Research Statement  Debanjana Kundu
1. Introduction
My primary research is on the Iwasawa theory of elliptic curves. I use tools from Galois cohomology, module theory,
algebraic and analytic number theory, and arithmetic statistics to answer questions on the structure and growth of
Selmer groups, ne Selmer groups, and class groups in innite (and nite) eld extensions.

Organization: In 2, basic denitions and notations are introduced. In 3, I discuss some of my results on growthquestions of ne Selmer groups. These results indicate that the ne Selmer group interpolates between the classgroup and the Selmer group. In 4, I explain some of my results which lie at the intersection of arithmetic statistics
and Iwasawa theory. If you are interested in knowing about my ongoing/ future projects, please send me an email
at dkundu@math.ubc.ca.

2. Basic Definitions and Notations
Let p be a xed prime. Consider the cyclotomic Zp -extension of Q, denoted by Qcyc . Set Γ := Gal
The Iwasawa algebra Λ = Λ(Γ) is the completed group algebra Zp JΓK := lim←−nγ of Γ; there is the following isomorphism of ringsn
Qcyc /Q ≃ Zp .

Zp [Γ/Γp ]. Fix a topological generator∼Λ−→ Zp JT K
γ 7→ 1 + T.

Let M be a conitely generated cotorsion Λ-module. The Structure Theorem of Λ-modules asserts that the Pontryagin
dual of M , denoted by M∨

, is pseudo-isomorphic to a nite direct sum of cyclic Λ-modules. In other words, there isa map of Λ-modulesM ∨ −→ 

s
M





Λ/(pmi ) ⊕ 

i=1with nite kernel and cokernel.

Here, mitMΛ/(hj (T ))

j=1> 0 and hj (T ) is a distinguished polynomial (i.e., a monic polynomial∨
The characteristic ideal of M
is (up to a unit) generated by thewith non-leading coecients divisible by p).

characteristic element,(p)fM (T ) := p

P

i mi

Y

hj (T ).

j(p)

The µ-invariant of M is dened as the power of p in fM (T ). More precisely,(µ(M ) = µp (M ) :=0Ps

if s = 0i=1 mi

if s > 0.

The λ-invariant of M is the degree of the characteristic element, i.e.

λ(M ) = λp (M ) :=tXdeg hj (T ).

j=1

3. Research Focus I: Fine Selmer Groups
The notion of ne Selmer group was formally introduced by J. Coates and R. Sujatha in [4] even though it had beenstudied by K. Rubin [49] and B. Perrin-Riou [45, 46], under various guises in the late 80's and early 90's. This is a
subgroup of the classical Selmer group obtained by imposing stronger vanishing conditions at primes above p (theprecise denition is reviewed in 3.1 below). A deep result of K. Kato shows that the ne Selmer group of an elliptic
curve over Qcyc is always Λ-cotorsion, regardless of whether the elliptic curve E/Q is ordinary at p or not, a factthat is not true for classical Selmer groups. The ne Selmer group is a fundamental object in the study of Iwasawa

theory and plays a crucial role in the reformulation of the Iwasawa Main Conjecture for elliptic curves without p-adic

L-functions (see [16, Conjecture 12.10] and [56, Conjecture 7]).

2
3.1.

Denition of ne Selmer groups. Suppose F is a number eld. Throughout, E/F is a xed elliptic curve.

Fix a nite set S of primes of F containing p, the primes dividing the conductor of E, as well as the archimedean
primes.

Denote by FS , the maximal algebraic extension of F unramied outside S .

For every (possibly innite)FS /L . Write S (L) for the set of primes of L above S . If L isa nite extension of F and w is a place of L, write Lw for its completion at w ; when L/F is innite, it is the unionof completions of all nite sub-extensions of L.
extension L/F contained in FS , write GS (L) = Gal

Denition. Let L/F be an algebraic extension. The p-primary ne Selmer group of E over L is dened as



M


Sel0 (E/L) = ker H 1 GS (L) , E[p∞ ] →H 1 Lv , E[p∞ ]  .
v∈SSimilarly, the p-ne Selmer group of E over L is dened as




M


Sel0 (E[p]/L) = ker H 1 GS (L) , E[p] →H 1 Lv , E[p]  .
v∈SEquivalently, the p-primary ne Selmer group is a subgroup of the classical Selmer group with additional vanishingconditions at v | p. More precisely,M
0 → Sel0 E/F → Sel E/F →E(Fv ) ⊗ Qp /Zp .
v|p
It is easy to observe that if F∞ /F is an innite extension,
Sel0 E/F∞ = lim Sel0 E/L ,
−→
L



Sel0 E[p]/F∞ = lim Sel0 E[p]/L ,−→L

where the inductive limits are taken with respect to the restriction maps and L runs over all nite extensions of F
contained in F∞ .

3.2.

µ = 0 Conjecture for ne Selmer groups. In [42], B. Mazur initiated the study of Iwasawa theory ofclassical Selmer groups of elliptic curves. Even over the cyclotomic Zp -extension Qcyc /Q, when the Selmer group isΛ-cotorsion (i.e., at an odd prime p of good ordinary reduction), he constructed examples of elliptic curves wherethe µ-invariant of the Selmer group is positive. Thus, providing evidence that class groups and Selmer groups oftendier in growth behaviour in innite extensions. On the other hand, when E/F is an elliptic curve with good ordinary
reduction at p and the residual Galois representation ρ : Gal



F /F



→ GL2 (Fp ) is irreducible, R. Greenberg has

conjectured that the associated µ-invariant of the Selmer group over Fcyc is trivial (see [9, Conjecture 1.11]).
Motivated by the classical Iwasawa µ = 0 conjecture for the cyclotomic Zp -extension and the observation that thegrowth behaviour of the ne Selmer group parallels that of the class group [4, Lemma 3.8], CoatesSujatha formulated
the following conjecture . Henceforth, this will be referred to as Conjecture A.

Conjecture A. Let p be an odd prime and E/F be an elliptic curve. Over Fcyc /F , the ne Selmer group Sel0 (E/Fcyc )is a conitely generated Zp -module. Equivalently, Sel0 (E/Fcyc ) is Λ-cotorsion and the associated µ-invariant, denotedby µfine (E/Fcyc ), is 0.
Much of my research has been driven by trying to understand this conjecture. Even though we are far from proving
this conjecture in full generality, I have made some modest contributions towards this (see Corollary 4.3). In my
doctoral thesis, I provided a large class of examples where the conjecture is true (see [25, 32]). More precisely,

Theorem 3.1. Let E/F be the base-change of a rational elliptic curve E/Q . Suppose that it has rank 0 over F andthat the ShafarevichTate group of E/F is nite. If E has CM by an order if an imaginary quadratic eld K , assume
further that the Galois closure of F , denoted by F c , contains K . Then the Selmer group Sel E/Fcyc is trivial for aset of prime numbers of density at least [F c1:Q] . In particular, Conjecture A holds for E/F at all such primes.
The key diculty in extending this result to elliptic curves dened over F is that we rely on [43] to show thatanomalous primes have density 0. Since these results are proven for normalized weight 2 eigencuspforms, we needto invoke the Modularity Theorem. This has now been extended to higher rank elliptic curves and primes of goodsupersingular reduction (see 4).

3
3.3.

Close relationship with class groups. In [40], M. F. Lim and V. K. Murty showed for the rst time thatthe class groups of number elds and ne Selmer groups are closely related in some nite extensions of number elds
and in Zp -extensions (other than the cyclotomic one) where primes are nitely decomposed.

Arbitrarily large µ-invariant. In my doctoral thesis, I explored similar questions in Z/pZ-extensions, nonp-adic analytic extensions, and other p-adic Lie extensions. A particular question I was interested in studying waswhether the growth properties of ne Selmer groups mimics that of class groups even in those p-adic Lie extensions
where primes are not nitely decomposed. K. Iwasawa had shown in [15] the existence of Zp -extensions of certain
number elds where the classical µ-invariant can be made arbitrarily large. In [23], I proved that an analogous result
holds for ne Selmer groups by comparing p-ranks of ne Selmer groups and class groups in towers of number elds,
and observing that the µ-invariant is closely related to the p-rank. More precisely,
3.3.1.

Theorem 3.2. Let F = Q(ζp ) be the cyclotomic eld of p-th roots of unity for p > 2. Let E/F be an elliptic curve

such that E(F )[p] ̸= 0. Given an integer N ≥ 1, there exists a cyclic Galois extension L/F of degree p and a
non-cyclotomic Zp -extension L∞ /L such that µfine (E/L∞ ) ≥ N .
In [24], I developed a strategy to show that the (generalized) µ-invariant of ne Selmer groups can be arbitrarily large
in extensions where the (generalized) µ-invariant associated to the class group is arbitrarily large. Using results of[11], I provided explicit examples of commutative and non-commutative p-adic Lie extensions with arbitrarily large(generalized) µ-invariant of ne Selmer groups. A striking feature in all these examples is that there are innitely
many primes which are innitely decomposed in these extensions. This raises the following question:

Question 3.3. Should one expect that for any Zp -extension, where primes are nitely decomposed the classicalIwasawa µ-invariant (i.e., associated to class groups) is 0? More generally, if F∞ /F is a (uniform) pro-p p-adic Lie
extension where primes are nitely decomposed, is the (generalized) µ-invariant trivial?
3.3.2. Anti-cyclotomic Zp -extension. In [41, Conjecture B], A. Matar extended Conjecture A to the anti-cyclotomic
Zp -extension, Kac , of an imaginary quadratic eld, K . He provided computational evidence for the same when the
mod-p representation of E is irreducible. In contrast, when the residual representation in reducible, we proved thefollowing result in [33] which again underlines the relationship between class groups and ne Selmer groups.

Theorem 3.4. Let E be an elliptic curve dened over an imaginary quadratic eld K . Assume that
(i )

(ii )

E(K)[p] ̸= 0 andthe Heegner hypothesis is satised.

Then the classical (anti-cyclotomic) Iwasawa µ-invariant, µ(Kac /K) = 0 if and only if Sel0 (E/Kac ) is a cotorsion
Λ-module with µfine (E/Kac ) = 0.
3.3.3.

p ̸= q Iwasawa theory. In [54, 55], L. C. Washington proved that for distinct primes p and q , the p-part ofthe class number stabilizes in the cyclotomic Zq -extension of an abelian number eld. These results were extended
by J. Lamplugh in [37] to other Zq -extensions where primes are nitely decomposed.

More precisely, if p, q are

distinct primes ≥ 5 that split in an imaginary quadratic eld K of class number 1 and F/K is a prime-to-p abelian
extension which is unramied at p, then the p-class group stabilizes in the Zq -extension of F which is unramied
outside precisely one of the primes above q . Using a theorem of H. Hida on the non-vanishing modulo p of algebraic

L-functions, we have extended these results to a class of anti-cyclotomic Zp -extensions in joint work with A. Lei [27].

Theorem 3.5. Let K be an imaginary quadratic eld of class number 1. Let p and q be distinct primes (≥ 5) which
split in K . Let g be a xed ideal of OK coprime to pq such that g is a product of split primes. Let F = R(gq) bea prime-to-p extension of K and R(gq ∞ )ac /F be the anti-cyclotomic Zq -extension. Then, there exists an integer N
such that for all n ≥ N ,ordp h (Fn ) = ordp h (FN ) ,where Fn is the n-th layer of the anti-cyclotomic Zq -extension.
Main idea of the proof: Proving the theorem involves the following main steps:
Step 1: A special case of a theorem of Hida [14] guarantees the non-vanishing modulo p of algebraic L-functions.
Step 2: Using a result of Lamplugh, show that certain p-primary Galois modules stabilize in the said anti-cyclotomic
Zq -extension. The theorem follows by an application of the Iwasawa Main Conjecture, which is known in this settingby the work of K. Rubin [48].
Even in this p ̸= q setting, it is possible to relate the growth of the p-part of the class group to the p-part of ne
Selmer group of a xed elliptic curve E over a Zq -tower. More precisely,

4

Theorem 3.6. Let p and q be distinct odd primes. Let F be any number eld and let E/F be an elliptic curve suchthat E(F )[p] ̸= 0. Let F∞ /F be any Zq -extension such that the primes aboveof bad reduction
 q and the primesof E are nitely decomposed. If there exists N ≫ 0 such that ordp h (Fn ) = ordp h (FN ) for all n ≥ N , thenSel0 (E/Fn ) = Sel0 (E/FN ).
In particular, the above theorem applies to the setting studied by Washington [54, 55].

A related question for exploration by early graduate students: I am hopeful that the strategy explainedin Section 3.3.1 can be combined with the results in this section. More precisely, I think it should be possible toshow that in Zq -extensions where primes of bad reduction of E split completely, the p-primary ne Selmer group has
unbounded (but predictable) growth.
The result I describe below is not related to the theory of ne Selmer groups. However, the questions arose naturally
while working on [27]. It is a classical problem to study the divisibility of the algebraic part of (Hecke) L-values by agiven prime p as one varies the (Hecke) characters of q -power conductor. W. Sinnott introduced the idea of relating
non-vanishing of such L-values modulo p to Zariski density (modulo p) of special points of the algebraic variety
underlying the L-values. Related questions have been studied by A. Burungale, T. Finis, Hida, and M. L. Hsieh on
anti-cyclotomic characters. In [28], which is joint work with Lei, we studied a generalization of the aforementioned
results to Hecke characters (not necessarily anti-cyclotomic) of q -power conductor over an imaginary quadratic eld.

Theorem 3.7. Let K be an imaginary quadratic eld over which p and q are both split and H be its Hilbert classeld. Suppose that q does not divide the class number of K and that both the prime ideals above q are principal in K .
Let E/H be an elliptic curve with CM by OK which has good reduction at primes above p and q . Let φ be the Hecke
character over K attached to E of conductorSf. Let g denote a principal integral ideal of K that is divisible by f andcoprime to pq . Let F = H(Egq ) and F∞ = n≥1 H(Egqn ) be the Z2q -extension over F . Let π be the uniformizer ofthe local eld Fv where v | p. Let ρ be a character of Gal(H/K) satisfying a technical hypothesis. Then, for a Zariski
dense set of nite-order characters κ of Gal F∞ /F ,



alg
k
κρφ
=0
ordπ Lfor k = 1, 2, . . . , p − 1.

Main idea of the proof: The proof consists of the following main ingredients:
Step 1: Establish a theory of Gamma transform of "elliptic function measures" on Z2q , which are measures that arise
from a rational function on an elliptic curve.

Step 2: Show that the π -adic valuations of the aforementioned Gamma transforms have the same p-adic valuation
2
for almost all nite characters on Zq .
Step 3: Show that by choosing an appropriate elliptic function measure arising from a rational function on a CM
elliptic curve, the Gamma transforms of this measure is related to the special values of L-series of interest. The
construction of this elliptic function measure is signicantly more involved than the analogous step in the work of
Lamplugh (since he assumes that K has class number 1). This step crucially uses the work of E. de Shalit.

Step 4: Show that the π -adic valuation discussed in Step 2 is zero.
3.4.

Close relationship with classical Selmer groups. Since the ne Selmer group is a subgroup of the classicalSelmer group, unsurprisingly these two arithmetic objects often show some similarity in their growth behaviour.
One instance where this has been observed, in a limited scope, is in the formula of the cyclotomic λ-invariants via a
Kida-type formula, see [26].
3.4.1.

Control Theorems. In [42], Mazur conjectured that the classical Selmer group Sel(E/Fcyc ) is Λ-cotorsion,and also provided the rst theoretical evidence towards the same.

Using theControl Theorem, he veried theconjecture when Sel(E/F ) is nite. This condition is satised precisely when the ShafarevichTate group over F is
nite and the elliptic curve E/F has MordellWeil rank 0.

Till date, this conjecture is known only when E is anelliptic curve over Q and F is an abelian extension of Q; see [16].
Via the Iwasawa Main Conjecture, the Selmer group Sel(E/Fcyc ) can be related to a p-adic L-function. Therefore,
Mazur's Control Theorem provides a channel to extract information on Sel(E/F ) from the main conjecture thereby
providing an invaluable approach towards studying the Birch and Swinnerton-Dyer Conjecture (see [16, 48, 52]). The
Control Theorem connects the Selmer groups at the nite layers with that over the innite tower, allowing one todeduce properties of this arithmetic object over the innite tower from those at the nite layers, and vice versa.

5
In joint work with M. F. Lim [31], we proved Control Theorems for ne Selmer groups. More precisely,(a) established estimates on the Zp -coranks of the kernel and cokernel of the restriction maps′

rF∞ /F ′ : Sel0 (E/F ′ ) −→ Sel0 (E/F∞ )Gal(F∞ /F )′for a p-adic Lie extension F∞ /F with intermediate sub-elds F /F .

′(b) showed how the module theoretic structure of Sel0 (E/F∞ ) determines the growth of Zp -coranks of Sel0 (E/F ) in′intermediate sub-elds F .

Zdp -extensions, multi-false-Tateextensions, and the trivializing extension obtained by adjoining to F all the p-power division points of theelliptic curve, E. In each of these cases, it is possible to show (under appropriate assumptions) that the kerneland cokernel of the restriction map are nite, and also establish growth estimates for their orders.

(c) obtained sharper results by specializing to three cases of p-adic Lie extensions:

Our results though stated only for elliptic curves go through for abelian varieties rather formally.

This has been

written down in [19] and has been used extensively in proving new asymptotic formulas for the growth of ideal classgroups and ne Selmer groups in multi-Zp -extensions. More recently, the ideas from our paper have been extended
to modular forms in [20] to study generalized Iwasawa invariants.
3.4.2.

Fine Selmer groups and duality. In [8], Greenberg established a criteria for when two nitely generated

Λ-modules are pseudo-isomorphic.

This result has been used to show that the Selmer groups of ordinary (resp.

non-ordinary) representations satisfy a functional equation in [8] (resp. [17, 1, 38]). A key ingredient in all theseworks is that the local Selmer conditions at p are exact annihilator of each other. Unfortunately, this is not truein the case of ne Selmer groups, since the local conditions at p are trivial. In joint work with J. Hatley, A. Lei,
and J. Ray [12], we investigated the link between ne Selmer groups of weight k modular forms and its dual. Moreprecisely, let f be a weight k(≥ 2) modular form and f be the conjugate modular form. Then,(a) for an integer i, several control theorems were proven for the ne Selmer groups of f (i) and f (k − i).
(b) via global duality and global Euler characteristic formulae, it could be shown that the criteria established by
Greenberg can be reinterpreted in terms of growth conditions on the localization maps.
(c) under hypotheses which could be veried computationally, using the control theorems it was shown that thegrowth conditions on certain localization maps suce to study the relation between the ne Selmer groups off (i) and f (k − i) over Qcyc /Q. In particular, if certain naturally arising growth conditions of localization maps
are satised, the µ-invariants of Sel0 (f (i)/Qcyc ) and Sel0 (f (k − i)/Qcyc ) are equal.

4. Research Focus II: Arithmetic Statistics and Iwasawa Theory
In a series of articles (with collaborators), I am exploring questions at the intersection of arithmetic statistics andIwasawa theory. The main goal is to understand the variation of the Iwasawa invariants as the triple (E, F, p) varies
such that E/F has good reduction at p. More precisely, I focus on studying the following interrelated problems.

Question 4.1.

(i) For a xed elliptic curve E/F , how do the Iwasawa invariants vary as p varies over all odd

primes p at which E has good reduction?
(ii) For a xed prime p and xed number eld F , how do the Iwasawa invariants vary as E varies over all elliptic
curves (with good reduction at p)?
(iii) Fix an elliptic curve E/Q with good reduction at p. How do the Iwasawa invariants of E/F vary when F varies
over a family of number elds?
4.1.

Iwasawa invariants in Qcyc /Q. In [34], we started exploring questions at the intersection of arithmetic statis-

tics and Iwasawa theory using the Euler characteristic. The Euler characteristic is dened as an alternating productof Galois cohomology groups. By the work of P. Schneider (see [50, 51]) and B. Perrin-Riou (see [44]), this invariant
is known to be given by the p-adic BSD formula for primes of good ordinary reduction. Thus, it captures information

about the size of the TateShafarevich group, the Tamagawa number, the anomalous primes, and the (global) torsion
points of the elliptic curve; thereby providing information about the Iwasawa invariants.
4.1.1.

As a rst step, Theorem 3.1 was extended to higher rank elliptic curves and to supersingular primes.

When E has supersingular reduction at p, the p-primary Selmer group, Sel(E/Qcyc ), is not Λ-cotorsion. This makes
the analysis of the algebraic structure of the Selmer group particularly dicult. Instead, one considers the plus andminus Selmer groups, denoted by Sel± (E/Qcyc ), which were introduced by S. Kobayashi in [21] and are known to beΛ-cotorsion. The Iwasawa invariants µ± and λ± associated with the ±-Selmer group are dened analogously. Even6in the supersingular case, there is sucient computational evidence suggesting that often the associated µ-invariants
vanish. Under standard hypotheses on the ShafarevichTate group, it is easy to show that the λ-invariant associatedto a (Λ-cotorsion) Selmer group is always at least as large as the MordellWeil rank of E. For rank 0 elliptic curves

over Q, the precise ±-Euler characteristic formula (associated to ±-Selmer groups) has been obtained by Lei and
Sujatha in [39]. This recent result was used crucially in proving the following theorem.

Theorem 4.2. Let E/Q be a xed rank 0 elliptic curve. Assume niteness of the ShafarevichTate group over Q.
Then, for all but nitely many primes primes of good supersingular reduction Sel± (E/Qcyc ) is trivial. In particular,
Sel0 (E/Qcyc ) is trivial for all but nitely many supersingular primes.
±

The nal assertion holds because for a prime of supersingular reduction, Sel0 (E/Qcyc ) is a subgroup of Sel(E/Qcyc ).

Combining Theorems 3.1 and 4.2, the next result is immediate as there are only a nite number of bad primes for E.

Corollary 4.3. Let E/Q be a xed rank 0 elliptic curve. Assume niteness of the ShafarevichTate group over Q.
Then, Conjecture A holds for density 1 primes.

In the higher rank setting, answering the question is more dicult. This is because of the presence of the (normalized)
p-adic regulator term in the Euler characteristic formula. In any case, the following theorem can be proven.

Theorem 4.4. Let E/Q be an elliptic curve such that its MordellWeil rank, rE ≥ 1. Then there exists an explicitly
determined set of good ordinary primes such that µ(E/Qcyc ) = 0 and λ(E/Qcyc ) = rE .

Remark 4.5. (i) Numerical data suggests that this explicitly determined set from the above theorem is a density
1 subset of the set of good ordinary primes. The obstruction in attaining an unconditional result is due to ourlimited knowledge on how often the normalized p-adic regulator is a unit.
(ii) An analogue of Theorem 4.4 in the supersingular setting can be proven analogously, provided one assumes theconjectural Euler characteristic formula in this setting.

In the direction of Question 4.1(ii), the following result from [34] allows distinguishing between when the λ-invariant

is exactly equal to the MordellWeil rank and when it is strictly greater than the rank.

Theorem 4.6. Let p ≥ 5 be a xed prime number. Let E(X) be the set of isomorphism classes of all elliptic curves

over Q with height ≤ X . Let J (X) be the subset of E(X) containing rank 0 elliptic curves E with good reduction atp, and Z(X) be a subset for which either of the following hold:
(i )

(ii )

if E has good ordinary reduction at p, then Sel(E/Qcyc ) = 0 or
if E has good supersingular reduction at p, then Sel± (E/Qcyc ) = 0.

Then,
#J (X)#Z(X)≥ lim sup− ϵ(p).
#E(X)
X→∞ #E(X)X→∞

lim sup

Here, ϵ(p) is an explicitly determined positive constant which approaches 0 (quickly) as p → ∞.
1p ), see [5].
By Goldfeld's Conjecture, it is expected that 1/2 the elliptic curves have rank 1. Therefore, one expects that

On average, the proportion of elliptic curves over Zp with good reduction at p (ordered by height) is (1 −#J (X)
1
=
#E(X)
2
X→∞

lim sup



1
1−
.
p

Theorem 4.6 indicates that for a positive proportion of elliptic curves Sel(E/Qcyc ) = 0; the proportion approaches

1/2 as p → ∞. In [35], it was possible to rene the results and prove that given any integer n, there is an explicit
lower bound for the density of the set of elliptic curves with good ordinary reduction at p for which λ + µ ≥ n. This
lower bound depends on p (and n), is strictly positive, and becomes smaller as p or n become larger. More precisely,

Theorem 4.7. Let n > 0 be an integer and p be an odd prime number. Assume that the ShafarevichTate group isnite for all elliptic curves E/Q . The set of elliptic curves E/Q with good ordinary reduction at p and the additional
property that µ + λ ≥ n, has positive density which can be explicitly determined.
4.2.

Iwasawa invariants in anti-cyclotomic Zp -extensions. In [13], the goal was to study problems raised in

Question 4.1 for rank 0 elliptic curves with good ordinary reduction at p over the anti-cyclotomic Zp -extensions ofan imaginary quadratic eld in both the denite and the indenite setting.

7
4.2.1.

Denite Case: Heegner hypothesis is not satised. This setting was studied by R. Pollack and T. We-ston in [47], and in their joint work with C.-H. Kim [18]. Here, the number of bad inert primes is odd, preventing
the existence of Heegner points. Consequently, there should be few rational points. Their work conrms this andshows that under various hypotheses, the anti-cyclotomic Selmer group is cotorsion. Hence, the story is somewhat
similar to the cyclotomic one. In particular, it is possible to prove an Euler characteristic formula for Sel(E/Kac ).
In the direction of Question 4.1(i), it is possible to prove that for non-CM elliptic curves, the exact order of growth
for the number of primes at which µ = 0 is closely related to the LangTrotter Conjecture.
In response to Question 4.1(ii) it can be shown that for rank 0 elliptic curves the answer is primarily dependent on
the variation of ShafarevichTate groups, which can be studied via the heuristics of C. Delaunay.
Question 4.1(iii) however is more subtle.

This question is largely dependent on the divisibility by p of the orderof the ShafarevichTate group upon base-change to Q(√−d) (as d > 0 varies). Even though it appears dicult toprovide (unconditional) theoretical results, there is computational data which suggests that often large values of p

do not divide the order of the ShafarevichTate group.

Renements in the case of supersingular reduction. In [36], we revisited Question 4.1(iii) with F. Sprung.

The goal of this paper was to understand the average behaviour of Iwasawa invariants at primes of supersingularreduction. The key dierence of this work from [13] is that it does not use the Euler characteristic formula. Instead,
it focuses on measuring how mild the assumptions in [18] and [47] are from a statistical point of view. A bit more
precisely, it asserts that the proportion of such imaginary quadratic elds is halved for each prime of bad reductionthat is split that would violate the key hypothesis of [18] were it inert. The main result is the following.

Theorem 4.8. Fix a pair (E/Q , p) so thatQr
E/Q is an elliptic curve with square-free conductor NE = i=1 qi , and(ii ) p > 3 is a prime at which E has good supersingular reduction, ρE,p is surjective, and k < r .
√Then the proportion of imaginary quadratic elds such that gcd disc Q( −d) , pNE = 1, the prime p splits in√√Q( −d), and Sel± (E/Q( −d)ac ) with associated µ-invariant equal to zero is at least is Λ-cotorsion is at least


pNE
∗
Q
·
1
−
c
.
p
2k+2 (p + 1) qi |NE (qi + 1)(i )Here, the constant c∗p is related to the CohenLenstra heuristics.
Remark 4.9. An analogous result√can be proven in the good ordinary setting as well. The only dierence is that p
can be either split or inert in Q( −d). So, the proportion is doubled.
4.2.2.

Indenite Case: Heegner hypothesis holds. When the Heegner hypothesis holds, the p-primary Selmer

group Sel(E/Kac ) is not Λ-cotorsion. The theory in this setting is vastly dierent. Many of the arguments used inproving the earlier results fail, and their analogues are often false. Importantly, in this setting, there is no known
formula for the Euler characteristic of Sel(E/Kac ). This issue can be circumvented by relating this Selmer group to

an auxiliary Selmer group which is Λ-cotorsion and then using recent progress towards the anti-cyclotomic Iwasawa
Main Conjectures made by A. BurungaleF. CastellaC.-H. Kim [3] to obtain an Euler characteristic formula forthe auxiliary Selmer group. It appears that answering Questions 4.1(i)-(iii) systematically in the indenite setting
is deeply intertwined with the theory of the BDP p-adic L-function and is currently out of reach. However, it waspossible to provide some partial answers and supplement the results with computational data.
4.3.

Iwasawa invariants in non-commutative p-adic Lie extensions. In [29], which is joint work with A. Lei

and A. Ray, we have extended the study of average Iwasawa invariants to the non-commutative setting. What makes
the task of determining Iwasawa invariants in this situation more challenging is that in a non-commutative p-adic
Lie extension F∞ /F , it is possible for primes other than p to ramify. Given a triple (E, p, F∞ ), the main task in thisarticle was study the variation of the algebraic structure of the Selmer group Sel(E/F∞ ) in three dierent contexts.
(a) Fix the pair (E, p) and let F∞ vary over a family of admissible extensions.
(b) Fix the pair (p, F∞ ) and let E vary over a subset of elliptic curves E/Q of rank 0.
(c) Fix an elliptic curve E and associate to each prime p, an extension F∞ in a natural way. Then, vary p over theprimes at which E has good ordinary reduction.

Using crucially the Euler characteristic formula, we studied these three questions in three distinct settings.

8(1) First, consider the Zp -extension of imaginary quadratic elds. This is a 2-dimensional abelian extension and2

a metabelian extension over Q. This case parallels the cyclotomic theory and in fact, the Euler characteristics

2for the Zp -extension and the cyclotomic Zp -extension coincide.
(2) The next step is to work with the simplest non-commutative 2-dimensional p-adic Lie extension, which isthe false Tate curve extension. Given primes p and ℓ, write1

F∞ := Q(µp∞ , ℓ pn : n = 1, 2, · · · )
F∞ /F ≃ Zp ⋊ Zp . The three questions of interest in this case can be explained as follows:(a) Fix an elliptic curve E of conductor NE and a prime p of good ordinary reduction of E. Consider thefamily of false Tate curve extensions obtained by varying ℓ ∤ NE p. The rst question involves studying
for what proportion of primes ℓ is the Selmer group trivial over F∞ .
(b) Fix the primes p and ℓ, and let E vary over all elliptic curves dened over Q ordered by height. The next
and set G = Galquestion is answered by calculating an upper bound for the proportion of elliptic curves for which theSelmer group is not trivial.

(c) Finally, x a rank 0 elliptic curve E/Q , a good prime ℓ, and let p vary over the primes at which E has goodordinary reduction. It is possible to show that for at least half of the primes p, the G-Euler characteristic
coincides with the Γ-Euler characteristic. When E has good supersingular reduction at ℓ, this happens
for exactly two-third of the primes p. For such primes p, the Selmer group over the false Tate curve
extension is trivial if and only if that over the cyclotomic Zp -extension of Q(µp ) is trivial.
(3) Finally, it is worth understanding the case of the trivializing extension F∞ /Q, generated by the p-primary′
torsion points of a non-CM elliptic curve (denoted by A, E0 and E in the three questions). Since G :=
Gal F∞ /Q is not a pro-p extension, it is not possible to exploit the theory of Akashi series and translate
the results on the G -Euler characteristic formula to conclusions regarding the pseudonullity of the Selmer
group over the innite extension. The main results in this direction are the following:(a) Fix a rank 0 elliptic curve E of conductor NE and a prime p of good ordinary reduction of E. There isa family of extensions obtained by varying a non-CM elliptic curve A/Q . For density 0 (but innitely
many) such elliptic curves A, the G -Euler characteristic is trivial.

(b) Fix p and a non-CM elliptic curve E0/Q . This xes the p-adic Lie extension QE0,p∞ /Q. As E varies

over all elliptic curves dened over Q and ordered by height, it is possible to calculate an upper boundfor the proportion of elliptic curves for which the G -Euler characteristic is not trivial.

′′
does not have CM, consider the Selmer group of Eover the p-adic Lie extension Q E [p ] /Q as p varies. For all but nitely many primes, the G -Euler
characteristic is equal to the Γ-Euler characteristic. Moreover, [34, Conjecture 3.17] predicts that thislatter quantity is trivial most of the time.

(c) For a pair of elliptic curves (E, E ) such that E

′

4.4.

∞

Diophantine stability. Questions pertaining to rank growth are of much interest to arithmetic geometers.

In [2], the aim was to study questions pertaining to Diophantine stability using tools and techniques from Iwasawa
theory. To answer questions on rank jump of elliptic curves upon base change, the natural thing to do was to study
the growth of a more tractable arithmetic object, i.e., the p-primary Selmer groups upon base change. In particular,
the following two questions were investigated using a Kida-type formula for λ-invariants (proven in [10]).

Question 4.10.

(i) Given an elliptic curve E/Q with trivial p-primary Selmer group, for what proportion of degreep cyclic extensions does the p-primary Selmer group remain trivial upon base-change.
(ii) Given p ̸= 2, 3, for how many elliptic curves over Q does there exist at least one Z/pZ-extension where thep-primary Selmer group remains trivial upon base-change.

The answer to the rst question is the following.

Theorem 4.11. Given an elliptic curve E/Q and a prime p ≥ 7 with µ(E/Qcyc ) = λ(E/Qcyc ) = 0, there are innitely
many Z/pZ-extensions of Q where the λ-invariant does not increase; in particular, the rank does not jump. Moreover,there are innitely many Z/pZ-extensions of Q where the MordellWeil group does not grow.

The assertion on rank growth follows from the fact that the MordellWeil rank is at most as large as the λ-invariant.
The nal assertion is an immediate consequence of a recent result of [7] on torsion growth. The proof shows thatthe λ-invariant does not jump in many Z/pZ-extensions. Unfortunately, the method falls short of proving a positive
proportion as predicted by a conjecture of C. DavidJ. FearnleyH. Kisilevsky.
A natural follow-up question is when does the p-primary Selmer group grow upon base-change. After establishing acriterion for either the rank to jump, or the order of the ShafarevichTate group to increase upon base-change, thenext result is proven by exploiting the relationship between λ-invariants and the Euler characteristic formula.

9

Theorem 4.12. Let p ≥ 5 be a xed prime and E/Q be an elliptic curve with good ordinary reduction at p. Supposethat the image of the residual representation is surjective, Sel(E/Q) is trivial, and µ(E/Qcyc ) = λ(E/Qcyc ) = 0. Then,
there is a set of primes of the form q ≡ 1 (mod p) with density at least (p−1)p2 (p+1) such that the p-primary Selmer
group becomes non-trivial in the unique Z/pZ-extension contained in Q(µq ).
The following theorem answers Question 4.10(ii).

Theorem 4.13. For a positive proportion of rank 0 elliptic curves dened over Q, there exists at least one Z/pZextension over Q disjoint from Qcyc , such that the p-primary Selmer group upon base-change is trivial.
4.5.

Hilbert's 10th Problem. In [30], which is joint work with A. Lei and F. Sprung, we studied the analogue ofHilbert's 10th Problem for rings of integers of number elds, which asks the following question:
Is Z a Diophantine subset of the ring of integers of a number eld L?
The strategy to prove such results is inspired by the work of N. Garcia-FritzH. Pasten, see [6]. However, there aresome important dierences which will be explained after mentioning the key ingredients that go into the proof.

Main idea of the proof : The proof has the following three main steps:

Step 1: The following assertion is a straightforward corollary of a result by A. Shlapentokh: if there exists a rank 0
elliptic curve over Q such that in the quadratic extension K/Q the rank jumps, and in the extension F/Q the rank

remains 0, then Hilbert's 10th problem has a negative solution for the composite number eld, L = K.F .
If F/Q is integrally Diophantine then by the Transitivity Property, L/Q is also integrally Diophantine.

Step 2: Find a rank 0 elliptic curve E/Q satisfying certain mild conditions and two families of number elds:• a family (of cubic number elds) such that rank of E/Q does not jump.
• another family of (quadratic) extensions such that rank jumps.

Step 3: Determine `how big' these families are. To count the number eld extensions where the rank is stabilized,
the authors of [6] crucially used Iwasawa theory. As is shown in [30], it is possible to provide a direct argument. To
count the number of quadratic extensions with rank jump requires the work of D. KrizC. Li [22] (or the work of
A. Smith [53] when working with the congruent number curves).

Improving the results of Garcia-FritzPasten. Here are the main dierences between this recent work and [6]:First, using a result from a previous work [29, Section 8 and Appendix] it is possible to rene the results of [6] andimprove the densities of both P and Q.

Theorem 4.14. There are explicit Chebotarev sets of primes P and Q, of density 169 and 487 , such√that√for all p ∈ P

and q ∈ Q, the analogue of Hilbert's 10th Problem is unsolvable for the ring of integers of L = Q( 3 p, −q).

Second, it was possible to provide a direct proof of the main theorem of [6] by proving the vanishing of certain

3-Selmer groups (rather than the niteness of 3∞ -Selmer group). This allows signicant weakening of the hypotheses
and the possibility to nd many auxiliary elliptic curves (not necessarily of positive minimal discriminant).

Theorem 4.15. Let
D = {7, 39, 95, 127, 167, 255, 263, 271, 303, 359, 391, 447, 479, 527, 535, 615, 623, 655, 679, 695}.
91
For all D ∈ D, there are explicit Chebotarev sets of primes P (independent of D) and QD , of density 16
and 12such that for
all p ∈ P and q ∈ QD , the analogue of Hilbert's 10th Problem is unsolvable for the ring of integers of√ √L = Q( 3 p, Dq).
By working with a pair of auxiliary elliptic curves, it is possible to improve signicantly the density of P , at theexpense of a smaller set of D and a lower density for the sets QD .

Theorem 4.16. Let D ∈ {7, 615}. There are explicit Chebotarev sets of primes P (independent of D) and QD , of1
density 103
for all p ∈ P and q ∈ QD , the analogue of Hilbert's 10th problem is unsolvable for the
128 and 36 , such that√ √ring of integers of L = Q( 3 p, Dq).
Finally, the more direct approach permits working with elliptic curves with good supersingular reduction at 3. This
provide the opportunity to use the congruent number curve.

11
Theorem 4.17. There is an explicit Chebotarev
set of primes P with density 16
such that Hilbert's 10th Problem is√ √unsolvable for the ring of integers of L = Q( 3 p, q) whenever q is a congruent number.

10
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