Universal murmuration and Hecke augmentation

Chung Pang Mok; Shenghao Hua

REVIEW

Universal murmuration and Hecke augmentation

Shenghao Hua, Chung Pang Mok

Number Theory · Submitted 1 Sep 2026 · 20 references

Pith's verdict: Solid trace-formula extension and honestly controlled experiments, but Proposition 4.3's validity for every fixed k is asserted rather than proved and needs to be pinned down before the main theorem is fully trusted.

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arXiv ID2609.00777 (version 1)

Pith IDMULWOBL6, Pith's permanent record of this paper and its review

CitationChung Pang Mok, Shenghao Hua. "Universal murmuration and Hecke augmentation." arXiv:2609.00777 (2026). https://pith.science/paper/MULWOBL6

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Editor's note

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The real news here is Theorem 4.1: a genuine derivation of the root-number-weighted symmetric-power trace for every fixed k≥3, with the square-case cutoff corrected, all wrapped around an explicit profile G(y)=M(y)/√y. The universal Conjecture 1.1 is a natural and useful organizing principle, and the paper is careful to label it as conjectural. The empirical section is also better than most of this genre: it separates the asymptotic statement from the representation claim and the prediction claim, runs matched monomial, conductor, solver, shuffle, and conductor-quintile controls, and reports the small, task-dependent gain without overclaiming. The five-assignment stability and the explicit formula (3.5) are honest diagnostics.

Now the soft spot, and I think the reader's report underweights it. Proposition 4.3 is load-bearing for every k≥2, but the citation [9, Theorem 2.1] only treats k=2, and the reference to Skoruppa–Zagier does not itself show the reduction to this exact shape for general k—especially the term −Σ_{j=0}^k P^j and the cutoff 2P^{k/2}/√N. No derivation and no numerical check are supplied. If that formula degenerates for some k, the parabolic and hyperbolic averages collapse and the common profile for k≥3 does not follow. I do not think this is fatal: the Skoruppa–Zagier framework is general enough that the formula is probably valid, and the paper's internal bookkeeping for k is otherwise consistent. But it is more than a disclosure issue. The authors should either prove Proposition 4.3 from [18] in an appendix, cite a source that states it for general k, or at minimum verify it numerically for k=3,4,5 on small levels. Until then, Theorem 4.1 is conditional on an unstated lemma.

Minor points: the missing package-version metadata in the main run is disclosed and partly remedied by the confirmation run; the single-block experiment lacks validation records, but it is explicitly labeled exploratory; the finite-data conductor transport is unstable, and the paper says so. None of these change the verdict.

Bottom line: this deserves a serious referee. I would send it to review and ask for Proposition 4.3 to be secured. For readers working on murmuration or arithmetic statistics, the conjecture and the k≥3 computation are worth engaging with now.

Referee report

Major objections (2)

[§4.1, Proposition 4.3] The manuscript applies the Skoruppa–Zagier trace formula to every fixed $k\ge 2$, but the cited reference [9, Theorem 2.1] only treats the case $k=2$. The statement of Proposition 4.3 includes the term $-\sum_{j=0}^k P^j$ and the range $0<r\le 2P^{k/2}/\sqrt{N}$; the dependence on $k$ enters through $Q=P^k$. The authors assert it is a “special case” and reference [18] and [9], but they provide no derivation, no explicit check that the general formulas in [18] reduce to this form for arbitrary $k$, and no numerical verification even for small $k$. Since the entire proof of Theorem 1.2 for $k\ge 2$ (and therefore the central claim of the paper) rests on this identity, the argument is incomplete. The authors must either supply a complete derivation, cite a known general theorem that covers all $k$, or provide computational evidence that the identity holds for the range of $k$ used in the proof. This is a load‑bearing gap.
[§4.4, completion of the proof] The error analysis in the completion of Theorem 4.1 uses the parabolic and hyperbolic estimates that are derived from Proposition 4.3. If Proposition 4.3 is not established for all $k$, then the entire error bound is conditional. Additionally, the dimension estimate at the end of §4.4 (taken from Zubrilina) is applied to the sum that includes the elementary divisor $\sum_{j=0}^k P^j$; this term depends on $k$ and its contribution is handled separately. That part is fine, but the overall logical dependence on the unverified trace formula remains.

Minor objections (6)

[Abstract and §1] The abstract states that the theorem extends Zubrilina’s prime case and Kundu–Müller’s square case, but Remark 1.3 notes that Kundu–Müller’s final displayed profile omitted a factor of 2. This should be made clearer in the abstract or introduction to avoid confusing readers about what is new.
[§2.2, Proposition 2.1] The notation $M_C$ is defined differently for the two families; it would help to give a unified definition inline, or at least to remark that the two cases are parallel.
[§3.2, Proposition 3.1] The proof states that “the constant function and $x_p$ are already present in the affine model.” This is true only if the affine model includes an intercept. It would be clearer to state explicitly that the model contains a constant term.
[§5.1, Data description] The text says “The unit of analysis is therefore an isogeny class representative.” This is fine, but it differs from the family in Theorem 4.1 (which averages all weight‑2 newforms). A brief remark on the scope difference (as in Table 3) would be helpful earlier.
[§6.2, Figure 3] The denominator 8750 is the midpoint of the conductor window. The caption says “$\sqrt{8750}\,(\mu_+ - \mu_-)/2$” but the plot shows a scaled contrast. The scaling by $\sqrt{8750}$ is motivated by Theorem 1.2, but the theorem uses the lower endpoint $X$ of an asymptotically short window. The discrepancy should be noted in the text, not only in the appendix.
[§6.3, Table 5] The “Change with interval” column includes bootstrap intervals. It would be useful to state whether these are percentile intervals or normal‑approximation intervals. From Appendix C it seems they are percentile intervals; that should be stated in the main text.

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The paper

The paper claims

Prime-power coefficients share one murmuration profile

The reading

This paper proves that for weight 2 newforms, the root-number-weighted average of the k-th symmetric-power coefficient at a prime P, when placed at the effective position P^k / X, is given by the same function G as the prime coefficient (k=1) case. This means prime powers sample the same leading murmuration pattern. The result is proved for every fixed k>=2 under short-window and growth conditions. The paper also leverages this connection to suggest using Hecke-derived coordinates (H2, H3) as features in machine learning, which gives a modest but consistent improvement in predicting root numbers of elliptic curves and a smaller improvement in distinguishing rank 0 versus rank 1.

What carries the argument

The Skoruppa–Zagier trace formula (Proposition 4.3) expresses the root-number-weighted Hecke trace at a prime power as a combination of class numbers and a polynomial term. The proof averages this formula over short intervals of squarefree levels using estimates for class numbers and Hooley's squarefree progression estimate, isolating the leading parabolic and hyperbolic contributions that yield a common profile independent of k.

Extended reading notes

Core claim

Theorem 1.2: Under the level-aspect and specified growth conditions, the expression $\sqrt$(X) * E_{X,Y}[ ε(f) $λ^{{sym_k}}$_f(P) ] = G(P^k/X) + o_k(1) for a continuous explicit function G, showing every fixed prime power samples the same leading murmuration function when placed at the effective position P^k/X.

Load-bearing premise

The proof depends on the Skoruppa–Zagier trace formula (Proposition 4.3) being valid for every fixed k at squarefree level, and on the short-window condition that the level interval length Y grows slower than X (Y ~ $X^{{1-δ1}}$ with δ1 < 1/8). If the trace formula does not hold for k≥3 in the squarefree case, or if the window is too wide, the common profile for k≥2 would not follow.

Editorial extensions

If this is right

The effective position P^k/X allows prime-power coefficients to be treated on the same footing as primes in murmuration predictions, unifying their statistical behavior.
Hecke coordinates H2 and H3 provide a principled way to incorporate higher-power information without additional coefficient queries, motivating structured feature augmentation.
The explicit form of G gives a precise asymptotic for the expected correlation between root sign and symmetric-power coefficients in large families, guiding further arithmetic statistics.
The result suggests that the same G function may govern all prime powers simultaneously for families where trace formulas extend to arbitrary indices.

Reading between the lines

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

The uniformity across fixed k suggests that the murmuration function G may be universal in the sense that it arises from a single functional equation, not merely from the trace formula. Proving this would require averaging over all k simultaneously rather than fixing k.
The Hecke augmentation framework could be extended to other automorphic families (e.g., Maass forms or higher weight) if analogous trace formulas for prime powers are available, linking arithmetic statistics to machine learning more broadly.
The empirical gain, though modest, indicates that the root-number signal is not purely linear in the prime coefficient but has a structured polynomial component, supporting the theoretical picture of a common profile.
The comparison between fixed-k theorem and finite-data experiment suggests that the asymptotic regime (short level windows) may not be necessary for observable predictive gains; practical benefit may appear even in moderate conductor ranges.

Original abstract

Prime coefficients of elliptic curves exhibit murmurations, statistical patterns that support the prediction of arithmetic labels. We conjecture that root number weighted averages of unitary normalized coefficients at primes and prime powers sample the same leading profile when placed at the effective position $p^k/X$, where $X$ is the conductor scale. For weight $2$ newforms of squarefree level in the level aspect, we prove this principle for every fixed $k$ under suitable short-window and growth conditions, extending Zubrilina's prime case and the square-case analysis of Kundu and M\"uller. Experiments with elliptic curve isogeny class representatives show that the resulting prime power features improve root number prediction and give a smaller gain in distinguishing ranks $0$ and $1$.

The compiled record

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

Prime coefficients of elliptic curves exhibit murmurations, statistical patterns that support the prediction of arithmetic labels. We conjecture that root number weighted averages of unitary normalized coefficients at primes and prime powers sample the same leading profile when placed at the effective position $p^k/X$, where $X$ is the conductor scale. For weight $2$ newforms of squarefree level in the level aspect, we prove this principle for every fixed $k$ under suitable short-window and growth conditions, extending Zubrilina's prime case and the square-case analysis of Kundu and M\"uller. Experiments with elliptic curve isogeny class representatives show that the resulting prime power features improve root number prediction and give a smaller gain in distinguishing ranks $0$ and $1$.

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	Prime coefficients of elliptic curves exhibit murmurations, statistical patterns that support the prediction of arithmetic labels.	background 1.00	0.13	0.08	asserted 0.90
s1	We conjecture that root number weighted averages of unitary normalized coefficients at primes and prime powers sample the same leading profile when placed at the effective position $p^k/X$, where $X$ is the conductor scale.	contribution 0.93	0.23	0.27	speculative 0.99
s2	For weight $2$ newforms of squarefree level in the level aspect, we prove this principle for every fixed $k$ under suitable short-window and growth conditions, extending Zubrilina's prime case and the square-case analysis of Kundu and M\"uller.	contribution 0.59	0.95	0.67	asserted 1.00
s3	Experiments with elliptic curve isogeny class representatives show that the resulting prime power features improve root number prediction and give a smaller gain in distinguishing ranks $0$ and $1$.	result 0.91	0.90	0.08	asserted 1.00

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

not examined

full text
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references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
C citations
R cross-paper relations

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

Figures

**Figure 1.** Rank-conditioned murmuration profiles in the four feature representations

**Figure 2.** Basic murmuration profiles from ordinary prime coefficients

**Figure 3.** Balanced root-sign contrasts for ordinary prime coefficients and the coordinates H2 and H3 obtained from Hecke relations in 7500 ≤ N ≤ 10000. The horizontal coordinate is the effective position p k/8750. Error bars are pointwise 95 percent normal intervals based on standard errors. The second and third panels sample the same oscillatory pattern as the first panel at the available prime-power locations

**Figure 4.** Balanced root-sign contrasts for k = 1, . . . , 12 in 7500 ≤ N ≤ 10000. At each of the first 200 base primes, Dk(p) = {µk,+(p) − µk,−(p)}/2, where µk,±(p) is the mean of Hk(xp) = ap k /pk/2 over curves with root sign ±1 after ramified entries are omitted. Every panel uses the same vertical scale. The displayed value is Rk = (200−1 P200 j=1 Dk(pj ) 2 ) 1/2

**Figure 5.** Balanced accuracy for the four feature panels and five prediction tasks. The first three rank rows have a high ordinary prime baseline. The root-number row leaves more room for a visible absolute gain

**Figure 6.** Test error counts for the ordinary prime panel and the full augmented panel. The root number and rank 0 versus rank 1 tasks show fewer errors after augmentation. The three-class task and rank 0 versus rank 2 show the task dependence of the result. Add H2 Add H3 Add H2 and H3 Added block 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Change in balanced accuracy percentage points Root number Add H2 Add H3 Add H2 and H3 Added block 0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Change in balanced accuracy percentage points Rank 0 against rank 1 Stability across repeated fitting splits

**Figure 7.** Change in balanced accuracy relative to the ordinary prime panel over five stratified assignments from one fixed balanced population. Each point is one complete fit with regularization selected on validation data. The red segment is the mean. The joint H2 and H3 panel improves root-number prediction in all five assignments. and ties in one. Its mean change is 0.2900 percentage points. Its mean full accuracy is 96.5575 percent and it corresponds to about 23.2 fewer errors per 8000 test curves. Applying formula (3.5) to the mean accuracies gives relative error reductions of 2.56…

**Figure 8.** Root-number change in balanced accuracy as derived coordinates are added to X1. Natural prefixes follow effective index order. Random subsets use only five repetitions at each displayed size. The orange bands show the empirical 2.5th and 97.5th percentiles from five subsets. They are descriptive empirical ranges. The curve is irregular and contains several local reversals. 0 5 10 15 20 Added derived coordinates 0.2 0.0 0.2 0.4 0.6 0.8 Change in balanced accuracy percentage points Quadratic block H2 0 2 4 6 8 Added derived coordinates Cubic block H3 0 5 10 15 20 25 30 Added derived…

**Figure 9.** Change in balanced accuracy for rank 0 versus rank 1 as derived coordinates are added to X1. Natural prefixes follow effective index order. Orange bands show the empirical 2.5th and 97.5th percentiles from five random subsets. They are descriptive empirical ranges. The response is gradual and contains local reversals. percentage points, and all five subsets give a positive change. Random joint subsets at size 24 have mean gain 0.5975 percentage points and are positive in all five cases. Cubic subsets alone follow a different pattern. At size 6, their mean root-number change is…

**Figure 10.** Matched polynomial and conductor controls. The upper panels show changes relative to X1 for five random assignments; horizontal segments give means. The lower panels hold out complete conductor quintiles and plot each change at the median conductor of its test fold. preserving class balance

**Figure 11.** Balanced accuracy for the single-block experiment. Every column k uses 1000 features Hk(xp). Binary chance is 0.5, and three-class chance is 1/3. The labels are arithmetic invariants of the original elliptic curve. 6.9. Limitations. The empirical gain depends on the task. The full panel lowers threeclass rank balanced accuracy from 95.5750 percent to 94.6833 percent. Rank 0 versus rank 2 changes from 99.6875 percent to 99.6250 percent. Rank 1 versus rank 2 rises from 99.7125 percent to 99.8625 percent at a baseline already near the ceiling. Across tasks, the effect of additional…

Figures reproduced from arXiv:2609.00777 by the authors.

References

Works cited (20)

[18]

Skoruppa, Nils-Peter; Zagier, Don. Jacobi forms and a certain space of modular forms.Invent. Math. 94(1988), no. 1, 113–146. 13

View on Pith 1988
[9]

Kundu, Debanjana; M¨ uller, Katharina. Murmurations of modular forms andp-power coefficients.Math. Proc. Cambridge Philos. Soc.180 (2026), no. 1, 65–92. 1, 3, 8, 12, 13, 14, 15

View on Pith 2026
[1]

Cremona, John E. The ecdata repository: elliptic-curve data overQ. Release 2026-04-22.https:// github.com/JohnCremona/ecdata/tree/2026-04-22. 27

View on Pith 2026
[2]

Bober, Jonathan W.; Booker, Andrew Richard; Lee, Min; Lowry-Duda, David. Murmurations of mod- ular forms in the weight aspect.Algebra Number Theory20 (2026), no. 2, 299–332. 1, 5, 6, 7

View on Pith 2026
[3]

Booker, Andrew Richard; Lee, Min; Lowry-Duda, David; Seymour-Howell, Andrei; Zubrilina, Nina. Murmurations of Maass forms. (Preprint). arXiv: 2409.00765. 1, 5, 6

View on Pith Pith review arXiv
[4]

Cowan, Alex. Murmurations and ratios conjecture. (Preprint). arXiv: 2408.12723. 1

View on Pith Pith review arXiv
[5]

He, Yang-Hui; Lee, Kyu-Hwan; Oliver, Thomas; Pozdnyakov, Alexey. Murmurations of elliptic curves. Exp. Math.34 (2025), no. 3, 528–540. 1, 30

View on Pith 2025
[6]

He, Yang-Hui; Lee, Kyu-Hwan; Oliver, Thomas; Pozdnyakov, Alexey. Murmurations: a case study in AI-assisted mathematics. (Preprint). arXiv: 2603.09680. 1

View on Pith arXiv
[7]

Hooley, Christopher. A note on square-free numbers in arithmetic progressions.Bull. London Math. Soc.7 (1975), 133–138. 18

View on Pith 1975
[8]

Iwaniec, Henryk; Luo, Wenzhi; Sarnak, Peter. Low lying zeros of families ofL-functions.Publ. Math. Inst. Hautes ´Etudes Sci.91 (2000), 55–131. 1

View on Pith 2000
[10]

Lee, Kyu-Hwan; Oliver, Thomas; Pozdnyakov, Alexey. Murmurations of Dirichlet characters.Int. Math. Res. Not. IMRN2025, no. 1, Paper No. rnae277. 1

View on Pith
[11]

Mok, Chung Pang. Pseudorandom vector generation using elliptic curves and applications to Wiener processes.Finite Fields Appl.85 (2023), Paper No. 102129. 5

View on Pith 2023
[12]

Mok, Chung Pang; Zheng, Huimin. Monte Carlo Integration Using Elliptic Curves.Chin. Ann. Math. Ser. B46(2) (2025), 241–260. 5

View on Pith 2025
[13]

Mok, Chung Pang; Zheng, Huimin. Pseudorandomness of Sato–Tate Distributions for Elliptic Curves. Journal of Experimental Mathematics1 (1) (2025), 186–206. 5

View on Pith 2025
[14]

Pozdnyakov, Alexey. Predicting root numbers with neural networks.Int. J. Data Sci. Math. Sci.2 (2024), no. 1, 15–37. 1

View on Pith 2024
[15]

Sarnak, Peter Clive. Letter to Drew Sutherland and Nina Zubrilina on murmurations and root numbers. https://publications.ias.edu/sarnak/paper/2726, 2023. 1

View on Pith 2023
[16]

Sarnak, Peter; Zubrilina, Nina. Convergence to the Plancherel measure of Hecke eigenvalues.Acta Arith. 214 (2024), 191–213. 1

View on Pith 2024
[17]

Sawin, William F; Sutherland, Andrew V. Murmurations for elliptic curves ordered by height. (Preprint). arXiv: 2504.12295. 1

View on Pith arXiv
[19]

Sutherland, Andrew V. Letter to Michael Rubinstein and Peter Sarnak. Unpublished letter, 2022. 1

View on Pith 2022
[20]

Zubrilina, Nina. Murmurations.Invent. Math.241 (2025), no. 3, 627–680. 1, 3, 9, 12, 14, 15, 20, 21, 22, 27, 45 UNIVERSAL MURMURATION 45 AppendixA.Technical supplements to the level aspect proof This appendix provides three auxiliary details for Section 4: the exceptional local factor, the parameter inequalities, and the endpoint contribution in the prime-...

View on Pith 2025

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] [18]

Skoruppa, Nils-Peter; Zagier, Don. Jacobi forms and a certain space of modular forms.Invent. Math. 94(1988), no. 1, 113–146. 13

View on Pith 1988

[2] [9]

Kundu, Debanjana; M¨ uller, Katharina. Murmurations of modular forms andp-power coefficients.Math. Proc. Cambridge Philos. Soc.180 (2026), no. 1, 65–92. 1, 3, 8, 12, 13, 14, 15

View on Pith 2026

[3] [1]

Cremona, John E. The ecdata repository: elliptic-curve data overQ. Release 2026-04-22.https:// github.com/JohnCremona/ecdata/tree/2026-04-22. 27

View on Pith 2026

[4] [2]

Bober, Jonathan W.; Booker, Andrew Richard; Lee, Min; Lowry-Duda, David. Murmurations of mod- ular forms in the weight aspect.Algebra Number Theory20 (2026), no. 2, 299–332. 1, 5, 6, 7

View on Pith 2026

[5] [3]

Booker, Andrew Richard; Lee, Min; Lowry-Duda, David; Seymour-Howell, Andrei; Zubrilina, Nina. Murmurations of Maass forms. (Preprint). arXiv: 2409.00765. 1, 5, 6

View on Pith Pith review arXiv

[6] [4]

Cowan, Alex. Murmurations and ratios conjecture. (Preprint). arXiv: 2408.12723. 1

View on Pith Pith review arXiv

[7] [5]

He, Yang-Hui; Lee, Kyu-Hwan; Oliver, Thomas; Pozdnyakov, Alexey. Murmurations of elliptic curves. Exp. Math.34 (2025), no. 3, 528–540. 1, 30

View on Pith 2025

[8] [6]

He, Yang-Hui; Lee, Kyu-Hwan; Oliver, Thomas; Pozdnyakov, Alexey. Murmurations: a case study in AI-assisted mathematics. (Preprint). arXiv: 2603.09680. 1

View on Pith arXiv

[9] [7]

Hooley, Christopher. A note on square-free numbers in arithmetic progressions.Bull. London Math. Soc.7 (1975), 133–138. 18

View on Pith 1975

[10] [8]

Iwaniec, Henryk; Luo, Wenzhi; Sarnak, Peter. Low lying zeros of families ofL-functions.Publ. Math. Inst. Hautes ´Etudes Sci.91 (2000), 55–131. 1

View on Pith 2000

[11] [10]

Lee, Kyu-Hwan; Oliver, Thomas; Pozdnyakov, Alexey. Murmurations of Dirichlet characters.Int. Math. Res. Not. IMRN2025, no. 1, Paper No. rnae277. 1

View on Pith

[12] [11]

Mok, Chung Pang. Pseudorandom vector generation using elliptic curves and applications to Wiener processes.Finite Fields Appl.85 (2023), Paper No. 102129. 5

View on Pith 2023

[13] [12]

Mok, Chung Pang; Zheng, Huimin. Monte Carlo Integration Using Elliptic Curves.Chin. Ann. Math. Ser. B46(2) (2025), 241–260. 5

View on Pith 2025

[14] [13]

Mok, Chung Pang; Zheng, Huimin. Pseudorandomness of Sato–Tate Distributions for Elliptic Curves. Journal of Experimental Mathematics1 (1) (2025), 186–206. 5

View on Pith 2025

[15] [14]

Pozdnyakov, Alexey. Predicting root numbers with neural networks.Int. J. Data Sci. Math. Sci.2 (2024), no. 1, 15–37. 1

View on Pith 2024

[16] [15]

Sarnak, Peter Clive. Letter to Drew Sutherland and Nina Zubrilina on murmurations and root numbers. https://publications.ias.edu/sarnak/paper/2726, 2023. 1

View on Pith 2023

[17] [16]

Sarnak, Peter; Zubrilina, Nina. Convergence to the Plancherel measure of Hecke eigenvalues.Acta Arith. 214 (2024), 191–213. 1

View on Pith 2024

[18] [17]

Sawin, William F; Sutherland, Andrew V. Murmurations for elliptic curves ordered by height. (Preprint). arXiv: 2504.12295. 1

View on Pith arXiv

[19] [19]

Sutherland, Andrew V. Letter to Michael Rubinstein and Peter Sarnak. Unpublished letter, 2022. 1

View on Pith 2022

[20] [20]

Zubrilina, Nina. Murmurations.Invent. Math.241 (2025), no. 3, 627–680. 1, 3, 9, 12, 14, 15, 20, 21, 22, 27, 45 UNIVERSAL MURMURATION 45 AppendixA.Technical supplements to the level aspect proof This appendix provides three auxiliary details for Section 4: the exceptional local factor, the parameter inequalities, and the endpoint contribution in the prime-...

View on Pith 2025