On the analytic rank of the twin prime elliptic curve $y^2=x(x-2)(x-p)$

Kirti Joshi

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On the analytic rank of the twin prime elliptic curve $y^2=x(x-2)(x-p)$

Kirti Joshi

Number Theory · Submitted 17 Aug 2025 · 11 references

Pith's verdict: A clean, self-contained root number computation for the twin prime family, with the expected (and counter) examples; the sole load-bearing external input is a Halberstadt table entry that should be reproduced.

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arXiv ID2508.12340 (version 1)

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CitationKirti Joshi. "On the analytic rank of the twin prime elliptic curve $y^2=x(x-2)(x-p)$." arXiv:2508.12340 (2025). https://pith.science/paper/VNHR676J

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Editor's note

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Here are the two things to know. The paper gives an explicit global root number formula for the twin prime elliptic curve Ep:y^2=x(x-2)(x-p) for all twin prime pairs, and uses it to prove analytic rank at least one for p≡3,5 mod 8. It also gives a counterexample to Beers' rank-two conjecture for p≡1 mod 8: E73 has rank zero. The main theorem is new and the computation is, as far as I checked, sound.

What I like. The derivation is self-contained: invariants, reduction types, local root numbers, and the T(p) residue table all check out. I re-verified T(p)=-2p^3+7p^2+10p-12 and the mod 16 table. The paper is careful to distinguish the analytic-rank statement from the part that depends on the parity or Birch–Swinnerton-Dyer conjectures. The numerical data in Section 5 are clearly presented, and the author credits Hatley for the rank bound and the generators.

Where the soft spots are. The one load-bearing step is the local root number W2 taken from Halberstadt's 1998 table: type III reduction with (v2(Δ),v2(c4),v2(c6))=(6,4,≥7), where W2=+1 iff T≡7,11 mod 16. The paper cites the table but does not reproduce the row or define T's normalization explicitly. If that entry were misread, the signs in the theorem would flip for exactly the congruence classes used. The internal checks are consistent—the announced W2=-(2/p) matches the table and the rank-zero data for p=5,7,73—so this is a verification risk, not an identified error. I would ask the referee to require a reproduction of the relevant Halberstadt row and an explicit definition of T.

The computational claims (165 twin primes with p≡1 mod 8 up to 50000, 14 with analytic rank two) are reported with Sage/LMFDB but without code or a precise algorithm. They are not load-bearing for the main theorem, but a responsible referee should ask for the data or a reproducible script.

Citation pattern is fine: Hatley, Rohrlich, Halberstadt, Silverman are the right references, and the acknowledgments are honest. There are minor typos, but nothing that touches the main argument.

Who this is for: people working on root numbers of elliptic curve families or on Beers' conjecture. It is a modest but real advance, not a sweeping one. It deserves a serious referee. Send it out; just make sure the referee checks the Halberstadt table entry and asks for the computational details.

Referee report

Significance

The result, if correct, is a clean and unconditional contribution to an existing conjecture: it gives a parameter-free formula for the global root number of an infinite family of elliptic curves indexed by twin primes and derives a nontrivial analytic-rank consequence from it. The derivation is transparent and the algebraic simplifications in §4.1 are checkable by hand; the numerical counterexample E_73 and the associated Sage/LMFDB computations are concrete and falsifiable. The paper is honest about the dependence on Hatley's result and on the standard table of Halberstadt, and it explicitly separates the proven statements from conjectures. The main limitation is that the local root number at 2 rests on an externally quoted table entry, which is a reproducibility issue rather than an apparent mathematical error.

Minor objections (5)

[Section 4.1, before Eq. (4.1.8)] The displayed definition of T(p) is garbled and, as printed, does not imply Eq. (4.1.8), because c_6 already contains the factor 2^6 and the printed denominator appears as '27'. The intended denominator is presumably 2^{v_2(c_6)} = 2^7, equivalently T = c'_4 − 2c'_6. Please correct this and, to make the sign rule verifiable, reproduce the relevant row of [Halberstadt, 1998, Table 1].
[Section 3.2, Proposition 3.2.1(3)] The sentence 'the exponent of the conductor f = v_2(Δ) = 5' is internally inconsistent, since v_2(Δ) = 6; it should state f = v_2(Δ) − 1 = 5. The conductor exponent is not used later, but the statement should be corrected.
[Introduction, page 2, and Remark 4.1.15] The sentence claiming that the Birch and Swinnerton-Dyer conjecture, together with Theorem 1.1.2 and [Hatley, 2009], implies that the Mordell-Weil rank is exactly one is only justified for p ≡ 5 mod 8; for p ≡ 3 mod 8, BSD plus Proposition 2.2.1 gives rank 1 or 2. Remark 4.1.15 correctly invokes the parity conjecture instead, so the earlier sentence should be rephrased for consistency.
[Remark 4.1.14] The statement that the twin-prime hypothesis 'cannot be relaxed' is stronger than the listed examples support; the examples show that the conclusion fails for those particular non-twin primes. A phrase such as 'does not hold for these primes' would be more accurate.
[Table 5.5.2] The table includes p = 5 even though Theorem 1.1.2 assumes p ≥ 7; please add a note indicating that p = 5 is included for illustration.

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The paper

The paper claims

This paper proves that for every twin-prime pair $(p,p-2)$ with $p\geq 7$, the global root number of $E_p:y^2=x(x-2)(x-p)$ equals $(-1)^{(p-1)/2+((p-2)^2-1)/8}$, so $p\equiv 3$ or $5\pmod 8$ forces $L(E_p,1)=0$.

The reading

The paper proves a formula for the global root number of the twin-prime elliptic curve $E_p:y^2=x(x-2)(x-p)$: it is $(-1)^{(p-1)/2+((p-2)^2-1)/8}$, which is $+1$ for $p\equiv 1,7\pmod 8$ and $-1$ for $p\equiv 3,5\pmod 8$. The immediate consequence is that when $p\equiv 3$ or $5\pmod 8$, the functional equation forces $L(E_p,1)=0$, so the analytic rank is at least one. This confirms the part of the motivating rank conjecture covering those residue classes and complements previously known upper bounds. The paper also shows that the unconditional rank-two prediction for $p\equiv 1\pmod 8$ is false, since $E_{73}$ has rank zero, and replaces it with the assertion that the rank is zero or two in that residue class.

What carries the argument

The carrying object is the decomposition of the global root number into local root numbers, $W_{E_p}=W_\infty W_2 W_p W_{p-2}$. The pieces are: $W_\infty=-1$; at the twin primes $W_p=-(-2/p)$ and $W_{p-2}=-(2/(p-2))$ from split multiplicative reduction; and at $2$, Tate's algorithm followed by a table of local signs for type III reduction (the Kodaira symbol for additive reduction with a double line) gives $W_2=-(2/p)$, determined by $T(p)=((p+2)^2-6p)-2(p+2)((p+2)^2-9p)\bmod 16$. The product simplifies to $(-1/p)(2/(p-2))$ and hence to the stated exponent; the twin-prime hypothesis fixes the 2-adic valuations of the invariants so that exactly one table entry applies.

Extended reading notes

Core claim

The central discovery is an exact formula for the sign in the functional equation of $E_p$: $W_{E_p}=(-1)^{(p-1)/2+((p-2)^2-1)/8}$, equal to $1$ for $p\equiv 1,7\pmod 8$ and $-1$ for $p\equiv 3,5\pmod 8$. The proof computes the local root numbers at $\infty$, $2$, $p$, and $p-2$ separately. At $p$ and $p-2$ the curve has multiplicative reduction whose split or non-split character is governed by Legendre symbols; at $2$ the reduction is of Kodaira type III with fixed 2-adic valuations, so the local sign is read from a published table of such signs. Multiplying the four local signs collapses to $(-1/p)(2/(p-2))$, which is the displayed exponent. Because the curve is modular, a negative root number in the $3,5\bmod 8$ cases implies $L(E_p,1)=0$, i.e. analytic rank at least one. A separate computation shows $E_{73}$ has rank zero, disproving the rank-two prediction for $p\equiv 1\pmod 8$ and leading to the corrected conjecture that the rank is zero or two in that case.

Load-bearing premise

The proof leans on a published table of local signs at 2: for type III reduction with the exact 2-adic valuation pattern of this curve, the local sign is $+1$ exactly when $T(p)\equiv 7,11\pmod{16}$; if that table entry were wrong or misread, the formula for $W_2$, and with it the main theorem, would fail.

Editorial extensions

If this is right

For every twin-prime pair with $p\equiv 3$ or $5\pmod 8$, the analytic rank of $E_p$ is at least one.
Under the Birch and Swinnerton-Dyer conjecture, those curves have Mordell-Weil rank exactly one, since the previously known upper bound of one for $p\equiv 5\pmod 8$ is now matched by a lower bound, and the $p\equiv 3\pmod 8$ case is newly covered.
The rank-two prediction for $p\equiv 1\pmod 8$ is false: $E_{73}$ has rank zero, and among 165 twin primes with $7\leq p\leq 50000$ in that residue class, 14 have rank two and the rest have rank zero.
The twin-prime hypothesis really is necessary for the lower bound: the paper lists non-twin primes $p\equiv 3$ or $5\pmod 8$ where the same curve has rank zero.
The root-number formula gives $W_{E_p}=+1$ for $p\equiv 1,7\pmod 8$, consistent with the known rank-zero result for $p\equiv 7\pmod 8$ and with the corrected statement that rank zero or two occurs for $p\equiv 1\pmod 8$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same assembly of local root numbers could yield explicit root-number formulas for other families $y^2=x(x-a)(x-b)$ with $a,b$ in fixed prime configurations, as long as the 2-adic valuation pattern stays rigid; the twin-prime condition here acts mainly as that rigidity condition.
The data showing rank mostly zero and rarely two for $p\equiv 1\pmod 8$ raise an arithmetic-statistics question the paper does not address: what is the asymptotic density of rank-two twin-prime curves, and can a Selmer-group distribution account for the observed 14-out-of-165 proportion?
If the Birch and Swinnerton-Dyer conjecture holds, the corrected conjecture is equivalent to $L'(E_p,1)\neq 0$ for all twin primes $p\equiv 3,5\pmod 8$; numerical nonvanishing checks of the first derivative over a larger range would be a concrete test.
The paper's correction separates parity-even rank 0 from rank 2 purely by root number plus descent data; extending the tables beyond $p=50000$ could reveal whether rank two for $p\equiv 1\pmod 8$ persists with positive density or eventually disappears.

Original abstract

Let $p\geq 7$ and suppose $(p,p-2)$ are twin prime numbers, in [Hatley, 2009], the elliptic curve $E_p:y^2=x(x-2)(x-p)$ was considered in the context of a conjecture by Jason Beers about the Mordell-Weil ranks of $E_p/\mathbb{Q}$. I show that for $p\equiv 3,5\bmod 8$, the analytic rank of $E_p$ is at least one (Theorem 1.1.2) in line with Beers' predictions. This is done by finding a formula (Theorem 4.1.1) for the global root number of $E_p$ for all twin prime pairs. I also show that Beers' conjecture, that for $p\equiv 1\bmod 8$ the rank of $E_p$ is two, is false as stated because $E_{73}$ has rank zero. In the light of Theorem 4.1.1, Beers' conjecture needs to be modified: if $p\equiv 1\bmod 8$ then the rank of $E_p$ is zero or two (Conjecture 5.3.1).

The compiled record

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,638 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.30 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.96 · compares with prior results 0.88 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Let $p\geq 7$ and suppose $(p,p-2)$ are twin prime numbers, in [Hatley, 2009], the elliptic curve $E_p:y^2=x(x-2)(x-p)$ was considered in the context of a conjecture by Jason Beers about the Mordell-Weil ranks of $E_p/\mathbb{Q}$. I show that for $p\equiv 3,5\bmod 8$, the analytic rank of $E_p$ is at least one (Theorem 1.1.2) in line with Beers' predictions. This is done by finding a formula (Theorem 4.1.1) for the global root number of $E_p$ for all twin prime pairs. I also show that Beers' conjecture, that for $p\equiv 1\bmod 8$ the rank of $E_p$ is two, is false as stated because $E_{73}$ has rank zero. In the light of Theorem 4.1.1, Beers' conjecture needs to be modified: if $p\equiv 1\bmod 8$ then the rank of $E_p$ is zero or two (Conjecture 5.3.1).

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Let $p\geq 7$ and suppose $(p,p-2)$ are twin prime numbers, in [Hatley, 2009], the elliptic curve $E_p:y^2=x(x-2)(x-p)$ was considered in the context of a conjecture by Jason Beers about the Mordell-Weil ranks of $E_p/\mathbb{Q}$.	background 1.00	0.04	0.03	asserted 0.99
s1	I show that for $p\equiv 3,5\bmod 8$, the analytic rank of $E_p$ is at least one (Theorem 1.1.2) in line with Beers' predictions.	result 0.99	0.96	0.60	asserted 1.00
s2	This is done by finding a formula (Theorem 4.1.1) for the global root number of $E_p$ for all twin prime pairs.	method 0.95	0.94	0.11	asserted 1.00
s3	I also show that Beers' conjecture, that for $p\equiv 1\bmod 8$ the rank of $E_p$ is two, is false as stated because $E_{73}$ has rank zero.	result 0.97	0.92	0.19	asserted 1.00
s4	In the light of Theorem 4.1.1, Beers' conjecture needs to be modified: if $p\equiv 1\bmod 8$ then the rank of $E_p$ is zero or two (Conjecture 5.3.1).	interpretation 0.80	0.61	0.29	speculative 0.80

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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sources read

arXiv listing abstract v1

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full text
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (11)

[1]

Juli\'an Aguirre, \'Alvaro Lozano-Robledo, and Juan Carlos Peral. Elliptic curves of maximal rank. In Proceedings of the `` S egundas J ornadas de T eor\'ia de N \'umeros'' , Bibl. Rev. Mat. Iberoamericana, pages 1--28. Rev. Mat. Iberoamericana, Madrid, 2008

View on Pith 2008
[2]

Christophe Breuil, Brian Conrad, Frederic Diamond, and Richard Taylor. On the modularity of elliptic curves over Q : Wild 3-adic exercises. Journal of the American Mathematical Society, 14: 0 843--939, 10 2001. doi:10.1090/S0894-0347-01-00370-8

View on Pith doi:10.1090/s0894-0347-01-00370-8 2001
[3]

Emmanuel Halberstadt. Signes locaux des courbes elliptiques en 2 et 3. Comptes Rendus de l'Acad\'emie des Sciences - Series I - Mathematics, 326 0 (9): 0 1047--1052, 1998. doi:https://doi.org/10.1016/S0764-4442(98)80060-8. URL https://www.sciencedirect.com/science/article/pii/S0764444298800608

View on Pith doi:10.1016/s0764-4442(98)80060-8 1998
[4]

Jeffrey Hatley. On the rank of the elliptic curve y^2=x(x-p)(x-2) , 2009. URL https://arxiv.org/abs/0909.1614

View on Pith Pith review arXiv 2009
[5]

Emmanuel Kowalski. Expos\'e B ourbaki 1084 : G aps between prime numbers and primes in arithmetic progressions. Ast\'erisque, 2018

View on Pith 2018
[6]

The LMFDB Collaboration . The L -functions and modular forms database. https://www.lmfdb.org, 2025. [Online; accessed 1 June 2025]

View on Pith 2025
[7]

Bernadette Perrin-Riou. Travaux de K olyvagin et R ubin. Ast\'erisque, 189/90: 0 69--106, 1990

View on Pith 1990
[8]

David E. Rohrlich. Variation of the root number in families of elliptic curves. Compositio Mathematica, 87 0 (2), 1993. URL https://www.numdam.org/item/CM_1993__87_2_119_0/

View on Pith 1993
[9]

Sage Developers . S age M athematics S oftware , 2015. available from http://www.sagemath.org

View on Pith 2015
[10]

Joseph Silverman. The arithmetic of elliptic curves, volume 106 of Graduate T ext in Mathematics . Springer- V erlag, Berlin, 1985

View on Pith 1985
[11]

Joseph H. Silverman. Advanced topics in the arithmetic of elliptic curves, volume 151 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1994. ISBN 0-387-94328-5. URL https://doi.org/10.1007/978-1-4612-0851-8

View on Pith doi:10.1007/978-1-4612-0851-8 1994

How the literature uses it

What this paper does with the arXiv papers it cites

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 10 sentences without a confident label

Builds on 1 paper

The purpose of this brief note is to complement the result of [ref] by proving the following theorem.

On the Rank of the Elliptic Curve y^2=x(x-p)(x-2) · Section · 0.93

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

Juli\'an Aguirre, \'Alvaro Lozano-Robledo, and Juan Carlos Peral. Elliptic curves of maximal rank. In Proceedings of the `` S egundas J ornadas de T eor\'ia de N \'umeros'' , Bibl. Rev. Mat. Iberoamericana, pages 1--28. Rev. Mat. Iberoamericana, Madrid, 2008

View on Pith 2008

[2] [2]

Christophe Breuil, Brian Conrad, Frederic Diamond, and Richard Taylor. On the modularity of elliptic curves over Q : Wild 3-adic exercises. Journal of the American Mathematical Society, 14: 0 843--939, 10 2001. doi:10.1090/S0894-0347-01-00370-8

View on Pith doi:10.1090/s0894-0347-01-00370-8 2001

[3] [3]

Emmanuel Halberstadt. Signes locaux des courbes elliptiques en 2 et 3. Comptes Rendus de l'Acad\'emie des Sciences - Series I - Mathematics, 326 0 (9): 0 1047--1052, 1998. doi:https://doi.org/10.1016/S0764-4442(98)80060-8. URL https://www.sciencedirect.com/science/article/pii/S0764444298800608

View on Pith doi:10.1016/s0764-4442(98)80060-8 1998

[4] [4]

Jeffrey Hatley. On the rank of the elliptic curve y^2=x(x-p)(x-2) , 2009. URL https://arxiv.org/abs/0909.1614

View on Pith Pith review arXiv 2009

[5] [5]

Emmanuel Kowalski. Expos\'e B ourbaki 1084 : G aps between prime numbers and primes in arithmetic progressions. Ast\'erisque, 2018

View on Pith 2018

[6] [6]

The LMFDB Collaboration . The L -functions and modular forms database. https://www.lmfdb.org, 2025. [Online; accessed 1 June 2025]

View on Pith 2025

[7] [7]

Bernadette Perrin-Riou. Travaux de K olyvagin et R ubin. Ast\'erisque, 189/90: 0 69--106, 1990

View on Pith 1990

[8] [8]

David E. Rohrlich. Variation of the root number in families of elliptic curves. Compositio Mathematica, 87 0 (2), 1993. URL https://www.numdam.org/item/CM_1993__87_2_119_0/

View on Pith 1993

[9] [9]

Sage Developers . S age M athematics S oftware , 2015. available from http://www.sagemath.org

View on Pith 2015

[10] [10]

Joseph Silverman. The arithmetic of elliptic curves, volume 106 of Graduate T ext in Mathematics . Springer- V erlag, Berlin, 1985

View on Pith 1985

[11] [11]

Joseph H. Silverman. Advanced topics in the arithmetic of elliptic curves, volume 151 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1994. ISBN 0-387-94328-5. URL https://doi.org/10.1007/978-1-4612-0851-8

View on Pith doi:10.1007/978-1-4612-0851-8 1994