REVIEW
Second derivatives of $p$-adic $L$-functions and the Shafarevich--Tate group of rank-two CM elliptic curves
Number Theory · Submitted 8 Sep 2026 · 63 references
Pith's verdict: Solid Iwasawa theory for CM rank-two curves, with computational payoff and an honest formalization.
T0 · AI review
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Referee report
The paper
The paper claims
Second p-adic L-derivative controls the Shafarevich–Tate group of rank-two CM elliptic curves.
The reading
What carries the argument
The central object is the normalised second Taylor coefficient ~c_2(p) of the Mazur–Tate–Teitelbaum p-adic L-function L_p(E,T), whose p-adic unitness is the condition of interest. The argument reduces this condition, via the Katz measure and the Bannai–Kobayashi theory of polynomial moments, to the vanishing modulo p of a combination of three Eisenstein–Kronecker class sums (the bracket B(p)) that are, up to periods, critical values of the Grössencharacter $ψ_E^{{2p-1}}$. On the algebraic side, Rubin’s main conjecture for K and Schneider’s leading-term theorem link ~c_2(p) to the cyclotomic regulator and the size of the Shafarevich–Tate group.
Extended reading notes
Core claim
The paper proves that for a rank-two CM elliptic curve E/ℚ at a split good ordinary prime p outside an explicit finite set S_E, the following are equivalent: (1) the normalised second Taylor coefficient of the p-adic L-function of E is a p-adic unit; (2) the cyclotomic p-adic height pairing on E(ℚ) is nondegenerate with unit regulator and the p-primary part of the Shafarevich–Tate group vanishes; (3) an explicit combination of three critical Hecke L-values of weight 2p-1 (the bracket B(p)) has p-adic valuation exactly 2. This supplies a cyclotomic counterpart to the Coates–Liang–Sujatha criterion and shows that the second derivative of the p-adic L-function fully determines the arithmetic at p.
Load-bearing premise
The Bannai–Kobayashi theory (Lemma 6.5) identifies polynomial moments of the Katz measure with Eisenstein–Kronecker class sums; if at some prime this identification fails because an Euler factor has an unexpected p-adic valuation, the reduction to the criterion class c(p) and the whole unit condition analysis could break.
Editorial extensions
If this is right
- At any split prime outside S_E where the unit condition holds, the Shafarevich–Tate group is finite and its p-primary part is zero, the p-adic regulator is a unit, and the analytic Iwasawa invariants are μ=0, λ=2.
- The criterion reduces the problem of confirming the unit condition to computing three Hecke L-values, one of which alone (at s=p) is the input of the Coates–Liang–Sujatha criterion; the new criterion also detects regulator failures.
- For the five curves of Coates, Liang and Sujatha, the unit condition is verified at all but three of the 8,050 eligible primes below 30,000, and the previously open case p=577 for y^2=x^3+34x is settled in the affirmative.
- The equivalence of (1) and (2) is formalised in Lean 4, providing a machine-checked proof of Theorem A assuming the cited literature results.
Reading between the lines
- The three failures of the unit condition (at p=37 for y^2=x^3+33x, and at p=5 and p=15289 for y^2=x^3+39x) show that regulator failures occur at a Wieferich-like density, with about three expected among 8,050 primes under a random model. This suggests the set of such primes may be infinite but thin.
- The criteria of Coates–Liang–Sujatha and of this paper fail at disjoint sets of primes in the scanned range, raising the question whether for all but finitely many split primes at least one of the two criteria will succeed, which would give a horizontal statement of finiteness for the Shafarevich–Tate group.
- The bracket B(p) offers a candidate object analogous to the Heegner point in rank one: if the primes where the unit condition fails are exactly the prime divisors of a fixed algebraic number (independent of p), then Question 3 would be answered. This would parallel Gillard’s theorem for the zeroth jet.
Original abstract
For an elliptic curve $E/\mathbb{Q}$ of rank two with complex multiplication, Coates, Liang and Sujatha gave a criterion for the vanishing of $Sha(E/\mathbb{Q})[p^\infty]$ at a good ordinary prime $p$ and applied it to five such curves for $p < 30{,}000$. We prove a cyclotomic criterion of the same kind: outside an explicit set of primes, the normalised second Taylor coefficient of the Mazur-Tate-Teitelbaum $p$-adic $L$-function at the central point is a $p$-adic unit if and only if the cyclotomic $p$-adic regulator is a unit and $Sha(E/\mathbb{Q})[p^\infty] = 0$, and, by the theory of Bannai and Kobayashi, if and only if an explicit combination of three critical Hecke $L$-values of weight $2p - 1$ has valuation exactly two. Following the algorithm of Stein and Wuthrich, we compute the regulator for the same five curves at every good ordinary prime below $30{,}000$: it is a unit at all but three of the $8{,}050$ primes outside the excluded set. The one case that the criterion of Coates, Liang and Sujatha left open, $p = 577$ for $y^2 = x^3 + 34x$, is settled by the new criterion. A Lean 4 formalisation of the first equivalence, assuming stated results from the literature, is provided.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
For an elliptic curve $E/\mathbb{Q}$ of rank two with complex multiplication, Coates, Liang and Sujatha gave a criterion for the vanishing of $Sha(E/\mathbb{Q})[p^\infty]$ at a good ordinary prime $p$ and applied it to five such curves for $p < 30{,}000$. We prove a cyclotomic criterion of the same kind: outside an explicit set of primes, the normalised second Taylor coefficient of the Mazur-Tate-Teitelbaum $p$-adic $L$-function at the central point is a $p$-adic unit if and only if the cyclotomic $p$-adic regulator is a unit and $Sha(E/\mathbb{Q})[p^\infty] = 0$, and, by the theory of Bannai and Kobayashi, if and only if an explicit combination of three critical Hecke $L$-values of weight $2p - 1$ has valuation exactly two. Following the algorithm of Stein and Wuthrich, we compute the regulator for the same five curves at every good ordinary prime below $30{,}000$: it is a unit at all but three of the $8{,}050$ primes outside the excluded set. The one case that the criterion of Coates, Liang and Sujatha left open, $p = 577$ for $y^2 = x^3 + 34x$, is settled by the new criterion. A Lean 4 formalisation of the first equivalence, assuming stated results from the literature, is provided.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | For an elliptic curve $E/\mathbb{Q}$ of rank two with complex multiplication, Coates, Liang and Sujatha gave a criterion for the vanishing of $Sha(E/\mathbb{Q})[p^\infty]$ at a good ordinary prime $p$ and applied it to five such curves for $p < 30{,}000$. | background 1.00 | 0.04 | 0.03 | asserted 1.00 |
| s1 | We prove a cyclotomic criterion of the same kind: outside an explicit set of primes, the normalised second Taylor coefficient of the Mazur-Tate-Teitelbaum $p$-adic $L$-function at the central point is a $p$-adic unit if and only if the cyclotomic $p$-adic regulator is a unit and $Sha(E/\mathbb{Q})[p^\infty] = 0$, and, by the theory of Bannai and Kobayashi, if and only if an explicit combination of three critical Hecke $L$-values of weight $2p - 1$ has valuation exactly two. | result 0.60 | 0.92 | 0.89 | asserted 1.00 |
| s2 | Following the algorithm of Stein and Wuthrich, we compute the regulator for the same five curves at every good ordinary prime below $30{,}000$: it is a unit at all but three of the $8{,}050$ primes outside the excluded set. | result 0.75 | 0.95 | 0.10 | asserted 1.00 |
| s3 | The one case that the criterion of Coates, Liang and Sujatha left open, $p = 577$ for $y^2 = x^3 + 34x$, is settled by the new criterion. | result 0.73 | 0.88 | 0.13 | asserted 1.00 |
| s4 | A Lean 4 formalisation of the first equivalence, assuming stated results from the literature, is provided. | contribution 0.97 | 0.92 | 0.04 | asserted 1.00 |
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Discussion
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