Statistics for Iwasawa invariants of elliptic curves
aa r X i v : . [ m a t h . N T ] F e b STATISTICS FOR IWASAWA INVARIANTS OF ELLIPTICCURVES
DEBANJANA KUNDU AND ANWESH RAY
Abstract.
We study the average behaviour of the Iwasawa invariants forthe Selmer groups of elliptic curves, setting out new directions in arithmeticstatistics and Iwasawa theory. Introduction
Iwasawa theory began as the study of class groups over infinite towers ofnumber fields. In [21], B. Mazur initiated the study of Iwasawa theory of ellipticcurves. The main object of study is the p -primary Selmer group of an ellipticcurve E , taken over the cyclotomic Z p -extension of Q . Mazur conjectured thatwhen p is a prime of good ordinary reduction, the p -primary Selmer group iscotorsion as a module over the Iwasawa algebra, denoted by Λ . This conjecturewas settled by K. Kato, see [17, Theorem 17.4].Note that the Iwasawa algebra Λ is isomorphic to the power series ring Z p J T K .The algebraic structure of the Selmer group (as a Λ -module) is encoded bycertain invariants which have been extensively studied. First consider the casewhen E has good ordinary reduction at p . By the p -adic Weierstrass PreparationTheorem, the characteristic ideal of the Pontryagin dual of the Selmer groupis generated by a unique element f ( p ) E ( T ) , that can be expressed as a power of p times a distinguished polynomial. The µ -invariant is the power of p dividing f ( p ) E ( T ) and the λ -invariant is its degree. R. Greenberg has conjectured thatwhen the residual representation on the p -torsion subgroup of E is irreducible,then the µ -invariant of the Selmer group vanishes, see [13, Conjecture 1.11].Further, if it is known that the p -primary part of the Tate Shafarevich group X ( E/ Q ) is finite, then one can show that λ -invariant is at least as large asthe Mordell-Weil rank of E (see Lemma . ). However, this λ may indeed bestrictly larger than the rank, and one of our main objectives is to determine itsbehaviour on average .When E has supersingular reduction at p , the Selmer group is not Λ -cotorsion.This makes the analysis of the algebraic structure of the torsion part of theSelmer group is particularly difficult. Instead, we consider the plus and mi-nus Selmer groups introduced by S. Kobayashi in [18], which are known to be Λ -cotorsion. The Iwasawa invariants µ + and λ + (resp. µ − and λ − ) of theplus (resp. minus) Selmer group are defined in an analogous manner. In thesupersingular case as well, there is much computational evidence towards theconjecture that the µ -invariants µ + and µ − vanish. Once again, under standardhypothesis on the Tate-Shafarevich group, both λ + and λ − are known to begreater than or equal to the Mordell-Weil rank of E (see Lemma . ). he main goal of this article is prove results about the variation of the Iwa-sawa invariants as the pair ( E, p ) varies such that E has good reduction (or-dinary or supersingular) at p . More precisely, we analyze the following twoseparate but interrelated problems.(1) For a fixed elliptic curve E , how do the Iwasawa invariants vary as p varies over all odd primes p at which E has good reduction?(2) For a fixed prime p , how do the Iwasawa invariants vary as E varies overall elliptic curves (with good reduction at p )?Greenberg studied the first question when E / Q has rank zero and p variesover the primes of good ordinary reduction (see [13, Theorems 4.1 and 5.1]).In Theorem 3.8, we generalize this result to include the case of supersingularprimes. We show that a conjecture of J. Coates and R. Sujatha on the vanishingof µ -invariants of fine Selmer groups holds for density one primes (see Corol-lary 3.9). Under natural assumptions, we prove similar results for higher rankelliptic curves (see Theorems 3.12 and 3.15). The results in both ordinary andsupersingular cases lead us to make the following conjecture (which is provedfor elliptic curves of rank zero). Conjecture.
Let E / Q be an elliptic curve of rank r E . For of the primes p at which E has good ordinary reduction (resp. supersingular), µ = 0 and λ = r E (resp. µ + = µ − = 0 and λ + = λ − = r E ).The second question is at the intersection of arithmetic statistics and Iwasawatheory. The area of arithmetic statistics concerns the behaviour of number the-oretic objects in families, and offers a probabilistic model that seeks to explainnumerous phenomena in the statistical behaviour of Selmer groups. The inves-tigations in this paper show that there is promise in the analysis of the averagebehaviour of Iwasawa invariants. The main results we prove are Theorems . , . and . . Our results indicate that it is reasonable to expect that for a fixedprime p , as we vary over all rank 0 elliptic curves over Q with good ordinary(resp. supersingular) reduction at p ordered by height, a positive proportionof them have trivial p -primary Selmer group (considered over the cyclotomic Z p -extension of Q ). In fact, the results suggest that the proportion of ellip-tic curves of rank with trivial p -primary Selmer group approaches as p → ∞ (see Conjecture . ).2. Background and Preliminaries
Γ := Gal( Q cyc / Q ) ≃ Z p . The Iwasawa algebra Λ is the completedgroup algebra Z p J Γ K := lim ←− n Z p [Γ / Γ p n ] . After fixing a topological generator γ of Γ , there is an isomorphism of rings Λ ∼ = Z p J T K , by sending γ − to the formalvariable T .Let M be a cofinitely generated cotorsion Λ -module. The Structure Theoremof Λ -modules asserts that the Pontryagin dual of M, denoted by M ∨ , is pseudo-isomorphic to a finite direct sum of cyclic Λ -modules. In other words, there is map of Λ -modulesM ∨ −→ s M i =1 Λ / ( p m i ) ⊕ t M j =1 Λ / ( h j ( T )) with finite kernel and cokernel. Here, m i > and h j ( T ) is a distinguishedpolynomial (i.e. a monic polynomial with non-leading coefficients divisible by p ).The characteristic ideal of M ∨ is (up to a unit) generated by the characteristicelement, f ( p )M ( T ) := p P i m i Y j h j ( T ) . The µ -invariant of M is defined as the power of p in f ( p )M ( T ) . More precisely, µ p ( M ) := ( if s = 0 P si =1 m i if s > . The λ -invariant of M is the degree of the characteristic element, i.e. λ p ( M ) := t X j =1 deg h j ( T ) . E be an elliptic curve over Q with good reduction at p . It shall beassumed throughout that the prime p is odd. Let N denote the conductor of E and set S to denote the set of primes which divide N p . Let Q S be the maximalalgebraic extension of Q which is unramified at the primes v / ∈ S . Set E [ p ∞ ] tobe the Galois module of all p -power torsion points in E ( Q ) .First, consider the case when E has good (ordinary or supersingular) reduc-tion at p . Let v be a prime in S . For any finite extension L/ Q contained in Q cyc , write J v ( E/L ) = M w | v H ( L w , E ) [ p ∞ ] where the direct sum is over all primes w of L lying above v . Then, the p -primary Selmer group over Q is defined as follows Sel p ∞ ( E/ Q ) := ker H (cid:0) Q S / Q , E [ p ∞ ] (cid:1) −→ M v ∈ S J v ( E/ Q ) . This Selmer group fits into a short exact sequence(2.1) → E ( Q ) ⊗ Q p / Z p → Sel p ∞ ( E/ Q ) → X ( E/ Q )[ p ∞ ] → , see [6]. Here, X ( E/ Q ) is the Tate Shafarevich group X ( E/ Q ) := H ( Q / Q , E [ p ∞ ]) → Y l H ( Q l / Q l , E [ p ∞ ]) . Now, define J v ( E/ Q cyc ) = lim −→ J v ( E/L ) here L ranges over the number fields contained in Q cyc and the inductive limitis taken with respect to the restriction maps. Taking direct limits, the p -primarySelmer group over Q cyc is defined as follows Sel p ∞ ( E/ Q cyc ) := ker H (cid:0) Q S / Q cyc , E [ p ∞ ] (cid:1) −→ M v ∈ S J v ( E/ Q cyc ) . As mentioned in the introduction, when p is a prime of good supersingularreduction, the Pontryagin dual of Sel p ∞ ( E/ Q cyc ) is not Λ -torsion. In this case,one studies the plus and minus Selmer groups which we describe below.Let E / Q be an elliptic curve with supersingular reduction at p . Denote by Q cyc n the n -th layer in the cyclotomic Z p -extension, with Q cyc0 := Q . Set b E to bethe formal group of E over Z p . Let L be a finite extension of Q p with valuationring O L , set b E ( L ) denote b E ( m L ) , where m L is the maximal ideal in L . Write p for the (unique) prime above p in Q cyc , and for the prime above p in every finitelayer of the cyclotomic tower. First, define the plus and minus norm groups asfollows b E + ( Q cyc n, p ) := n P ∈ b E ( Q cyc n, p ) | tr n/m +1 ( P ) ∈ b E ( Q cyc m, p ) , for ≤ m < n and m even o , b E − ( Q cyc n, p ) := n P ∈ b E ( Q cyc n, p ) | tr n/m +1 ( P ) ∈ b E ( Q cyc m, p ) , for ≤ m < n and m odd o , where tr n/m +1 : b E ( Q cyc n, p ) → b E ( Q cyc m +1 , p ) denotes the trace map with respect tothe formal group law on b E . The completion Q cyc p is the union of completions S n ≥ Q cyc n, p , and set b E ± ( Q cyc p ) := S n ≥ b E ± ( Q cyc n, p ) . Define J ± p ( E/ Q cyc ) := H ( Q cyc p , E [ p ∞ ]) b E ± ( Q cyc p ) ⊗ Q p / Z p , where the inclusion b E ± ( Q cyc p ) ⊗ Q p / Z p ֒ → H ( Q cyc p , E [ p ∞ ]) is induced via the Kummer map. For v ∈ S \ { p } , set J ± v ( E/ Q cyc ) to be equalto J v ( E/ Q cyc ) . The plus and minus Selmer groups are defined as follows Sel ± p ∞ ( E/ Q cyc ) := ker H (cid:0) Q S / Q cyc , E [ p ∞ ] (cid:1) → M v ∈ S J ± v ( E/ Q cyc ) . For each choice of sign ‡ ∈ { + , −} , set f ( p ) , ‡ E ( T ) to denote the characteristicelement of Sel ‡ p ∞ ( E/ Q cyc ) ∨ , with µ ‡ p ( E ) , λ ‡ p ( E ) defined analogously.Let E be an elliptic curve with good (ordinary or supersingular) reductionat p ≥ . In order to state results in both ordinary and supersingular case atonce, we set for the remainder of the article the following notation, Sel ‡ p ∞ ( E/ Q cyc ) := ( Sel ± p ∞ ( E/ Q cyc ) if E has supersingular reduction at p, where ‡ = ± , Sel p ∞ ( E/ Q cyc ) if E has ordinary reduction at p. .3. Next, we introduce the fine Selmer group. Let E be an elliptic curve over Q and p be any odd prime. At each prime v ∈ S , set K v ( E/ Q cyc ) := M η | v H ( Q cyc η , E [ p ∞ ]) . The p -primary fine Selmer group of E is defined as follows Sel p ∞ ( E/ Q cyc ) := ker H ( Q S / Q cyc , E [ p ∞ ]) −→ M v ∈ S K v ( E/ Q cyc ) . By the result of Kato mentioned in the introduction, the Pontryagin dual of
Sel p ∞ ( E/ Q cyc ) is known to be a cotorsion Λ -module independent of the reduc-tion type at p . Further, it is conjectured that this fine Selmer group is a cotorsion Z p -module (i.e. the corresponding µ -invariant vanishes), see [7, Conjecture A].2.4. In what follows, M will be a cofinitely generated cotorsion Λ -module. Notethat H i (Γ , M) is always zero for i ≥ . Lemma 2.1.
Let M be a cofinitely generated cotorsion Λ -module. Then, corank Z p M Γ = corank Z p M Γ . Proof.
Note that H (Γ , M) may be identified with M Γ (see [22, Proposition1.7.7]). It follows from [15, Theorem 1.1] that corank Λ M = corank Z p H (Γ , M) − corank Z p H (Γ , M) . Since M is assumed to be cotorsion over Λ , the result follows. (cid:3) When the cohomology groups H (Γ , M) and H (Γ , M) are finite, the (classi-cal) Euler characteristic χ (Γ , M) is defined as the alternating product χ (Γ , M) = Y i ≥ (cid:16) i (Γ , M) (cid:17) ( − i . On the other hand, when the cohomology groups H i (Γ , M) are not finite, thereis a generalized version denoted by χ t (Γ , M) . Since H (Γ , M) is isomorphic tothe group of coinvariants H (Γ , M) = M Γ , there is a natural map Φ M : M Γ → M Γ sending x ∈ M Γ to the residue class of x in M Γ . We say that the truncated Eulercharacteristic is defined if the kernel and cokernel of Φ M are finite. In this case,the χ t (Γ , M) is defined to be the following quotient, χ t (Γ , M) := M ) M ) . It is easy to check that when χ (Γ , M) is defined, so is χ t (Γ , M) . In fact, χ t (Γ , M) = χ (Γ , M) . Express the characteristic element f ( p )M ( T ) as a polynomial, f ( p )M ( T ) = c + c T + · · · + c d T d . et r M denote the order of vanishing of f ( p )M ( T ) at T = 0 . For a, b ∈ Q p , wewrite a ∼ b if there is a unit u ∈ Z × p such that a = bu . Lemma 2.2 (S. Zerbes) . Let M be a cofinitely generated cotorsion Λ -module.Assume that the kernel and cokernel of Φ M are finite. Then,(1) r M = corank Z p (M Γ ) = corank Z p (M Γ ) .(2) c r M = 0 .(3) c r M ∼ χ t (Γ , M) . Proof.
See [40, Lemma 2.11]. (cid:3)
In particular, the classical Euler characteristic χ (Γ , M) is defined if and onlyif r M = 0 . When this happens, the constant coefficient c ∼ χ (Γ , M) .We specialize the discussion on Euler characteristics to Selmer groups of ellip-tic curves. Let ‡ be a choice of sign and recall the definition of Sel ‡ p ∞ ( E/ Q cyc ) .When E has good ordinary reduction at p , the choice of ‡ is irrelevant. De-note by χ ‡ t (Γ , E [ p ∞ ]) (resp. χ ‡ (Γ , E [ p ∞ ]) ) the truncated (resp. classical) Eulercharacteristic of the Selmer group Sel ‡ p ∞ ( E/ Q cyc ) . The invariants µ ‡ p ( E ) and λ ‡ p ( E ) simply refer to µ p ( E ) and λ p ( E ) when E has good ordinary reduction at p . When E has good ordinary reduction at p , we shall drop the sign ‡ from thenotation. It follows from Lemma . that the truncated Euler characteristic isalways an integer.3. Results for a fixed elliptic curve and varying prime
In this section, we study the variation of the classical and the truncated Eulercharacteristic as p varies. Fix a pair ( E, p ) such that(1) p is odd.(2) E is defined over Q and has good reduction at p .We record some lemmas which are required throughout this section. Recallthat when E has good ordinary reduction, Sel ‡ p ∞ ( E/ Q cyc ) simply refers to Sel p ∞ ( E/ Q cyc ) . Theorem 3.1.
Let ‡ ∈ { + , −} be a choice of sign. Let E / Q be an elliptic curvewith good reduction at p > . Then, there is a natural map Sel p ∞ ( E/ Q ) → Sel ‡ p ∞ ( E/ Q cyc ) Γ with finite kernel and cokernel. Proof.
When E has good ordinary reduction at p , this follows from Mazur’scontrol theorem, see [21] or [13, Theorem 1.2]. In the good supersingular re-duction case, the result follows from the proof of [10, Lemma 3.9] (see also [28,Proposition 5.1]). (cid:3) The following lemma gives a criterion for when the classical Euler character-istic χ ‡ (Γ , E [ p ∞ ]) is well defined. Lemma 3.2.
Let ‡ ∈ { + , −} be a choice of sign. Let E / Q be an elliptic curvewith good reduction at p > . The following are equivalent.(1) The classical Euler characteristic χ ‡ (Γ , E [ p ∞ ]) is well defined. Sel ‡ p ∞ ( E/ Q cyc ) Γ is finite.(3) The Selmer group Sel p ∞ ( E/ Q ) is finite.(4) The Mordell-Weil group E ( Q ) is finite, i.e. the Mordell-Weil rank is 0. Proof.
Recall that the Selmer group
Sel ‡ p ∞ ( E/ Q cyc ) is a cotorsion Λ -module. ByLemma . , Sel ‡ p ∞ ( E/ Q cyc ) Γ is finite if and only if Sel ‡ p ∞ ( E/ Q cyc ) Γ is finite. Thisshows that (1) and (2) are equivalent.Theorem . asserts that there is a natural map Sel p ∞ ( E/ Q ) → Sel ‡ p ∞ ( E/ Q cyc ) Γ , with finite kernel and cokernel. Hence, the conditions (2) and (3) are equivalent.By the work of V. Kolyvagin, X ( E/ Q ) is known to be finite when E ( Q ) isfinite (see [19]). Thus, it follows from (2.1) that conditions (3) and (4) areequivalent. (cid:3) Lemma 3.3.
Let ‡ ∈ { + , −} be a choice of sign. Let E / Q be an elliptic curvewith good reduction at p > . Assume that χ ‡ t (Γ , E [ p ∞ ]) is well defined and X ( E/ Q )[ p ∞ ] is finite. Let r E denote the Mordell-Weil rank of E ( Q ) . Then, r E is equal to the order of vanishing of f ( p ) , ‡ E ( T ) at T = 0 . In particular, we havethat λ p ( E ) ≥ r E . Proof.
Since it is assumed that X ( E/ Q )[ p ∞ ] is finite, it follows from (2.1) that r E = corank Z p Sel p ∞ ( E/ Q ) . By Lemma . , it suffices to show that r E = corank Z p Sel ‡ p ∞ ( E/ Q cyc ) Γ . Therefore the result follows from Theorem . . (cid:3) Lemma 3.4.
Let M be a cofinitely generated and cotorsion Λ -module such that φ M has finite kernel and cokernel. Let r M be the order of vanishing of f ( p )M ( T ) at T = 0 . Then, the following are equivalent.(1) χ t (Γ , M) = 1 ,(2) µ (M) = 0 and λ (M) = r M . Proof.
Suppose that χ t (Γ , M) = 1 . Write f ( p )M ( T ) = T r M g ( p )M ( T ) where g ( p )M ( T ) ∈ Λ and g M (0) = 0 . By Lemma . , (cid:12)(cid:12)(cid:12) g ( p )M (0) (cid:12)(cid:12)(cid:12) − p = χ t (Γ , M) = 1 . In particular, f ( p )M ( T ) and g ( p )M ( T ) are distinguished polynomials. Since g ( p )M (0) is a unit, it follows that g ( p )M ( T ) is a unit. Since g ( p )M ( T ) is a distinguishedpolynomial, it follows that g ( p )M ( T ) = 1 and f ( p )M ( T ) = T r M . Therefore, µ (M) = 0 and λ ( M ) = deg f ( p ) M ( T ) = r M .Conversely, suppose that µ (M) = 0 and λ ( M ) = r M . Since µ (M) = 0 , itfollows that f ( p )M ( T ) and g ( p )M ( T ) are distinguished polynomials. The degree of ( p )M ( T ) is λ M = r M . It follows that g ( p )M ( T ) is a constant polynomial and hence, g ( p )M ( T ) = 1 . By Lemma . , χ t (Γ , M ) = (cid:12)(cid:12) g M (0) (cid:12)(cid:12) − p = 1 . (cid:3) Proposition 3.5.
Let ‡ ∈ { + , −} be a choice of sign. Let E / Q be an ellipticcurve with good reduction at p > . Suppose that these additional conditionshold.(i) The truncated Euler characteristic χ ‡ t (Γ , E [ p ∞ ]) is defined.(ii) X ( E/ Q )[ p ∞ ] is finite.Let r E be the Mordell-Weil rank of E ( Q ) . Then the following statements areequivalent(1) χ ‡ t (Γ , E [ p ∞ ]) = 1 ,(2) µ ‡ p ( E ) = 0 and λ ‡ p ( E ) = r E . Proof.
Lemma . asserts that r E is equal to the order of vanishing of f ( p ) , ‡ E ( T ) at T = 0 . The assertion follows from Lemma . . (cid:3) Elliptic curves over Q with rank zero. In this subsection, we studythe variation of the classical Euler character as p varies over primes of goodreduction. Corollary 3.6.
Let E be an elliptic curve over Q with good reduction at p > for which the Mordell-Weil rank of E is zero. The following are equivalent.(1) χ ‡ (Γ , E [ p ∞ ]) = 1 ,(2) µ ‡ p ( E ) = 0 and λ ‡ p ( E ) = 0 .(3) Sel ‡ p ∞ ( E/ Q cyc ) is finite.(4) Sel ‡ p ∞ ( E/ Q cyc ) = 0 . Proof.
Since E is assumed to have rank zero, the Tate-Shafarevich group X ( E/ Q ) is finite. It follows from Lemma . that the Euler characteristic χ ‡ (Γ , E [ p ∞ ]) is well defined. By Proposition . that (1) and (2) are equivalent. From theStructure Theorem of finitely generated Λ -modules, it is clear that (2) and (3)are equivalent conditions. It is known that Sel ‡ p ∞ ( E/ Q cyc ) contains no properfinite index submodules (see [13, Proposition 4.14] for the case when p is ordi-nary and [10, Theorem 3.14] for when p is supersingular). Hence, (3) and (4)are equivalent. (cid:3) Let E / Q be an elliptic curve. Denote by S bad the finite set of primes atwhich E has bad reduction. Denote by S good the primes at which E has goodreduction; write S good = S ord ∪ S ss . When E has good reduction at a fixedprime p , we set e E to denote the reduced curve over F p .The following result was initially proved by Greenberg for good ordinaryprimes. Theorem 3.7.
Let E be an elliptic curve over Q such that E ( Q ) is finite. Let Σ ⊂ S ord be the set of primes at which p divides e E ( F p ) . Let Σ ′ ⊂ S ord be thefinite set of primes p such that either p = 2 .(2) p divides X ( E/ Q ) .(3) p divides the Tamagawa product Q l ∈ S bad c l ( E ) .Then for all primes p ∈ S ord \ (Σ ∪ Σ ′ ) , we have that Sel p ∞ ( E/ Q cyc ) = 0 .Observe that Σ ′ is always a finite set. However, the set of primes Σ , referredto as the set of anomalous primes , is known to be a set of Dirichlet density zero.The conjecture of Lang and Trotter predicts that the proportion of anomalousprimes p < X is C · √ X log X for some constant C . It has recently been shownthat all but a density zero set of elliptic curves have infinitely many anomalousprimes [1, Corollary 4.3]. However, in special cases one can in fact show that C = 0 [29, §1.1]. By the Hasse inequality, a prime p ≥ is anomalous for anelliptic curve defined over Q if and only if a p = p + 1 − e E ( F p ) = 1 . When p ≥ and an elliptic curve has 2-torsion, a p is even and hence C = 0 . For moreexamples where the set Σ is finite, and detailed discussion on this subject, werefer the reader to [21, 23, 26]. Proof of Theorem . . Since the Mordell-Weil rank of E is assumed to be zero,the Euler characteristic χ (Γ , E [ p ∞ ]) is well defined. The following formula forthe Euler characteristic is well known(3.1) χ (Γ , E [ p ∞ ]) ∼ X ( E/ Q )[ p ∞ ] × (cid:0)Q l ∈ S bad c l ( E ) (cid:1) × (cid:16) e E ( F p ) (cid:17) (cid:0) E ( Q )[ p ∞ ] (cid:1) , see [6, Theorem 3.3]. Since the Euler characteristic is an integer, the resultfollows immediately from Corollary . . (cid:3) We prove a similar result for primes of supersingular reduction.
Theorem 3.8.
Let E be an elliptic curve over Q such that E ( Q ) is finite. Let Υ ⊂ S ss be the finite set of primes p such that either(1) p = 2 .(2) p divides X ( E/ Q ) .(3) p divides the Tamagawa product Q l ∈ S bad c l ( E ) .Then for all primes p ∈ S ss \ Υ , we have that Sel + p ∞ ( E/ Q cyc ) = 0 and Sel − p ∞ ( E/ Q cyc ) = 0 . Proof.
Since E ( Q ) is assumed to be finite, the Euler characteristic χ ± (Γ , E [ p ∞ ]) is well defined. Since E has supersingular reduction at p , E [ p ] is irreducible asa G Q -module; hence, E ( Q )[ p ] is the trivial group. Finiteness of E ( Q ) impliesthat E ( Q ) ⊗ Q p / Z p = 0 . Therefore p ∞ ( E/ Q ) = X ( E/ Q )[ p ∞ ] . By [20, Theorem 5.15], we know that(3.2) χ ± (Γ , E [ p ∞ ]) = X ( E/ Q )[ p ∞ ] × Y l ∈ S bad c l ( E ) . The result follows from Corollary . . (cid:3) he following result gives evidence for the conjecture of Coates and Sujathaon the vanishing of the µ -invariant of fine Selmer groups. In fact, more is true. Corollary 3.9.
Let E be a rank 0 elliptic curve defined over Q . Then, the p -primary fine Selmer group Sel p ∞ ( E/ Q cyc ) is trivial for density one primes. Proof.
Given any elliptic curve, it has good reduction at all but finitely manyprimes. When p is a prime of good (ordinary or supersingular) reduction, notethat the p -primary fine Selmer group is a subgroup of Sel ‡ p ∞ ( E/ Q cyc ) . The resultis immediate from Theorems 3.7 and 3.8. (cid:3) Elliptic curves over over Q with positive rank. Next, we considerthe case when the elliptic curve E has positive rank.When E has good ordinary reduction at p , there is a notion of the p -adicheight pairing, which is the p -adic analog of the usual height pairing, and wasstudied extensively in [31, 32]. This pairing is conjectured to be non-degenerate,and the p -adic regulator R p ( E/ Q ) is defined to be the determinant of thispairing. Theorem 3.10. [B. Perrin-Riou [25], P. Schneider [32]] Let E / Q be an ellipticcurve of Mordell-Weil rank r E with good ordinary reduction at p . Assume that(1) X ( E/ Q )[ p ∞ ] is finite.(2) the p -adic Height pairing is non-degenerate, i.e., the p -adic regulator isnon-zero.Express the characteristic element f ( p ) E ( T ) as T r E g ( p ) E ( T ) . Then, g ( p ) E (0) = 0 and g ( p ) E (0) ∼ (cid:16) R p ( E/ Q ) p rE (cid:17) × X ( E/ Q )[ p ∞ ] × (cid:0)Q l ∈ S bad c l ( E ) (cid:1) × (cid:16) e E ( F p ) (cid:17) (cid:0) E ( Q )[ p ∞ ] (cid:1) . The above theorem implies that the right hand side of the equation is aninteger since the left hand side is.
Corollary 3.11.
Let E / Q be an elliptic curve of Mordell-Weil rank r E withgood ordinary reduction at p . Assume that(1) X ( E/ Q )[ p ∞ ] is finite.(2) the p -adic Height pairing is non-degenerate.(3) the truncated Euler characteristic χ t (Γ , E [ p ∞ ]) is defined.Then, we have that χ t (Γ , E [ p ∞ ]) ∼ (cid:16) R p ( E/ Q ) p rE (cid:17) × X ( E/ Q )[ p ∞ ] × (cid:0)Q l ∈ S bad c l ( E ) (cid:1) × (cid:16) e E ( F p ) (cid:17) (cid:0) E ( Q )[ p ∞ ] (cid:1) . Proof.
The assertion a direct consequence of Theorem . and Lemma . . (cid:3) To analyze Iwasawa invariants on average, we shall apply Theorem . . Let E / Q be an elliptic curve. As before, we denote by S ord the set of primes p atwhich E has good ordinary reduction. Let Σ be the set of anomalous primesand Σ ′ the finite set of primes for which(1) p = 2 , p divides X ( E/ Q ) .(3) p divides the Tamagawa product Q l ∈ S bad c l ( E ) .Denote by v p the p -adic valuation on Q p normalized by v p ( p ) = 1 . Let Π ⊂ S ord be the set of primes p at which v p ( R p ( E/ Q )) ≥ r E . In other words, it is theset of primes for which p divides (cid:16) R p ( E/ Q ) p rE (cid:17) . When r E = 0 , the p -adic regulator R p ( E/ Q ) = 1 , therefore the set Π is empty. On the other hand, the set ofprimes Π need not be empty when r E ≥ . The following result generalizesTheorem . and is an easy consequence of Theorem . . Theorem 3.12.
Let E / Q be an elliptic curve such that r E ≥ . Then for allprimes p ∈ S ord \ (Σ ∪ Σ ′ ∪ Π) , we have that f ( p ) E ( T ) = T r E . In particular, µ p ( E ) = 0 and λ p ( E ) = r E . Proof.
Express the characteristic element f ( p ) E ( T ) as T r E g ( p ) E ( T ) . Theorem . asserts that g ( p ) E (0) = 0 and g ( p ) E (0) ∼ (cid:16) R p ( E/ Q ) p rE (cid:17) × X ( E/ Q )[ p ∞ ] × (cid:0)Q l ∈ S bad c l ( E ) (cid:1) × (cid:16) e E ( F p ) (cid:17) (cid:0) E ( Q )[ p ∞ ] (cid:1) . Thus, g ( p ) E (0) is a p -adic unit for p ∈ S ord \ (Σ ∪ Σ ′ ∪ Π) . Therefore, g ( p ) E ( T ) isa unit in Λ . Since f ( p ) E ( T ) is a product of a distinguished polynomial with apower of p , it follows that g ( p ) E ( T ) = 1 and f ( p ) E ( T ) = T r E . (cid:3) While it is known that the set Σ ′ is finite and Σ is a density zero set of primes,the same is not known for Π . For N ≥ , let Π ≤ N be the set of primes p ∈ Π for which ≤ p ≤ N . Calculations in [5] suggest that the set Π is likely tohave Dirichlet density zero, at least for elliptic curves of rank (see also [39]).We compute Π ≤ for the first 10 elliptic curves E/ Q of rank (ordered byconductor). The calculations in the following table were done on sage.Cremona Label Π ≤ Cremona Label Π ≤ . a ∅ . a ∅ . a { } . a { , } . d { } . a { } . a ∅ . c ∅ . b ∅ . a {29}. Remark 3.13.
For elliptic curves E / Q with Mordell-Weil rank r E = 1 , theabove theorem asserts that for primes outside a set (possibly of Dirichlet densityzero), the characteristic element f ( p ) E ( T ) = T . Under reasonable hypothesisthat the p -adic height pairing on E ( Q cyc n ) is non-degenerate for all n and that X ( E/ Q cyc n )[ p ∞ ] is finite for all n , it is easy to see that f ( p )Sel ( T ) = 1 (see[38, pp. 104-105]). Thus, for a large class of examples, the fine Selmer group el p ∞ ( E/ Q cyc ) is not only cofinitely generated as a Z p -module, but in fact itis finite. On the other hand, when r E > even though the fine Selmer groupshould be cofinitely generated as a Z p -module, it is not expected to be finite.Next, we consider the supersingular case. Let E / Q be an elliptic curve withgood supersingular reduction at p ≥ . Note that p divides a p := p +1 − e E ( F p ) .Hasse’s bound states that (cid:12)(cid:12) a p (cid:12)(cid:12) < √ p ; hence if p ≥ , it forces that a p = 0 .However, when p = 3 , it is indeed possible for a p = 0 . For simplicity, we shallassume that a p = 0 . In this setting, the Main conjectures were formulated in[18]. When a p = 0 , the Main conjecture has been proved in a preprint of X.Wan (see [36]). Let ‡ ∈ { + , −} be a choice of sign. Perrin Riou [24] formulateda p -adic L-function in the supersingular case, which is closely related to theplus and minus p -adic L -function defined by R. Pollack. The p -adic Birch andSwinnerton-Dyer conjecture formulated by D. Bernardi and Perrin-Riou in [2]predicts a formula for the leading term of the p -adic L-function. This conjectureis reformulated in terms of Pollack’s p -adic L -functions in [35].Let log p be a branch of the p -adic logarithm, χ the p -adic cyclotomic char-acter, and r E be the Mordell-Weil rank of E . Let ‡ ∈ { + , −} , and denote by R ‡ p ( E/ Q ) the signed p -adic regulator (defined up to p -adic unit). The conven-tion in loc. cit. is to choose a generator γ of the cyclotomic Z p extension, anddivide the regulator (defined w.r.t. the choice of γ ) by log p ( χ ( γ )) r E . We arehowever, not interested in the exact value of the regulator, but only the valueup to a p -adic unit. Therefore, we simply work with the fraction (cid:18) R ‡ p ( E/ Q ) p rE (cid:19) .Lemma . asserts that the order of vanishing of f ( p ) , ‡ E ( T ) at T = 0 is equalto r E . Express f ( p ) , ‡ E ( T ) as a product T r E g ( p ) , ‡ E ( T ) . The following conjecture isequivalent to the p -adic Birch and Swinnerton-Dyer conjecture. Conjecture 3.14.
Let E be an elliptic curve with good supersingular reductionat the prime p and ‡ ∈ { + , −} . Then, g ( p ) , ‡ E (0) ∼ R ‡ p ( E/ Q ) p r E ! × X ( E/ Q )[ p ∞ ] × Y l ∈ S bad c l ( E ) . Let Π ‡ ⊂ S ss be the set of primes p at which v p ( R ‡ p ( E/ Q )) ≥ r E . In otherwords, it is the set of primes for which p divides (cid:18) R ‡ p ( E/ Q ) p rE (cid:19) . The followingresult is proved using the same strategy as that of Theorem . , so we skip theproof. Theorem 3.15.
Assume that Conjecture . holds. Let E / Q be an ellipticcurve with Mordell-Weil rank r E ≥ . Let p be a prime at which E has goodsupersingular reduction. Then for all primes p ∈ S ss \ (Σ ∪ Σ ′ ∪ Π ‡ ) , we havethat f ( p ) , ‡ E ( T ) = T r E . In particular, µ ‡ p ( E ) = 0 and µ ‡ p ( E ) = r E .It seems reasonable to make the following conjecture. onjecture 3.16. Let E / Q be an elliptic curve of rank r E with good (ordinaryor supersingular) reduction at p . For of the primes, µ ‡ p ( E ) = 0 and λ ‡ p ( E ) = r E .4. Results for a fixed prime and varying elliptic curve
In this section, we fix a prime p ≥ and study the variation of Iwasawainvariants as E ranges over all elliptic curves of rank zero with good reductionat p . Recall that any elliptic curve E over Q admits a unique Weierstrassequation(4.1) E : y = x + Ax + B where A, B are integers and gcd( A , B ) is not divisible by any twelfth power.Since p ≥ , such an equation is minimal. We order elliptic curves by heightand expect that similar results shall hold when they are ordered by conductoror discriminant. Recall that the height of E satisfying the minimal equation(4.1) is given by H ( E ) := max (cid:16) | A | , | B | (cid:17) .Let E be set of isomorphism classes of all elliptic curves over Q . Let J bethe set of elliptic curves E over Q satisfying the following two properties(1) E has rank zero,(2) E has good reduction (ordinary or supersingular) at p .The set J is the (disjoint) union of two sets J ord and J ss , consisting of rank0 elliptic curves with ordinary and supersingular reduction at p , respectively.For X > , denote by E ( X ) be the set of isomorphism classes of elliptic curvesover Q of height < X . If S is a subset of E , write S ( X ) = S ∩ E ( X ) . It isconjectured that when ordered by height, discriminant or conductor, that halfof the elliptic curves E over Q have rank (see for example [12, ConjectureB]). If E ∈ J has good ordinary reduction at p , then the Euler characteristicformula (3.1) states that(4.2) χ (Γ , E [ p ∞ ]) ∼ X ( E/ Q )[ p ∞ ] × (cid:0)Q l c l ( E ) (cid:1) × (cid:16) e E ( F p ) (cid:17) (cid:0) E ( Q )[ p ∞ ] (cid:1) . Note that in the above equation, E ( Q )[ p ∞ ] = 1 if p ≥ . On the other hand,if E ∈ J has good supersingular reduction at p , then according to (3.2), wehave that(4.3) χ ± (Γ , E [ p ∞ ]) ∼ X ( E/ Q )[ p ∞ ] × Y l c l ( E ) . Denote by c ( p ) l ( E ) the p -part of c l ( E ) , given by c ( p ) l ( E ) := p v p ( c l ( E )) . The keyobservation in this section is that to analyze the variation of the Euler charac-teristic (and hence µ and λ -invariants) of elliptic curves, it suffices to study theaverage behaviour of the following quantities for fixed p and varying E ∈ J ,(1) s p ( E ) := X ( E/ Q )[ p ∞ ] ,(2) τ p ( E ) := Q l c ( p ) l ( E ) , δ p ( E ) := (cid:16) e E ( F p )[ p ] (cid:17) . Definition 4.1.
Let E ( X ) , E ( X ) , and E ( X ) be the subset of elliptic curvesin E ( X ) for which p divides s p ( E ) , τ p ( E ) and δ p ( E ) respectively.Note that no assumptions are made on the rank of elliptic curves in E ( X ) or E i ( X ) . On the other hand, for elliptic curves E ∈ J , the rank is zero.The primary goal is to obtain upper bounds for d ( i ) p := lim sup X →∞ E i ( X ) E ( X ) for i = 2 , (with no constraints on the rank of the elliptic curves).In [9], C. Delauney gave heuristics for the average number of elliptic curveswith s p ( E ) = 1 . These heuristics are stated in terms of elliptic curves orderedby conductor. However, they indicate that d (1) p goes to as p → ∞ ratherfast. Since there is still not much known about this particular question, we areunable to make further clarifications about the behaviour of d (1) p . However, weexpect that the analysis of this part of the formula is the most difficult.Let κ = ( a, b ) ∈ F p × F p be such that the discriminant ∆( κ ) := 4 a + 27 b is nonzero. The elliptic curve E κ : y = x + ax + b defined over F p is smooth.Let d ( p ) be the number of pairs κ = ( a, b ) ∈ F p × F p such that(1) ∆( κ ) = 0 .(2) E κ : y = x + ax + b has a point over F p of order p .For the primes p in the range ≤ p < , computations on sage show that d ( p ) ≤ and d ( p ) = 1 for p ∈ { , , } . The estimate (4.4) follows from themethod of M. Sadek [30], or the results of J. Cremona and Sadek, see [8]. Theorem 4.2.
Let p ≥ be a fixed prime number. Then(4.4) d (2) p ≤ X l = p ( l − l p +2 , where the sum is taken over prime numbers l = p , and(4.5) d (3) p ≤ ζ (10) · d ( p ) p . Note that ζ (10) = π is approximately equal to . . This quantityarises since the proportion of Weierstrass equations ordered by height whichare minimal is ζ (10) (see [8]). To avoid confusion, we state the results for goodordinary and good supersingular elliptic curves separately. First, we state theresult for elliptic curves with good ordinary reduction at p . Theorem 4.3.
Let p ≥ be a fixed prime number. Let Z ord denote the setof rank 0 elliptic curves E with good ordinary reduction at p for which thefollowing equivalent conditions are satisfied(1) χ (Γ , E [ p ∞ ]) = 1 ,(2) Sel p ∞ ( E/ Q cyc ) = 0 . hen, lim sup X →∞ Z ord ( X ) E ( X ) ≥ lim sup X →∞ J ord ( X ) E ( X ) − d (1) p − X l = p ( l − l p +2 − ζ (10) · d ( p ) p . Proof.
It follows from Corollary . that χ (Γ , E [ p ∞ ]) = 1 and Sel p ∞ ( E/ Q cyc ) are equivalent. By the Euler characteristic formula (4.2), lim sup X →∞ Z ord ( X ) E ( X ) ≥ lim sup X →∞ J ord ( X ) E ( X ) − d (1) p − d (2) p − d (3) p . The result follows from Theorem . . (cid:3) Next, we prove an analogous result in the case when E varies over ellipticcurves with good supersingular reduction at p . Theorem 4.4.
Let p ≥ be a fixed prime number. Let Z ss be the rank 0elliptic curves E with good supersingular reduction at p , for which the followingequivalent conditions are satisfied(1) χ ± (Γ , E [ p ∞ ]) = 1 (2) Sel ± p ∞ ( E/ Q cyc ) = 0 .Then, lim sup X →∞ Z ss ( X ) E ( X ) ≥ lim sup X →∞ J ss ( X ) E ( X ) − d (1) p − X l = p ( l − l p +2 . Proof.
The proof is identical to that of Theorem . . It is a direct consequenceof Corollary . , Theorem . , and the Euler characteristic formula (4.3). (cid:3) We prove a result which applies for all elliptic curves with good reduction.
Theorem 4.5.
Let p ≥ be a fixed prime number. Let Z be the of rank 0elliptic curves E with good reduction at p , for which Sel ‡ p ∞ ( E/ Q cyc ) = 0 . Then, lim sup X →∞ Z ( X ) E ( X ) ≥ lim sup X →∞ J ( X ) E ( X ) − d (1) p − X l = p ( l − l p +2 − ζ (10) · d ( p ) p . Proof.
Let
Y ⊂ E consist of the elliptic curves E for which s p ( E ) τ p ( E ) δ p ( E ) =1 . It follows from the Euler characteristic formulas (4.2) and (4.3) that Z iscontained in E \ Y . By Theorem . that lim sup X →∞ Y ( X ) E ( X ) ≤ d (1) p + X l = p ( l − l p +2 + ζ (10) · d ( p ) p and the result follows. (cid:3) Remark 4.6.
On average, the proportion of elliptic curves over Z p with goodreduction at p (ordered by height) is (1 − p ) , see [8]. Also, it is expected that / the elliptic curves have rank when ordered by height. Therefore, it isreasonable to expect that lim sup X →∞ J ( X ) E ( X ) = 12 (cid:18) − p (cid:19) . euristics of Delauney suggest that d (1) p should approach zero quite rapidlyas p → ∞ . The result indicates that the proportion of elliptic curves for whichthe Selmer group is zero is > and the proportion approaches / as p → ∞ .We are led to make the following conjecture. Conjecture 4.7.
Let p be a fixed prime. Denote by J p the set of rank 0 ellipticcurves with good reduction at p , and by Z p the subset of elliptic curves for which Sel ‡ p ∞ ( E/ Q cyc ) = 0 . Then, lim inf p →∞ lim sup X →∞ Z p ( X ) J p ( X ) ! = 1 . Remark 4.8.
In the rank one case, such an analysis is difficult. This is becauseof the term arising from the p -adic regulator in the formula for the truncatedEuler characteristic. At the time of writing, the authors are not aware of anyresults or heuristics for the average behaviour of the p -adic valuation of R p ( E/ Q ) as E ranges over all elliptic curves of rank with good ordinary reduction at p .Theorem . is proved in the remainder of the section.4.1. Average results on Tamagawa numbers.
Let p ≥ be a fixed prime,and l be a prime different from p . In this section, we estimate the proportion ofelliptic curves E/ Q up to height X with Kodaira type I p at l . These estimatesare well known, but we include them for the sake of completeness, see [30, 8].Recall that when the Kodaira symbol at the prime l is I p , the Tamagawa number c l is divisible by p [34, p. 448]. Let E ( X ) be the set of isomorphism classes allelliptic curves over Q with height ≤ X . This is in one-to-one correspondencewith the set | A | ≤ √ X, | B | ≤ √ X ( A, B ) ∈ Z × Z : 4 A + 27 B = 0 for all primes q if q | A, then q ∤ B . Lemma 4.9 (A. Brumer) . With notation as above, E ( X ) = 4 X / ζ (10) + O (cid:16) √ X (cid:17) . Proof.
See [3, Lemma 4.3]. (cid:3)
Consider the set E I p l ( X ) , i.e. the set of elliptic curves over Q with bad reduc-tion at l , height ≤ X , and Kodaira type I p . The Kodaira symbol forces the badreduction to be of multiplicative type. It follows from Tate’s algorithm thatthis set is in one-to-one correspondence with(4.6) | A | ≤ √ X, | B | ≤ √ X ( A, B ) ∈ Z × Z : l ∤ A, l ∤ B, l p k A + 27 B ( A, B ) = (0 , ∈ Z /q × Z /q for any prime q . We include both upper and lower bounds, however, we only apply upper boundsin our analysis. The following calculations have been done in the preprint [30,Lemma 4.1]. We clarify the arguments and include them here for completeness. emma 4.10 (Sadek) . Let
X > and l i be the i -th prime. Let k > be thelargest positive integer such that L k = Q ki l i ≤ √ X . Then, l p ( l − k Y i =1 ( l i − $ √ Xl p +1 L k %$ √ Xl p +1 L k % − X / l p +2 L k l k ≤ E I p l ( X ) ≤ l p ( l − k Y i =1 ( l i − $ √ Xl p +1 L k %$ √ Xl p +1 L k % + √ X l p +1 L k l k + √ X l p +1 L k l k . Proof.
To obtain the estimate on the size of the set E I p l ( X ) , we use the descrip-tion of the set in (4.6). Observe that l p +1 ∤ A + 27 B , hence the condition A + 27 B = 0 is inherent in the definition of E I p l ( X ) .Consider the congruence equation A + 27 B ≡ l ) . It has l − non-singular solutions, which lift to l p − ( l − solutions modulo l p (see for example[37, §3.4.1]). Note that the description of the set in (4.6) says that l ∤ A and l ∤ B ; thereby allowing us to ignore the point (0 , .In view of (4.6), we are interested in solutions modulo l p that fail to satisfythe congruence equation modulo l p +1 . Since the l − non-singular solutions liftto l p − ( l − solutions modulo l p and l p ( l − solutions modulo l p +1 , it followsthat the number pairs ( A, B ) ∈ Z /l p +1 × Z /l p +1 such that:(1) ( A, B ) (0 ,
0) mod l ,(2) A + 27 B ≡ l p ,(3) A + 27 B l p +1 ,is equal to l · ( l p − ( l − − l p ( l −
1) = l p ( l − . Therefore, E I p l ( X ) has l p ( l − pairs of residue classes in Z /l p +1 × Z /l p +1 . Next,we need to count the number of lifts of each such pair under the additionalcondition that ( A, B ) = (0 , ∈ Z /l i × Z /l i for each prime l i . Note that thenumber of pairs ( A, B ) satisfying this additional condition is ( l i − .It follows that the number of pairs ( A, B ) in the box [ − √ X, √ X ] × [ −√ X, √ X ] such that ( A, B ) = (0 , ∈ Z /l p +1 × Z /l p +1 , ( A, B ) = (0 , ∈ Z /l i × Z /l i and A + 27 B ≡ l p is l p ( l − k Y i =1 (cid:0) l i − (cid:1) $ √ Xl p +1 L k %$ √ Xl p +1 L k % . Our estimate so far might include pairs ( A, B ) such that it is (0 , ∈ Z /q × Z /q when l k < q ≤ √ X . So, we must exclude the integral pairs which reduce to (0 , ∈ Z /q × Z /q for l k < q ≤ √ X . Therefore, we need to remove l p ( l − k Y i =1 (cid:0) l i − (cid:1) X l k The following result follows from the previous lemmas in this section. Theorem 4.11. With notation as above, lim sup X →∞ E I p l ( X ) E ( X ) ≤ ( l − l p +2 . Proof. The result follows from using the upper bound of E I p l ( X ) and Lemma4.9. (cid:3) Remark 4.12. When the elliptic curves are ordered by conductor (rather thanheight), the same bounds have been obtained in [33, Theorem 1.6]. .2. Average results on anomalous primes. We fix a prime p ≥ . Let W consist of tuples ( A, B ) ∈ Z × Z , where ( A, B ) is identified with the (minimal)Weierstrass equation y = x + Ax + B. Denote by W ( X ) the set of Weierstrass equations for which the height is ≤ X .Note that E ( X ) is a subset of W ( X ) such that(4.7) lim X →∞ E ( X ) W ( X ) = 1 ζ (10) (see [8]). Thus . of Weierstrass equations are globally minimal.Let κ = ( a, b ) ∈ F p × F p with ∆( κ ) = 0 . Let E κ be the elliptic curve definedby the Weierstrass equation E κ : y = x + ax + b. Note that κ is not uniquely determined by E κ . Lemma 4.13. Let κ be a pair and W κ ( X ) ⊂ W ( X ) be the subset of Weierstrassequations y = x + Ax + B such that the pair ( A, B ) reduces to κ . Then, lim sup X →∞ W κ ( X ) E ( X ) ≤ ζ (10) p . Proof. Observe that W (0 , is the lattice in Z × Z with lattice basis ( p, and (0 , p ) . Since W κ is simply a translation of W (0 , , it follows that lim X →∞ W κ ( X ) W ( X ) = 1 p . The result follows from (4.7). (cid:3) Denote by S the set of pairs κ = ( a, b ) ∈ F p × F p such that E κ contains apoint of order p over F p . Recall that d ( p ) := S . Let W ′ ( X ) ⊂ W ( X ) be theset of Weierstrass equations y = x + Ax + B which reduce to E κ for some κ ∈ S . Theorem 4.14. We have that d (3) p ≤ ζ (10) · d ( p ) p . Proof. It follows from Lemma . that lim sup X →∞ W ′ ( X ) E ( X ) ≤ ζ (10) · d ( p ) p . Recall that E ( X ) ⊆ W ′ ( X ) . The result follows. (cid:3) Acknowledgments DK thanks J. Balakrishnan for helpful discussions. She acknowledges thesupport of the PIMS Postdoctoral Fellowship. 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