a representation of the dual group) have a pole ats = 1. The second step involves a comparison of this data for two different groups and aims to determine functorialtransfers. The first step was completed in Altuğ (2015) where he worked with the group GL2 (Q). In a joint projectwith M. Emory, M. Espinosa Lara and T. A. Wong, we are utilizing this method to generalize the first step toGL2 (K) where K is a number field.
Undergraduate ResearchOver the past year, I have been working with two undergrads for their bachelors thesis. For me, it is important tomake mathematics as inclusive as possible and I want to encourage students from all walks of life to participate inmath research. It is rather hard to prove anything original in number theory in a short period of time. Therefore,depending on the background of the student and the time they can devote, I often find it more realistic to expectthat they will read an interesting topic in number theory or perform some numerical analysis. However, I have a few
project ideas for motivated undergraduate students:
1. Benford’s Law and elliptic curves: Benford’s Law is an observation that in many real-life sets of numerical data,
the leading digit is likely to be small. I think it might be interesting to understand whether the coefficients
arising in the Fourier expansion of an elliptic curve over Q satisfy Benford’s Law or not.
2. Bounds for Fibonacci period growth: It is well-known that any sequence of integers which satisfy a recurrence
relation becomes periodic when reduced mod n for any positive integer n. It would be interesting to investigate
the behaviour of the length of the period of the Fibonacci sequence or the Lucas sequence modulo n.

References
Salim Ali Altuğ. Beyond endoscopy via the trace formula: 1. Poisson summation and isolation of special representations. Compos. Math., 151(10):1791–1820, 2015.
Lea Beneish, Debanjana Kundu, and Anwesh Ray. Rank jumps and growth of Shafarevich–Tate groups for elliptic
curves in Z/pZ-extensions. 2021. arXiv preprint arXiv:2107.09166.
Kestutis Cesnavicius. p-Selmer growth in extensions of degree p. J. London Math. Soc., 95(3):833–852, 2017.
J. Coates and R. Sujatha. Fine Selmer groups of elliptic curves over p-adic Lie extensions. Math. Ann., 331(4):809–839, Apr 2005.
Kenkichi Iwasawa. Riemann-Hurwitz formula and p-adic Galois representations for number fields. Tohoku Mathematical Journal, Second Series, 33(2):263–288, 1981.
Debanjana Kundu. Growth of p-fine Selmer groups and p-fine Shafarevich-Tate groups in Z/pZ extensions. J.
Ramanujan Math. Soc., pages 1–10, 2020a.
Debanjana Kundu. Growth of fine Selmer groups in infinite towers. Canadian Math. Bull., pages 1–15, 2020b.
Debanjana Kundu. Growth of Selmer groups and fine Selmer groups in uniform pro-p extensions.
mathématiques du Québec, pages 1–16, 2020c.

AnnalesDebanjana Kundu. An analogue of Kida’s formula for fine Selmer groups of elliptic curves. J. Number Theory, 2021.
Debanjana Kundu and Meng Fai Lim. Control theorems for fine Selmer groups. 2021. accepted for publication (J.
Théor. Nombres Bordeaux).
Debanjana Kundu and Anwesh Ray. Statistics for Iwasawa invariants of elliptic curves. Trans. Am. Math. Soc.,
2021a.
Debanjana Kundu and Anwesh Ray. Anticyclotomic µ-invariants of residually reducible Galois representations.
Journal of Number Theory, 2021b. arxiv 2103.02092.
Debanjana Kundu, Antonio Lei, and Anwesh Ray. Arithmetic statistics and noncommutative Iwasawa theory.
Documenta Math., 27:89–149, 2022a.
Debanjana Kundu, Antonio Lei, and Florian Sprung. Studying Hilbert’s 10th problem via explicit elliptic curves.
2022b. arXiv preprint 2207.07021.
Robert P Langlands. Beyond endoscopy. Contributions to automorphic forms, geometry, and number theory, 611:
697, 2004.
Meng Fai Lim and V Kumar Murty. The growth of fine Selmer groups. J. Ramanujan Math. Soc., 31:79–94, 03 2016.
Barry Mazur. Rational points of abelian varieties with values in towers of number fields. Invent. Math., 18(3):183–266, Dec 1972.