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Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$

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Pith's one-line read Under the Generalized Riemann Hypothesis, the average analytic rank of the quartic twists $y^2=x^3-dx$ is at most $13/6$, and with a quartic Patterson conjecture it improves to $3/2$.

desk verdict A serious conditional theorem paper: under GRH it gets one-level density at support 3/5 for quartic twists and rank bound 13/6; the stronger support-1 and 3/2 results rest on an unproved conjecture from the authors' own earlier work. read the letter →

arxiv 2608.06286 v1 pith:3HBIQ2UC submitted 2026-08-06 math.NT

Chantal David , Lucile Devin , Alessandro Fazzari , Ezra Waxman This is my paper · ORCID
classification math.NT MSC 11G0511M5011L05
keywords averageanalyticrankquartictwistsellipticcurvesone-leveldensitylow-lyingzerosGausssumsPattersonconjecturegeneralizedRiemannhypothesis
topics The Riemann Hypothesis
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper studies the family of elliptic curves $E_d: y^2=x^3-dx$ as $d$ ranges over odd fourth-power-free integers, and asks how large the average analytic rank can be. Under the Generalized Riemann Hypothesis, the authors compute the one-level density of low-lying zeros for the associated $L$-functions and deduce that the average analytic rank is at most $13/6$. Assuming, in addition, a quartic analogue of Patterson's conjecture on the distribution of quartic Gauss sums at prime elements, the support of the density computation widens to $(-1,1)$ and the average rank bound improves to $3/2$. Both results imply that a positive proportion of the twists have analytic rank $1$, and the conjectural half also yields a positive proportion of twists with analytic rank $0$. If the paper is right, the low-lying zeros of this CM family match the expected Katz–Sarnak symmetry in the admissible support ranges, and the average rank is significantly smaller than the trivial bound.

What carries the argument

The machinery is the explicit formula for the Hecke-character $L$-functions $L(s,\xi_d)$, which splits the one-level density into an archimedean term, an inert-primes term, and a split-primes term; the split term is the hard part. The split term is reduced, through Poisson summation and a sieving step, to bounding sums $H_\beta(X,Y,r)$ of quartic Gauss sums $g_4(r,c)$ weighted by von Mangoldt's function over $\mathbb Z[i]$. Vaughan's identity decomposes these sums into Type I and Type II pieces; Type I pieces are handled by a Lindelöf-on-average bound across metaplectic $L$-functions (obtained via the quadratic large sieve), and Type II pieces by factoring the quartic Gauss sums and exploiting the oscillation of quadratic characters through the same sieve. The support constraint $\nu<3/5$ arises solely from the Type II sums.

What would settle it

For a fixed residue class $\beta\in\{1,1+\lambda^3\}$, compute the sums in Conjecture 1.4 for Gaussian integers $c\equiv\beta\bmod 4$ up to large $X$; if the $\ell=0$ sum fails to grow like a constant times $X^{3/4}$, the $3/2$ average rank bound collapses. Separately, evaluate the one-level density with a test function whose Fourier support crosses $\nu=0.6$: under the paper's claims it must match the Katz–Sarnak density up to $O(1/\log D)$ as $D\to\infty$.

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Extended reading notes

Core claim

The central discovery is that the one-level density of the family $\{L(s,E_d)\}$ for $d$ odd fourth-power-free obeys, on average, the formula $D_{\mathcal F^*}(\phi,w,D)=\hat\phi(0)+\frac12\int_{\mathbb R}\hat\phi(u)\,du+O(1/\log D)$, where the Fourier support of $\phi$ is contained in $(-3/5,3/5)$ under GRH and in $(-1,1)$ under a quartic Patterson conjecture. This matches the expected Katz–Sarnak symmetry for these supports. From this, the authors derive the average analytic rank bounds of $13/6$ (under GRH) and $3/2$ (under the conjecture), and consequently positive proportions of twists with minimal analytic rank consistent with parity.

Load-bearing premise

For the stronger rank bound, the load-bearing premise is Conjecture 1.4, which asserts that quartic Gauss sums at prime elements in $\mathbb Z[i]$ are equidistributed with main term of size $X^{3/4}$; if that distribution is wrong, only the $13/6$ bound under GRH remains.

Editorial extensions

If this is right

  • Under GRH alone, at least $5/12$ of the negative-root-number twists have analytic rank $1$.
  • If Conjecture 1.4 holds, the proportions improve: at least $3/4$ of the negative-root-number twists have rank $1$, and at least $1/4$ of the positive-root-number twists have rank $0$.
  • The low-lying zeros of the family follow the expected Katz–Sarnak symmetry for test functions whose Fourier transform is supported in $(-3/5,3/5)$ under GRH, and in $(-1,1)$ under the conjecture.
  • As the paper notes, if the density formula held for every even Schwartz test function, the average analytic rank would be $1/2$, the minimalist value predicted by Goldfeld's conjecture.
  • The average rank bounds of $13/6$ and $3/2$ are direct consequences of Theorem 1.2, obtained by substituting admissible test functions into the density formula.

Reading between the lines

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

  • The quartic Patterson conjecture could be tested numerically for Gaussian primes of moderate norm; a failure of the predicted $X^{3/4}$ growth in Conjecture 1.4 would directly invalidate the $3/2$ rank bound.
  • The Type II limitation suggests that a different factorization or sieve for quartic Gauss sums could push the admissible support further, which would also benefit non-vanishing results for quartic Hecke characters.
  • The bounds for quartic twists parallel Heath-Brown's results for quadratic twists, strengthening the heuristic expectation that Goldfeld's minimalist conjecture holds across all twist families, not only the quadratic one.
  • A natural extension is to apply the same Vaughan-identity and large-sieve decomposition to cubic twists over $\mathbb Q(\sqrt{-3})$, where the analogue of Conjecture 1.4 is Patterson's original cubic conjecture and a similar rank bound could be pursued.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper studies the one-level density of low-lying zeros of the L-functions L(s,E_d) for the quartic-twist family E_d:y^2=x^3-dx with d odd and fourth-power-free. The main result, Theorem 1.2, states that under GRH for these L-functions the density equals bφ(0)+(1/2)∫bφ(u)du for test functions with Fourier support in (-3/5,3/5), and that under a quartic analogue of Patterson's conjecture (Conjecture 1.4) the support extends to (-1,1). Theorem 1.1 derives average analytic rank bounds of 13/6 and 3/2, respectively, and Corollary 1.3 gives positive proportions of rank-1 twists in F^- and, under Conjecture 1.4, rank-0 twists in F^+. The proof proceeds via the explicit formula, a sieving step (Lemma 4.1), Poisson summation leading to quartic Gauss sums, a Vaughan-type decomposition into Type I and Type II sums, and a Lindelöf-on-average estimate (Proposition 7.6) for the Dirichlet series of quartic Gauss sums, proved in Section 8 using the quadratic large sieve over Z[i].

Significance. If correct, this is a substantial contribution to the study of higher-order twist families of elliptic curves: it gives the first one-level density result for quartic twists with support beyond 1/2 and, conditionally on Conjecture 1.4, reaches the optimal support (-1,1) with the expected symmetry density. The GRH-only 3/5-support result is an unconditional-in-the-conjecture statement that yields a nontrivial average-rank bound of 13/6 and a positive proportion of rank-one twists. The strategy of bounding quartic Gauss-sum sums via Vaughan's identity, the quadratic large sieve over Z[i], and a Lindelöf-on-average estimate for metaplectic L-functions is innovative and carefully structured. The paper is also transparently conditional: Conjecture 1.4 is explicitly isolated, and the main new technical ingredient, Proposition 7.6, is proved in detail. The manuscript does not contain machine-checked proofs or reproducible code, but the analytic arguments are laid out in sufficient detail for expert verification.

major comments (2)
  1. [§4, Lemma 4.1 and Remark 4.2] There is a discrepancy in the sieving step that is load-bearing for the passage from sums over F* to sums over all integers. Lemma 4.1 states a remainder of O(D^{1+ε}/y), and Remark 4.2 says to apply the lemma with y=D^ε; with that choice the stated error becomes O(D), which is not an admissible error for the target bound O(D) in Remark 3.4. The proof of Lemma 4.1 appears to yield the stronger O(D^{1+2(ν+1)ε}/y^{1+ε}) (because the sum over ℓ≥y of ℓ^{-(2+ε)} contributes y^{-(1+ε)}), and with y=D^{Cε} for a sufficiently large fixed C this would be fine. The authors should correct the stated error term in Lemma 4.1, specify the correct choice of y in Remark 4.2, and reconcile the numerology with the condition δ>30ε.
  2. [§8, proof of Proposition 8.3] The proof applies 'the quadratic large sieve [23, Theorem 1]' to a bilinear sum in which both n and v range over Gaussian integers and the character is the quadratic symbol (v/n)_2 over Z[i]. Reference [23] is a large sieve for real Dirichlet characters over the rationals; the estimate being used is precisely the Gaussian quadratic large sieve stated earlier in Section 6 as [36, Thm. 1]. Please replace the citation and state the exact theorem used, or give a direct justification from [23]. This step is central to the proof of Proposition 7.6 and hence to the 3/5-support result.
minor comments (5)
  1. [§5, Eq. (5.6)] The sum in (5.6) runs over all integers m with |m|<16ηM D^{2ν-1}, but Hβ(X,Y,r) and R_{X,Y}(t) are defined for Y>0, while the argument mD/(16M) is negative for m<0. Please either define R_{X,Y} for all real Y or replace mD/(16M) by |m|D/(16M) and note that the m=0 term vanishes because g4(0,c)=0 for c≠1.
  2. [§4, Lemma 4.1] The compatibility condition 'A≡B mod 4' in the statement of Lemma 4.1 could be made more explicit: it is the condition for the two congruences d≡A mod 16 and d≡B mod 4 to have a solution, and it is equivalent to a≡b d2 mod 4.
  3. [§4, Remark 4.6] The phrase 'uniformly for all ε>0 and M<D^ε' should be read as 'for each fixed ε>0, uniformly for M<D^ε'; otherwise the existence of a single δ>0 for all ε simultaneously is not what is meant and is not what the subsequent argument proves.
  4. [§7, proof of Proposition 7.4] The sentence 'It remains to justify that Gβ(s,r|r,a) is c for Re(s)>1/2' appears to be missing the word 'analytic' in the source text; please correct.
  5. [Throughout] There are several typographical issues to correct, including 'V aughan' in the Section 6 heading, 'P´ olya–Vinogradov' in Remark 4.2, 'fourt -power' in Section 2.1, and 'similiar' in the proof of Corollary 1.3.

Circularity Check

1 steps flagged · score 4.0 of 10

The (−1,1)-support one-level density and 3/2 average-rank bound are conditional on Conjecture 1.4, an unproved quartic Patterson conjecture imported from the authors' earlier paper [11]; the GRH-only (−3/5,3/5) result is independently derived.

  1. self citation load bearing [Section 1.1 (Conjecture 1.4, Theorem 1.2) and Section 4 (Proof of Theorem 1.2 assuming Conjecture 1.4, eq. (4.11))]
    "The following conjecture is a slight generalization of [11, Conjecture 1.2]... Assuming also Conjecture 1.4, the support condition can be relaxed to ν<1. ... By partial summation and Conjecture 1.4, the sum over c∈Z[i] in (4.11) is bounded by ..."

    The extended support and the 3/2 rank bound are obtained by assuming Conjecture 1.4, which the paper labels 'Conjecture 1.2 from [11]', a paper sharing a co-author (David) with the present paper. The proof of the second part of Theorem 1.2 explicitly reduces (4.9)/(4.11) to this conjecture, so the 'prediction' of support (−1,1) rests entirely on an unverified, self-cited input. This is load-bearing self-citation, but not a target-equivalent reduction: the conjecture concerns quartic Gauss-sum averages, not one-level densities, and the GRH-only 3/5 theorem is proved without it. Hence partial circularity only.

full rationale

The paper's primary GRH-conditional derivation is self-contained: Theorem 1.2 for ν<3/5 is obtained through the explicit formula, Poisson summation, Vaughan's identity, the quadratic large sieve, and the new Lindelöf-on-average estimate (Proposition 7.6), with GRH as the only external benchmark. No fitted parameter is renamed as a prediction, and no equation in the 3/5 chain is equivalent to its input by construction. The sole load-bearing self-citation is Conjecture 1.4, imported from the authors' [11] and used to extend the support to (−1,1) and lower the rank bound to 3/2; that extension is explicitly conditional and does not affect the independent 3/5 result. Because the conjecture is openly assumed rather than smuggled, and because the main theorem has independent content, the circularity score is 4 rather than higher.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No data-fitted constants appear; auxiliary optimization parameters such as the Vaughan cutoff U and the support parameter ν are chosen to balance error terms, not fitted to data. No new particles, forces, or entities are introduced. The strongest result imports an unproved conjecture from [11] and inherits GRH, so the ledger is dominated by background axioms rather than invented objects.

assumptions (3)
  • domain assumption Generalized Riemann Hypothesis for L(s,E_d) for all d in F*
    Assumed in Theorems 1.1 and 1.2; used in Lemma 4.1 to replace sums over fourth-power-free d with sums over all d, and to control remainder terms via Lindelöf-type bounds for L'/L.
  • domain assumption Quartic analogue of Patterson's conjecture (Conjecture 1.4)
    Imported from [11, Conjecture 1.2]; needed to relax the Fourier support to (-1,1) and to obtain the 3/2 average rank bound. It is an unproved distributional conjecture for quartic Gauss sums, not established in this paper.
  • standard math Standard analytic background: quartic reciprocity, meromorphic continuation and functional equations for metaplectic theta functions, quadratic large sieve, Vaughan identity
    Invoked in Sections 2, 6, 7, and 8 from references [15], [24], [33], [36], and others; treated as established literature rather than proved here.

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Pith. "Pith review of Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$." pith.science (2026). https://pith.science/paper/3HBIQ2UC

@misc{pith2026260806286,
  author       = {Pith},
  title        = {Pith review of: Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3HBIQ2UC}},
  note         = {Machine review of arXiv:2608.06286}
}
abstract

We study the average analytic rank in the family of $L$-functions $L(s, E_d)$ associated with the elliptic curves $E_d : y^2=x^3-dx$, as $d$ varies over fourth-power-free odd integers. Since this is a family of curves with complex multiplication, we have $L(s, E_d)=L(s - \frac12, \xi_d)$, where $\xi_d$ is a Hecke character over $\mathbb{Z}[i]$. Assuming the Generalized Riemann Hypothesis, we compute the one-level density of the low-lying zeros of this family for test functions whose Fourier transform is supported in $(-\frac35, \frac35)$. As a consequence, we obtain the upper bound $\frac{13}{6}$ for the average analytic rank $r(E_d)$ over the family. Under the additional assumption of a conjecture on the distribution of quartic Gauss sums at prime elements (a quartic analogue of Patterson's conjecture for cubic Gauss sums), we extend the admissible support to $(-1, 1)$ and improve the upper bound for the average analytic rank to $\frac32$. Both results imply that a positive proportion of twists satisfy $r(E_d) =1$, while the second also yields a positive proportion of twists with $r(E_d)=0$.

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Works this paper leans on

47 extracted references · 43 canonical work pages

  1. [11]

    David, A

    C. David, A. Dunn, A. Hamieh and H. Liu,Quartic Gauss sums over primes and metaplectic theta functions, to appear in Algebra & Number Theory

    work page

  2. [23]

    D. R. Heath-Brown,A mean value estimate for real character sums, Acta Arith.72(3) (1995), 235–275

    work page 1995

  3. [1]

    Alp¨ oge, M

    L. Alp¨ oge, M. Bhargava, A.Shnidman,Integers expressible as the sum of two rational cubes(with an appendix by A. Burungale and C. Skinner), preprint, arXiv:2210.10730 (2022)

    arXiv 2022

  4. [2]

    B. J. Birch and N. M. Stephens,The parity of the rank of the Mordell-Weil group, Topology5(1966), 295–299

    work page 1966

  5. [3]

    BrumerThe average rank of elliptic curves I, Invent Math109(1992), 445–472

    A. BrumerThe average rank of elliptic curves I, Invent Math109(1992), 445–472

    work page 1992

  6. [4]

    Burungale and Ye Tian,A rank zerop-converse to a theorem of Gross–Zagier, Kolyvagin and Rubin, preprint, arXiv:2506.03465 (2025)

    A. Burungale and Ye Tian,A rank zerop-converse to a theorem of Gross–Zagier, Kolyvagin and Rubin, preprint, arXiv:2506.03465 (2025). 48

    work page arXiv 2025

  7. [5]

    Non-vanishing for quartic Hecke $L$-functions and ranks of elliptic curves

    C. Castillo, A. de Faveri and A. Dunn,Non-vanishing for quartic HeckeL-functions and ranks of elliptic curves, preprint, arXiv:2604.01316 (2026)

    work page Pith review arXiv 2026

  8. [6]

    P. J. Cho and K. Jeong,On the distribution of analytic ranks of elliptic curvesMath. Z.305(2023), no. 3, Paper No. 42, 20 p

    work page 2023

Show all 47 references
  1. [7]

    Comeau-Lapointe,One-level density of the family of twists of an elliptic curve over function fields, J

    A. Comeau-Lapointe,One-level density of the family of twists of an elliptic curve over function fields, J. Number Theory241(2022), 165–197

    2022

  2. [8]

    Davenport,Multiplicative Number Theory, Second Edition, Springer-Verlag, New York, 1980

    H. Davenport,Multiplicative Number Theory, Second Edition, Springer-Verlag, New York, 1980

    1980

  3. [9]

    David, A

    C. David, A. de Faveri, A. Dunn and J. Stucky,Non-vanishing for cubic HeckeL-functions, preprint, arXiv:2410.03048 (2024)

    2024

  4. [10]

    David, L

    C. David, L. Devin and E. Waxman,One-level densities in families of Gr¨ ossencharakters associated to CM elliptic curves, Mathematika72(2026), no. 1, Paper No. e70067

    2026

  5. [12]

    David, A

    C. David, A. Florea, and M. Lalin,Nonvanishing of L–functions associated to fixed order characters over function fields, preprint, arXiv:2506.07815 (2025)

    2025 arXiv

  6. [13]

    David and A

    C. David and A. G¨ ulo˘ glu,One-level density and non-vanishing for cubicL-functions over the Eisenstein field, Int. Math. Res. Not. IMRN2022, no. 23, 18833–18873

  7. [14]

    de Faveri, A

    A. de Faveri, A. Dunn and J. Hoffstein,Non-vanishing for cubic HeckeL-functions, preprint, arXiv:2607.07911 (2026)

    2026 arXiv

  8. [15]

    Diaconu,Mean square values of HeckeL–series formed withr–th order characters, Invent

    A. Diaconu,Mean square values of HeckeL–series formed withr–th order characters, Invent. Math. 157, 635–684 (2004)

    2004

  9. [16]

    Diaconu, B

    A. Diaconu, B. Ion, V. Pasol and A. Popa,On the second moment and non-vanishing of central values of Hecke L-functions ofr-th order characters, preprint, arXiv:2607.27131 (2026)

    2026 arXiv

  10. [17]

    Dunn and M

    A. Dunn and M. Radziwi l l,Bias in cubic Gauss sums: Patterson ’s conjecture, Ann. of Math. (2)200 (2024), no. 3, 967–1057

    2024

  11. [18]

    Fiorilli,A conditional determination of the average rank of elliptic curves, J

    D. Fiorilli,A conditional determination of the average rank of elliptic curves, J. London Math. Soc. (2) 94(2016), no. 3, 767–792

    2016

  12. [19]

    Fiorilli, J

    D. Fiorilli, J. Parks and A. S¨ odergren,Low-lying zeros of elliptic curve L-functions: beyond the ratios conjecture, Math. Proc. Camb. Philos. Soc.160(2016) no. 2, 315–351

    2016

  13. [20]

    Gao and L

    P. Gao and L. ZhaoOne level density of low-lying zeros of families of L-functions, Compos. Math.147 (2011) no. 1, 1–18

    2011

  14. [21]

    Gao and L

    P. Gao and L. Zhao,One-level density of low-lying zeros of quadratic and quartic HeckeL-functions, Can. J. Math.72(2020) no. 2, 427–454

    2020

  15. [22]

    Goldfeld,Conjectures on elliptic curves over quadratic fields, Number theory, Proc

    D. Goldfeld,Conjectures on elliptic curves over quadratic fields, Number theory, Proc. Conf., Carbondale 1979, Lect. Notes Math.751, (1979), 108–118

    1979

  16. [24]

    D. R. Heath-Brown,Kummer’s conjecture for cubic Gauss sums, Israel J. Math.120(2000), 97–124

    2000

  17. [25]

    D. R. Heath-Brown,The average analytic rank of elliptic curves, Duke Math. J.122(2004), no. 3, 591–623

    2004

  18. [26]

    D. R. Heath-Brown and S. J. Patterson,The distribution of Kummer sums at prime arguments, J. Reine Angew. Math.310(1979), 111–130

    1979

  19. [27]

    Ireland and M

    K. Ireland and M. Rosen,A classical introduction to modern number theory, Second edition, Graduate Texts in Mathematics, 84, Springer-Verlag, New York, 1990

    1990

  20. [28]

    Iwaniec and E

    H. Iwaniec and E. Kowalski,Analytic number theory, American Mathematical Society Colloquium Pub- lications 53, American Mathematical Society, Providence RI, 2004

    2004

  21. [29]

    Iwaniec, W

    H. Iwaniec, W. Luo, P. Sarnak,Low lying zeros of families ofL-functions, Publications Math´ ematiques de l’I.H. ´E.S., tome 91 (2000), 55–131

    2000

  22. [30]

    Katz and P

    N. Katz and P. Sarnak,Random Matrices, Frobenius Eigenvalues and Monodromy, AMS Colloq. Publ. 45 (1999)

    1999

  23. [31]

    Katz and P

    N. Katz and P. Sarnak,Zeroes of zeta functions and symmetry, Bull. Amer. Math. Soc. (N.S.)36(1999), no. 1, 1–26

    1999

  24. [32]

    Koymans and A

    P. Koymans and A. Smith,Sums of rational cubes and the3-Selmer group, preprint, arXiv:2405.09311 (2024). 49

    2024 arXiv

  25. [33]

    Lemmermeyer,Reciprocity laws: From Euler to Eisenstein, Springer Monogr

    F. Lemmermeyer,Reciprocity laws: From Euler to Eisenstein, Springer Monogr. Math. Springer-Verlag, Berlin, (2000)

    2000

  26. [34]

    Meisner and A

    P. Meisner and A. S¨ odergrenLow-lying zeros in families of elliptic curve L-functions over function fields, Finite Fields Appl.84,(2022), 46 p

    2022

  27. [35]

    S. J. Miller,One- and two-level densities for rational families of elliptic curves: evidence for the under- lying group symmetries. Compos. Math.140(2004) no. 4, 952–992

    2004

  28. [36]

    Onodera,Bound for the sum involving the Jacobi symbol inZ[i], Funct

    K. Onodera,Bound for the sum involving the Jacobi symbol inZ[i], Funct. Approx. Comment. Math. 41(2009), 71–103

    2009

  29. [37]

    S. J. Patterson,On the distribution of Kummer sums, J. Reine Angew. Math.303/304, (1978), 126–143

    1978

  30. [38]

    S. J. Patterson,The distribution of general Gauss sums and similar arithmetic functions at prime arguments, Proc. London Math. Soc. (3)54(1987), no. 2, 193–215

    1987

  31. [39]

    Phillips,Average analytic ranks of elliptic curves over number fields, Forum Math

    T. Phillips,Average analytic ranks of elliptic curves over number fields, Forum Math. Sigma13(2025), e40, 1–36

    2025

  32. [40]

    Sarnak, S

    P. Sarnak, S. W. Shin and N. Templier,Families ofL-functions and their symmetry, Proceedings of Simons Symposia,Families of Automorphic Forms and the Trace Formula, Springer-Verlag (2016), 531–578

    2016

  33. [41]

    Smith,The distribution ofℓ ∞-Selmer groups in degreeℓtwist families

    A. Smith,The distribution ofℓ ∞-Selmer groups in degreeℓtwist families. I, J. Am. Math. Soc.39,(2026), no. 1, 1–72

    2026

  34. [42]

    Smith,The distribution ofℓ ∞-Selmer groups in degreeℓtwist families

    A. Smith,The distribution ofℓ ∞-Selmer groups in degreeℓtwist families. II, J. Am. Math. Soc. 39,(2026), no. 2, 453–514

    2026

  35. [43]

    Smith,The Birch and Swinnerton-Dyer conjecture implies Goldfeld’s conjecture, preprint, arXiv:2503.17619 (2025)

    A. Smith,The Birch and Swinnerton-Dyer conjecture implies Goldfeld’s conjecture, preprint, arXiv:2503.17619 (2025)

    2025 arXiv

  36. [44]

    Soundararajan,Nonvanishing of quadratic DirichletL-functions ats= 1 2 , Ann

    K. Soundararajan,Nonvanishing of quadratic DirichletL-functions ats= 1 2 , Ann. of Math. (2)152 (2000), no. 2, 447–488

    2000

  37. [45]

    Suzuki,Some results on the coefficients of the biquadratic theta series, J

    T. Suzuki,Some results on the coefficients of the biquadratic theta series, J. Reine Angew. Math.340 (1983), 70–117

    1983

  38. [46]

    R. C. Vaughan,Sommes trigonom´ etriques sur les nombres premiers, Comptes Rendus de l’Acad´ emie des Sciences, S´ erie A, 285 (1977), 981–983

    1977

  39. [47]

    M. P. Young,Low-lying zeros of families of elliptic curves, J. Amer. Math. Soc.19(2006), no. 1, 205–250. Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve West, Montreal, H2G 1M8, Qu ´ebec, Canada. Email address:chntl.david@gmail.com Univ. Lit...

    2006

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