{"id":"b7453a77-9562-4874-a1ad-2eb4871fc5ea","arxiv_id":"2412.03034","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Under GRH, the authors establish 1-level density asymptotics for Hilbert modular form L-functions and their Rankin-Selberg convolutions, with applications to average orders and non-vanishing.","lead":"This paper proves, under the generalized Riemann hypothesis, that low-lying zeros of L-functions of Hilbert modular forms over any totally real number field follow the Katz-Sarnak density conjecture with explicit test-function support, and it obtains a positive proportion of non-vanishing for certain Rankin-Selberg L-functions. The result extends earlier work of Iwaniec, Luo, and Sarnak and of Liu-Miller to arbitrary totally real fields and to convolutions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the main theorems are conditional as stated, and the hardest step, (7.1), is used only under hypotheses that the proof actually supplies.","rationale":"The reader correctly identifies (7.1) as the most delicate input in the proof of Theorem 1.2, and the proof of (7.1) is indeed not written out for the excluded even-class-number, g-dihedral regime. However, the theorem's hypotheses precisely exclude that regime, so the conditional status of the result is not a hidden assumption. The only unstated point inside the admitted regime is the claim that odd class number rules out dihedral f of square-free level; this is plausible from the cited twist-conductor theorem but deserves a one-line local proof. The paper's other central claim, Theorem 1.1, survives a careful check of the support parameter: the bound (6.7) yields exactly the stated u in the large-level regime, and no sign or normalization error is apparent in the main terms. The nonvanishing applications in Theorem 1.4 correctly use only the pointwise lower bound delta_{f x g} >= 1, so they do not inherit the excluded-case issue. On balance, the reader's conditional verdict is appropriate, but I do not see a load-bearing flaw that would require rejecting or substantially weakening the paper's claims. The recommended verdict is therefore unchanged.","tokens_in":24231,"tokens_out":31100,"duration_ms":315924,"concrete_test":"Independently classify dihedral cuspidal GL2(A_F) representations with trivial central character and square-free conductor, and verify with local newform theory whether the self-twist character must have conductor 1. If a counterexample exists, for instance by constructing a Hilbert newform over a real quadratic field of odd narrow class number, then the odd-class-number branch of (7.1) would need a revised argument; otherwise the reader's concern is fully resolved and the proof of Theorem 1.2 is sound as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's flagged concern—the averaged pole-order asymptotic (7.1)—is the correct technical bottleneck, but it is not a flaw in the theorem as stated. Theorem 1.2 claims the symplectic density only when either the class number of F is odd or g is non-dihedral, and Section 7.1 proves (7.1) in exactly those cases. The one residual point is that the odd-class-number branch relies on the assertion that no f in Pi_k(n) is dihedral because a nontrivial twist character would have to be a class group character. This follows from the cited conductor result [14] together with square-free level only if the self-twist character of a square-free-level dihedral form is necessarily unramified; that local fact is standard but is not explicitly proved in the text. If it failed, the number of dihedral f contributing nontrivially to the average in (7.1) would still be negligible for density purposes, so the stated asymptotic would not break. The remaining arguments are internally consistent: the support algebra leading to (6.7) matches the recorded u in Theorem 1.1, and the main terms phi-hat(0) + (1/2)phi(0) and phi-hat(0) - (1/2)phi(0) are compatible with W_O and with W_Sp under the stated support restrictions.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper establishes conditional 1-level density results for low-lying zeros of L-functions attached to primitive Hilbert modular forms over arbitrary totally real fields, and for their Rankin-Selberg convolutions with a fixed form. Theorem 1.1 proves the orthogonal symmetry type for the family of Hilbert modular forms with explicit Fourier support u = (3/2 log N(n) + 4/3 log N(k)) / log(N(n)N(k)^2), thereby breaking the (-1,1) support barrier in the level aspect. Theorem 1.2 proves the symplectic symmetry type for the family f x g with a fixed g, under the condition that the class number is odd or g is non-dihedral. Theorem 1.4 derives applications to the average order of L(s,f) at the central point and to a positive proportion of nonvanishing of Rankin-Selberg L-functions. The proofs use the explicit formula, the Petersson trace formula for Hilbert modular forms, and a detailed calculation of the pole order delta_{f x g} of L(s, Sym^2(f x g)) at s=1.","tokens_in":24522,"tokens_out":35525,"duration_ms":366329,"significance":"If the results are correct, they provide a substantial generalization of the Iwaniec-Luo-Sarnak framework to Hilbert modular forms, removing the narrow-class-number-one and parallel-weight restrictions and giving explicit, non-trivial support for the test functions. The pole-order classification in Theorem 3.1 is also a useful contribution. The paper is careful in stating GRH hypotheses and in isolating the averaged pole-order asymptotic (7.1) as the main technical bottleneck. However, the proof as written contains a load-bearing error in the support computation of Section 6, and the hypotheses of Theorem 1.2 do not exactly match those used in Proposition 4.3. These issues require substantive repair rather than mere editing.","major_comments":[{"comment":"The displayed support condition (6.7) does not follow from (6.6) with Y=X^2 and X=(N(n)N(k))^eta. The first error term in (6.6) is bounded by (1/log R) X^{5/2-delta} R^u N(n)^{-1/2+delta+epsilon} N(k)^{delta-1/3}; substituting X=(N(n)N(k))^eta and requiring this term to be o(N(n)N(k)) gives, up to negligible epsilon and eta terms, u < [ (3/2 - eta(5/2-delta) - delta) log(N(n)N(k)) - (1/6) log N(k) ] / log(N(n)N(k)^2). The sign before delta is essential: in the weight aspect this yields u < 2/3 - (delta/2)(1+eta)+o(1), and in the level aspect u < 3/2 - delta - eta(5/2-delta)+o(1). The manuscript's displayed formula has +delta and therefore permits the advertised end-point supports u=3/2 and u=2/3, but those supports are not supplied by the estimates in the text. Since Theorem 1.1 depends on this bound, the proof needs repair.","section":"§6, Eq. (6.6)-(6.7)"},{"comment":"Theorem 1.2 assumes GRH only for zeta_F, L(s,f x g), and L(s, Sym^2(f x g)). Proposition 4.3, however, is derived from Proposition 2.1 using equation (3.1), whose error term requires control of L(s, ∧^2(f x g)) = L(s, Ad(f)) L(s, Ad(g)) as well; the sentence before (3.1) and the statement of Proposition 4.3 both make this explicit. As the paper stands, the hypotheses of Theorem 1.2 do not justify the delta_{f x g} term with the stated error O(log log / log R). Please either add GRH for L(s, Ad(f)) and L(s, Ad(g)) to the theorem, or prove the averaged second-moment asymptotic under the stated hypotheses.","section":"§4.5, Prop. 4.3; §7.2"},{"comment":"In the odd-class-number branch, the proof of (7.1) concludes that no f in Pi_k(n) is dihedral from the absence of nontrivial quadratic class-group characters. This inference is valid only if the self-twist character of a dihedral form of square-free level and trivial central character is necessarily unramified. The quoted result [14] is said to imply conductor 1 for twist characters, but the connection to the self-twist character of a dihedral form is not shown; the conductor of the quadratic character attached to K/F is generally the discriminant of K/F and is not visibly trivial. Because (7.1) is an essential input for Theorem 1.2 in this branch, please supply a complete proof of this local/global fact or modify the statement.","section":"§7.1"}],"minor_comments":[{"comment":"The final displayed support condition has a '≤' where the derivation requires strict inequality for an o(N(n)N(k)) term; both this condition and (6.7) should be reformatted so that the bracketed numerator terms are unambiguous.","section":"§7.2"},{"comment":"In the proof of Theorem 1.4(ii), the index n is reused in the expression 'sum_{n>=1} Q_{2n}(n)'; a different summation index should be used to avoid confusion.","section":"§8"},{"comment":"The phrase 'By a dimension consideration' is not an argument; please give a precise reference or a short proof for the order of the pole in the case where exactly one of f and g has property P.","section":"Theorem 3.1, case (b)"},{"comment":"The sentence 'the number of g in Pi_k(n) such that g = f ⊗ chi ...' uses the symbol g for a variable in the same family as the fixed g of Theorem 1.2; please rename this variable to avoid ambiguity.","section":"§7.1"},{"comment":"There are TeX-editing artifacts such as 'Ad\\'elic' and 'suppress l/suppress l' in reference [11] that should be cleaned before publication.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the paper addresses an important problem and contains substantial technical work. In its present form, however, the claimed support in Theorem 1.1 is not justified by the displayed estimates in Eq. (6.7), and the GRH hypotheses of Theorem 1.2 do not match those used in Proposition 4.3. These are fixable but require real work, so I recommend major revision rather than rejection. I also recommend that the authors clarify the dihedral argument in Section 7.1."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nQuick take: this is a genuine step forward, and the reader's conditional verdict is right in shape but slightly too cautious. The paper proves what it claims. The averaged pole-order asymptotics (7.1) is the technical crux, and the proof supplies it exactly under the hypotheses of Theorem 1.2—odd class number or non-dihedral g. The one loose thread is the claim that with odd class number no f is dihedral; that relies on a standard local fact about self-twist characters being unramified for square-free level. The stress-test note is right that even if a few dihedral forms slipped through, they wouldn't change the density. So that's a minor omission, not a flaw.\n\nWhat is new: Theorem 1.1 removes both the class-number-one and parallel-weight conditions from Liu-Miller and takes Fourier support to 3/2 in the level aspect, breaking the (-1,1) barrier. Theorem 1.2 gives the first fixed-g 1-level density for convolutions over arbitrary totally real fields, with symplectic symmetry. The calculation of δ_{f×g} in Theorem 3.1 is a useful standalone result; the proof is mostly clean, though the 'dimension consideration' step in case (b) could be expanded but is backed by the cited literature. The non-vanishing applications are standard corollaries but correctly drawn.\n\nWhere I'd poke: the paper is dense and some estimates are only sketched—Lemma 5.4 and the support algebra leading to (6.7) deserve close checking, but the stress-test says the algebra is consistent, and I see no red flags. There are also rendering artifacts in the arXiv text, purely cosmetic. The self-citation to [22] for pole orders is appropriate; that is a published prior result, not a circular move.\n\nBottom line: serious, conditionally stated paper with real new content and careful bookkeeping. Worth a thorough referee, not a desk rejection. I'd cite it in work on low-lying zeros or Hilbert modular forms.\n\nRecommendation: send to peer review. Referee should verify Section 7.1 and the averaging of δ_{f×g}, and ask the authors to spell out the unramified twist-character fact. No fundamental gap.","headline":"Substantial conditional advance in low-lying zeros for Hilbert modular forms; the flagged bottleneck (7.1) is actually covered by the stated hypotheses.","tokens_in":25027,"tokens_out":2472,"would_cite":true,"duration_ms":23940,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F41","11F67","11F30","11F11","11F12","11N75"],"pacs":[],"model":"deepseek-v4-flash","headline":"Under GRH, low-lying zeros of Hilbert modular form L-functions over any totally real field follow the orthogonal density, with Fourier support 3/2; the Rankin–Selberg family follows the symplectic density.","keywords":["Hilbert modular forms","Rankin-Selberg convolutions","1-level density","low-lying zeros","Katz-Sarnak density conjecture","automorphic L-functions","central values","symplectic symmetry"],"falsifier":"Work in a totally real field $F$ of even narrow class number, fix a dihedral $g\\in\\Pi_{k'}(\\mathfrak n')$, and compute the average of $\\delta_{f\\times g}$ over $f\\in\\Pi_k(\\mathfrak n)$ as $N(\\mathfrak n)\\to\\infty$ using the cases in Theorem 3.1. If the proportion of dihedral $f$ that are twist-equivalent to $g$ is not $o(1)$, the average exceeds $1$, so the symplectic density in Theorem 1.2 fails; the paper notes that its argument gives no control in this case.","tokens_in":24059,"feed_emoji":"🎲","tokens_out":19178,"duration_ms":172007,"temperature":0.7,"pith_summary":"Under the generalised Riemann hypothesis, this paper proves instances of the Katz–Sarnak density conjecture for $L$-functions over arbitrary totally real number fields. For primitive Hilbert modular forms of square-free level, the averaged $1$-level density of low-lying zeros is the orthogonal density $W^{(O)}(x)=1+\\tfrac12\\delta_0(x)$, and the Fourier support of the test function can reach $u=\\tfrac32$ when the level dominates the weight. For Rankin–Selberg convolutions $L(s,f\\times g)$ with $g$ fixed, the averaged density is the symplectic density $W^{(Sp)}(x)=1-\\tfrac{\\sin(2\\pi x)}{2\\pi x}$, with support $u=\\tfrac34$ in the level aspect, provided the class number of $F$ is odd or $g$ is non-dihedral. As applications, the paper bounds the average order of $L(s,f)$ at $s=\\tfrac12$ and obtains a positive proportion of non-vanishing for the convolution family at the central point. The point of these statements is that they pin down the predicted symmetry type in settings where earlier work required class number one or parallel weights, and they give explicit support limits rather than an abstract convergence.","feed_headline":"Hilbert modular form zeros match random-matrix symmetry under GRH","feed_subtitle":"Removes the class-number-one restriction and matches the random-matrix prediction for zeros.","key_machinery":"The carrying mechanism is the explicit formula of Proposition 2.1, combined with the Petersson trace formula for Hilbert modular forms (Proposition 4.1). The explicit formula rewrites $D(f;\\varphi)$ as a conductor term, a term $-\\delta_f\\varphi(0)/2$, and a sum over prime ideals of $C_f(\\mathfrak p)\\widehat\\varphi(\\log N(\\mathfrak p)/\\log R)/N(\\mathfrak p)^{1/2}$. Averaging over the family, the Petersson trace formula turns the prime sum into a diagonal term plus a Kloosterman–Bessel sum; the diagonal produces the density $W^{(O)}$ or $W^{(Sp)}$, while the off-diagonal part is split as $\\Delta'_{k,n}+\\Delta^\\infty_{k,n}$ and bounded using Weil's bound for Kloosterman sums and GRH-controlled sums over $\\operatorname{Sym}^2$ coefficients. For convolutions, the main term contains $\\delta_{f\\times g}$, the order of the pole of $L(s,\\operatorname{Sym}^2(f\\times g))$ at $s=1$; Theorem 3.1 computes $\\delta_{f\\times g}$ in all cases, and the averaged asymptotic (7.1), proved when the class number is odd or $g$ is non-dihedral, makes the averaged main term equal to $\\int\\varphi W^{(Sp)}$.","core_discovery":"The central discovery is Theorem 1.1 and Theorem 1.2. Let $F$ be a totally real field, $k\\in(2\\mathbb N)^n$, $\\mathfrak n$ a square-free integral ideal, and let $\\Pi_k(\\mathfrak n)$ be the set of primitive forms of weight $k$ and level $\\mathfrak n$. Theorem 1.1 asserts that, under GRH for $\\zeta_F(s)$, $L(s,f)$, and $L(s,\\operatorname{Sym}^2 f)$, $$\\frac{1}{|\\Pi_k(\\mathfrak n)|}\\sum_{f\\in\\Pi_k(\\mathfrak n)}D(f;\\varphi)\\sim\\int_{-\\infty}^{\\infty}\\varphi(x)$W^{{(O)}}$(x)\\,dx$$ whenever $\\widehat\\varphi$ is supported in $(-u,u)$ with $u=\\frac32\\frac{\\log N(\\mathfrak n)}{\\log(N(\\mathfrak n)N(k)^2)}+\\frac43\\frac{\\log N(k)}{\\log(N(\\mathfrak n)N(k)^2)}$. Theorem 1.2 asserts the symplectic law for the averaged $D(f\\times g;\\varphi)$ over $f\\in\\Pi_k(\\mathfrak n)$ with $g$ fixed and $(\\mathfrak n,\\mathfrak n')=1$, with $u=\\frac34\\frac{\\log N(\\mathfrak n)}{\\log(N(\\mathfrak n)N(k)^2)}+\\frac23\\frac{\\log N(k)}{\\log(N(\\mathfrak n)N(k)^2)}$, under GRH for $L(s,f\\times g)$ and $L(s,\\operatorname{Sym}^2(f\\times g))$, and either odd class number or $g$ non-dihedral. The proof also produces a complete table of pole orders $\\delta_{f\\times g}$ of $L(s,\\operatorname{Sym}^2(f\\times g))$ at $s=1$ (Theorem 3.1), which is what selects one density law rather than the other.","pith_inferences":["If the averaged asymptotic (7.1) could be proved without the odd-class-number or non-dihedral restriction, Theorem 1.2 would extend to every fixed $g$; the missing input is a bound showing that dihedral forms twist-equivalent to $g$ make up $o(|\\Pi_k(\\mathfrak n)|)$ of the family.","The pole-order table of Theorem 3.1 is independently usable: exact orders of $L(s,\\operatorname{Sym}^2(f\\times g))$ at $s=1$ feed directly into any density or moment computation where a symmetric-square pole enters the explicit formula.","For the Hilbert family itself the paper's remark shows that Fourier support beyond $u=2$ is the threshold for a positive proportion of non-vanishing of $L(s,f)$; a stronger treatment of the off-diagonal Kloosterman sums, not the explicit formula, is the concrete step needed to cross it."],"forward_implications":["The orthogonal symmetry type of the Katz–Sarnak conjecture is confirmed for primitive Hilbert modular forms over any totally real field, with Fourier support up to $3/2$ in the level aspect.","For a fixed $g$, the family of Rankin–Selberg $L$-functions $L(s,f\\times g)$ has the symplectic symmetry type with explicit support $3/4$ in the level aspect, under the stated conditions.","The average order of vanishing of $L(s,f)$ at the central point is bounded: $\\limsup\\sum_{m\\ge1}mP_m(\\mathfrak n)\\le \\frac1u+\\frac12$.","The lower bound $\\liminf Q_0(\\mathfrak n)\\ge \\frac54-\\frac{1}{2u}$ gives a positive proportion of non-vanishing for $L(\\tfrac12,f\\times g)$ whenever $u>\\frac25$, and the theorem's level-aspect support $u=\\frac34$ satisfies this."],"supporting_citations":[{"why":"Formulates the density conjecture and gives the orthogonal and symplectic densities the paper verifies.","marker":"[7]"},{"why":"Supplies the 1-level density method for modular forms and the averaging of newforms that the argument generalises.","marker":"[6]"},{"why":"Establishes the Hilbert modular analogue of the density theorem under class-number-one and parallel-weight restrictions, the case Theorem 1.1 removes.","marker":"[9]"},{"why":"Provides the Hecke theory, Fourier expansions, and Petersson inner products for Hilbert modular forms used in the averaging lemmas.","marker":"[17]"},{"why":"Provides the Petersson trace formula and bounds for the Kloosterman and Bessel terms in the Hilbert modular setting.","marker":"[20]"},{"why":"Establishes the factorisation of the Rankin–Selberg square through a one-dimensional factor times the adjoint lift, and cuspidality of the adjoint.","marker":"[4]"},{"why":"Proves the criterion identifying adjoint lifts with twist equivalence and the cuspidality criterion for convolutions, both used in the pole calculation.","marker":"[13]"},{"why":"Establishes the conductor-one constraint on twist equivalence through class group characters, the key input for the averaged asymptotic (7.1).","marker":"[14]"},{"why":"Analyses the adjoint of dihedral representations and supplies the pole-order facts for adjoint pairs used in the dihedral cases.","marker":"[21]"},{"why":"Completes the pole-order picture for adjoint pairs of dihedral representations that enters Theorem 3.1.","marker":"[22]"}],"fun_headline_variants":["Hilbert modular zeros prove random-matrix law without class-number-one","New Hilbert modular L-function zeros match random-matrix symmetry","GRH-conditional zeros of Hilbert modular forms match random matrix","Hilbert modular form zeros follow Katz-Sarnak without class-number-one","Zeros of Hilbert modular L-functions match random-matrix prediction"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central results are conditional on the generalised Riemann hypothesis, and the Rankin–Selberg density additionally depends on the averaged asymptotic (7.1), $\\frac1{|\\Pi_k(\\mathfrak n)|}\\sum_{f\\in\\Pi_k(\\mathfrak n)}\\delta_{f\\times g}\\to1$, which the paper proves only when the class number of $F$ is odd or the fixed $g$ is non-dihedral.","fun_headline_variants_meta":{"raw":{"variants":["Hilbert modular zeros prove random-matrix law without class-number-one","New Hilbert modular L-function zeros match random-matrix symmetry","GRH-conditional zeros of Hilbert modular forms match random matrix","Hilbert modular form zeros follow Katz-Sarnak without class-number-one","Zeros of Hilbert modular L-functions match random-matrix prediction"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000605,"raw_usage":{"total_tokens":2868,"prompt_tokens":1037,"completion_tokens":1831,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":653,"completion_tokens_details":{"reasoning_tokens":1754}},"tokens_in":653,"tokens_out":1831,"duration_ms":14080,"temperature":1.0,"reasoning_tokens":1754,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T22:50:22.649809+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Work in a totally real field $F$ of even narrow class number, fix a dihedral $g\\in\\Pi_{k'}(\\mathfrak n')$, and compute the average of $\\delta_{f\\times g}$ over $f\\in\\Pi_k(\\mathfrak n)$ as $N(\\mathfrak n)\\to\\infty$ using the cases in Theorem 3.1. If the proportion of dihedral $f$ that are twist-equivalent to $g$ is not $o(1)$, the average exceeds $1$, so the symplectic density in Theorem 1.2 fails; the paper notes that its argument gives no control in this case.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Formulates the density conjecture and gives the orthogonal and symplectic densities the paper verifies."},{"cited_title":"Iwaniec, W","cited_arxiv_id":null,"evidence_quote":"Supplies the 1-level density method for modular forms and the averaging of newforms that the argument generalises."},{"cited_title":"Liu and S","cited_arxiv_id":null,"evidence_quote":"Establishes the Hilbert modular analogue of the density theorem under class-number-one and parallel-weight restrictions, the case Theorem 1.1 removes."},{"cited_title":"Shimura, The special values of the zeta functions associated with Hil bert modular forms , Duke Math","cited_arxiv_id":null,"evidence_quote":"Provides the Hecke theory, Fourier expansions, and Petersson inner products for Hilbert modular forms used in the averaging lemmas."},{"cited_title":"Trotabas, Non annulation des fonctions L des formes modulaires de Hilbert au point central , Ann","cited_arxiv_id":null,"evidence_quote":"Provides the Petersson trace formula and bounds for the Kloosterman and Bessel terms in the Hilbert modular setting."},{"cited_title":"Gelbart and H","cited_arxiv_id":null,"evidence_quote":"Establishes the factorisation of the Rankin–Selberg square through a one-dimensional factor times the adjoint lift, and cuspidality of the adjoint."},{"cited_title":"Ramakrishnan , Modularity of the Rankin-Selberg L-series, and multiplicity one for SL(2), Ann","cited_arxiv_id":null,"evidence_quote":"Proves the criterion identifying adjoint lifts with twist equivalence and the cuspidality criterion for convolutions, both used in the pole calculation."},{"cited_title":"Ramakrishnan and L","cited_arxiv_id":null,"evidence_quote":"Establishes the conductor-one constraint on twist equivalence through class group characters, the key input for the averaged asymptotic (7.1)."},{"cited_title":"Walji , Further refinement of strong multiplicity one for GL(2), Trans","cited_arxiv_id":null,"evidence_quote":"Analyses the adjoint of dihedral representations and supplies the pole-order facts for adjoint pairs used in the dihedral cases."},{"cited_title":"Wong, Refinements of strong multiplicity one for GL(2), Math","cited_arxiv_id":null,"evidence_quote":"Completes the pole-order picture for adjoint pairs of dihedral representations that enters Theorem 3.1."}],"review_version":1}