{"id":"ca7a8ae6-0de6-4566-9c5f-4e6237dacbbe","arxiv_id":"1908.05980","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"U(p) and Ramanujan-type congruences for Hermitian Jacobi forms of any index and for degree-2 Hermitian modular forms over Q(i) are characterized by the mod p filtration of heat-operator iterates, via an isomorphism with matrix-index Jacobi forms.","lead":"A number theory paper develops congruence theory modulo primes for Hermitian Jacobi forms over the Gaussian integers and for degree-2 Hermitian modular forms, characterizing two classic classes of coefficient congruences. It supplies a new isomorphism between two spaces of Jacobi forms and uses it to extend prior index-1 results to arbitrary index.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Freeness transfer in Corollary 3.6 is the load-bearing unverified black box; the hypotheses of [22] for B=mI2 are not checked.","rationale":"I read the paper in good faith. The explicit isomorphism in Theorem 2.3 appears correct, and the arithmetic in Section 4, including the derivation of the two possible values in Theorem 4.3 from Lemma 4.1, is internally coherent. The strongest claim, Theorem 4.3, is genuinely conditional on the filtration theory built in Section 3. That theory depends on Corollary 3.6, which is not proved in the paper and is transferred from Raum--Richter without a stated verification of hypotheses. This is the same weakest assumption identified by the reader. I agree with the CONDITIONAL verdict: the concern is real but may be resolvable by checking the cited hypotheses or by adding a direct freeness proof for the Hermitian setting. I do not see an internal contradiction that would force REJECT, and the computational examples, while not reproducible from the text, are illustrative rather than load-bearing for Theorem 4.3. A secondary statement-level issue is that Theorem 3.3 asserts equality where the proof establishes only a congruence modulo p; this is worth correcting but does not change the main structural risk.","tokens_in":27692,"tokens_out":18050,"duration_ms":171227,"concrete_test":"Extract the exact hypotheses of [22, Thm 2.8, Prop 2.11, Thm 2.14] and check them for the lattice (Z^2, m(x^2+y^2)) with B = diag(m,m), first for p∤m and then for p|m. If the hypotheses fail for p|m, restrict Corollary 3.6, Proposition 3.7, and Theorem 3.3 to p∤m and check whether every later use is covered by the main theorems' assumptions. As an independent computational cross-check, compute dim J^1_{k,1}(F_p) for p=5 and a range of k, say 4 <= k <= 30, and compare with the prediction of freeness of rank 4 over M_*(F_5); any systematic deviation would indicate that unique mod-p reduction fails even in the range needed by Theorem 4.3.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The entire filtration and heat-cycle machinery rests on Corollary 3.6: J^1_{*,m}(Gamma_1(O), Z_(p)) is free of rank 4m^2 over M_*(SL_2(Z), Z_(p)) with an integral basis and unique mod-p coefficient reduction. This is obtained by transferring Lemma 3.4 and Lemma 3.5 from the matrix-index setting via Theorem 2.3. But Lemma 3.4 and Lemma 3.5 are only quoted as particular cases of [22, Thm 2.8, Prop 2.11, Thm 2.14]; the paper never states the hypotheses of those results and never verifies them for B = diag(m,m). The index form m(x^2+y^2) becomes degenerate modulo p when p divides m, and the statement of Corollary 3.6 covers all m, while Proposition 3.7 and Theorem 3.3 use it without a p∤m restriction. A failure of freeness or of unique coefficient reduction in that range would make the expansions phi = sum f_i psi_i and the modular-form filtration arguments from [30] invalid. The main congruence theorems assume p∤m, so a failure only at p|m could be repaired by restricting the statements, but the paper does not make that restriction and does not prove freeness directly. Even in the range p∤m, the transfer relies on an unverified black box; if the rank or the mod-p reduction property fails there, Theorem 4.3 loses its foundation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops a mod p theory for Hermitian Jacobi forms over Q(i) of arbitrary index m, and applies it to U(p) congruences and Ramanujan-type congruences for Hermitian Jacobi forms and degree-2 Hermitian modular forms. The main technical tool is an isomorphism (Theorem 2.3) between the space J^1_{k,m}(Γ_1(O)) of Jacobi forms on the Hermitian Jacobi group and the space J_{k,B}(Γ_2) of matrix-index Jacobi forms with B = diag(m,m). This isomorphism transfers the Raum–Richter freeness theorem for matrix-index Jacobi forms to J^1_{*,m}(Γ_1(O), Z_(p)) (Corollary 3.6). Using this freeness, the authors prove a Swinnerton-Dyer type theorem (Theorem 3.1), a heat-operator filtration theorem (Theorem 3.3), and then a precise formula (Theorem 4.3) for the filtration of the (p+2−k)-th heat iterate: if φ is a nonzero mod p Hermitian Jacobi form of weight k < p, index m with p ∤ m, then Ω(L^{p+2−k}_m(φ)) equals 2p+4−k or p+5−k according as φ | U(p) is nonzero or zero mod p. They also prove a non-existence theorem for Ramanujan-type congruences (Theorem 4.7). The second half of the paper transfers these results to symmetric Hermitian modular forms of degree 2: a filtration result for the heat operator (Proposition 7.7), a U(p) characterization (Theorem 8.2), and a Ramanujan-type non-existence theorem (Theorem 8.4), with SAGE-supported examples.","tokens_in":27920,"tokens_out":18534,"duration_ms":147308,"significance":"If the main theorems are correct, the paper provides a substantial generalization of earlier work of Richter and Senadheera for index 1 to arbitrary index, and the explicit filtration formula in Theorem 4.3 is a sharp, falsifiable prediction: it identifies U(p)-annihilation from a single filtration value. The isomorphism in Theorem 2.3 is elegant and likely to be useful beyond this paper. The examples are verified by explicit SAGE computations rather than numerical fits. The main caveats are that the entire Hermitian Jacobi theory rests on an unverified transfer of the Raum–Richter freeness theorem (Corollary 3.6), and that the Hermitian modular form results depend on an unproved filtration criterion (Corollary 7.3). These are gaps in verification and presentation, not obvious contradictions, and they are fixable.","major_comments":[{"comment":"The freeness of J^1_{*,m}(Γ_1(O), Z_(p)) as a module over M_*(SL_2(Z), Z_(p)) of rank 4m², with an integral basis and unique mod p coefficient reduction, is quoted as a particular case of [22, Thm 2.8, Prop 2.11, Thm 2.14]. The paper does not state the hypotheses of those results and does not verify them for B = diag(m,m). In particular, the index form m(x²+y²) becomes degenerate modulo p when p | m, and Corollary 3.6 is stated for all m while Proposition 3.7 and Theorem 3.3 use it without a p ∤ m restriction. Since Theorem 3.3 is the foundation of Theorem 4.3, this is a load-bearing unverified step. Please state the exact hypotheses of [22] and check them for B = diag(m,m), or restrict the statements to p ∤ m and adjust the main theorems accordingly; alternatively, give a direct proof of freeness in the Hermitian setting.","section":"§3, Lemma 3.4/3.5 and Corollary 3.6"},{"comment":"The criterion ℧(F) < k if and only if \\bar B divides \\bar P_F is asserted with the proof omitted as 'similar to [17, Theorem 7.5(i)]'. This corollary is used essentially in Lemma 7.4 and Proposition 7.7, so it is load-bearing for the Hermitian modular form results. A full proof, or a precise statement of the hypotheses under which the elliptic modular form argument transfers to this polynomial-ring setting, should be provided.","section":"§7, Corollary 7.3"},{"comment":"After establishing the existence of m with p ∤ m and Ω(φ_m) = ℧(F), the proof is concluded by 'using Proposition 7.7 and following a similar argument as in the proof of Theorem 4.3'. This omits the key point that for a Hermitian modular form F, ℧(D^j(F)) is the supremum of Ω(L^j_m(φ_m)) over all Fourier–Jacobi components m, and one must show that the chosen component φ_m continues to control ℧(D^j(F)) for every j in the heat cycle, including at the drop where other components could in principle contribute. Please supply the induction in detail.","section":"§8, Theorem 8.2"}],"minor_comments":[{"comment":"The reference to 'Theorem 7.7' should be 'Proposition 7.7'.","section":"§8, proof of Theorem 8.2"},{"comment":"There is an extra parenthesis in '℧(D^{p+2−k}(F )))'.","section":"§8, statement of Theorem 8.2"},{"comment":"The notation L^{p+1/2}_m appears without parentheses; it should be L^{(p+1)/2}_m. This occurs in several places, including the statement and proof of Proposition 4.5 and Theorem 5.1.","section":"§4 and §5"},{"comment":"The Sturm bound η(k,m) is asserted to hold in characteristic p with the comment that Das's proof goes through; since this is used to verify the examples, please provide a proof or a precise reference that covers the characteristic p case. Also, the letter p is used both for the prime and for the primes dividing 4m in the product, which is confusing.","section":"§5.2, Proposition 5.2"},{"comment":"The wording 'the low point is L^j_m(φ)' is slightly confusing because low points are defined as L^{i+1}_m(φ) after a high point L^i_m(φ); it should clarify that L^j_m(φ) is the representative of the low point in the periodic heat cycle.","section":"§4, Lemma 4.1"},{"comment":"In the examples, the prime is not fixed when writing φ^+_{4,1} ∈ HJ^+_{4,1}(Γ_J(O), Z_(p)); each example should specify the prime being considered.","section":"§5.1"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid contribution to the subject, but the reliance on unstated hypotheses of [22] and the omitted proof of Corollary 7.3 make it unsuitable for publication in its present form. I recommend major revision. The authors should be encouraged to state the exact Raum–Richter theorems and verify the hypotheses for B = diag(m,m), and to provide full proofs for Corollary 7.3 and the inductive step in Theorem 8.2. If these points are resolved, the paper is likely to be acceptable. The isomorphism in Theorem 2.3 and the explicit filtration formula in Theorem 4.3 are strong selling points."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a solid, useful extension of the mod p Jacobi-form program to Hermitian objects over Q(i), powered by a clean isomorphism between Gaussian-integer Jacobi forms and matrix-index Jacobi forms. The main congruence theorems are new and plausible, but the paper has a misstated key theorem and leans on an unverified freeness transfer that should be pinned down before publication.\n\nWhat is actually new: Theorem 2.3 is a neat explicit isomorphism between J^1_{k,m}(Gamma_1(O)) and J_{k,B}(Gamma_2) for B = diag(m,m). It is the right tool: it lets the authors import Raum–Richter's matrix-index theory into the Hermitian setting. From that they get arbitrary-index filtration theorems (3.1, 3.3) and then U(p) and Ramanujan-type characterizations for Hermitian Jacobi forms (4.3, 4.7), plus the degree-2 Hermitian modular form analogues (8.2, 8.4). These genuinely generalize the index-1 work of Richter–Senadheera and the Siegel cases of Choi–Choie–Richter and Dewar–Richter. I checked several internal arguments—the Fourier-index bijection in 2.3, the choice of rho in 2.6, the heat-cycle arithmetic in Lemma 4.1 and Theorem 4.3—and they hold up. The dependency chain is linear, not circular; the paper leans on known black boxes, not on its own conclusions.\n\nThe soft spots are real but mostly fixable. First, Theorem 3.3 states L_m(phi) = psi with psi a Hermitian Jacobi form over Z_(p), but the proof only establishes a congruence modulo p. As written, the equality is not true: the heat operator does not preserve the space integrally. This statement error sits in a key theorem and must be corrected to congruence. Second, the filtration machinery rests on Corollary 3.6, a freeness statement transferred from Raum–Richter via the isomorphism, but the paper never states the hypotheses of the cited theorems or verifies them for B = diag(m,m), especially the mod p coefficient-reduction property. The main theorems assume p does not divide m, so a failure only at p|m would be repairable by a restriction, but the authors should either prove the freeness directly in the Hermitian setting or give a precise citation with conditions. Third, some proofs are deferred to the literature (Sturm bound in characteristic p, Corollary 7.3) without checking hypotheses, and the SAGE examples are not reproducible from the text. Those are minor by comparison.\n\nWho is this for? Anyone working on mod p congruences for higher-degree automorphic forms. It deserves a serious referee: the core ideas are sound and the flaws are correctable.","headline":"A competent extension of the mod p Jacobi-form program to Hermitian forms, with a clean new isomorphism; the key heat-operator theorem is misstated and the freeness transfer needs verification, but the flaws are fixable.","tokens_in":28572,"tokens_out":4933,"would_cite":true,"duration_ms":47997,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F33","11F55","11F50"],"pacs":[],"model":"deepseek-v4-flash","headline":"For Hermitian Jacobi forms over Q(i), the mod p image under the U(p) operator is encoded exactly in the filtration of a single heat iterate.","keywords":["Hermitian Jacobi forms","Hermitian modular forms","mod p congruences","U(p) congruences","Ramanujan-type congruences","heat operator","filtration","Gaussian integers"],"falsifier":"Take $p=5$, $m=5$, $k=4$, and any Hermitian Jacobi form $\\varphi$ that is nonzero mod $5$; compute $\\varphi|U(5)$ mod $5$ and $\\Omega(L_5^3(\\varphi))$ (since $p+2-k=3$). The theorem predicts $\\Omega=10$ if $\\varphi|U(5)$ survives and $\\Omega=6$ if it vanishes; a single computation matching neither value would refute the dichotomy, and independently, checking whether an integral free basis exists for index $5$ directly tests the transferred freeness assumption.","tokens_in":27322,"feed_emoji":"🧮","tokens_out":14472,"duration_ms":117745,"temperature":0.7,"pith_summary":"This paper develops a theory of Hermitian Jacobi forms modulo a prime $p$ over the Gaussian integers, for arbitrary index $m$, and uses it to characterize two standard kinds of congruences. The central result is a dichotomy: if $p\\geq 5$, $k\\geq 4$, $p>k$, and $p\\nmid m$, then for a nonzero mod $p$ Hermitian Jacobi form $\\varphi$ the filtration $\\Omega(L_m^{p+2-k}(\\varphi))$ is exactly $2p+4-k$ when $\\varphi|U(p)$ is nonzero mod $p$, and exactly $p+5-k$ when $\\varphi|U(p)$ is zero mod $p$. In other words, a single number attached to one heat iterate decides whether a U(p)-type congruence holds, turning an infinite coefficient condition into a finite invariant. The same dichotomy is proved for symmetric Hermitian modular forms of degree 2, and a companion theorem rules out Ramanujan-type congruences (when the first heat iterate is nonzero mod $p$) except possibly when $p\\leq k$ or $p=2k-3$. If correct, the paper gives Hermitian analogues of the classical mod $p$ filtration arguments and supplies a practical filter for finding such congruences.","feed_headline":"Filtration value decides U(p) congruences in Hermitian Jacobi forms","feed_subtitle":"For p>k, the (p+2-k)-th heat iterate has filtration either 2p+4-k or p+5-k, telling whether phi|U(p) vanishes mod p.","key_machinery":"The load-bearing objects are the heat operator $L_m=-(1/\\pi^2)(2\\pi i m\\,\\partial_\\tau-\\partial_{z_1}\\partial_{z_2})$, its iterates forming the heat cycle $L_m(\\varphi),\\ldots,L_m^{p-1}(\\varphi)$, and the filtration $\\Omega(\\varphi)$, the least weight in which $\\varphi$ appears modulo $p$. The key fact is Theorem 3.3: applying the heat operator raises $\\Omega$ by $p+1$, with equality unless $p$ divides $(\\Omega(\\varphi)-1)m$. This is proved by writing Hermitian Jacobi forms in terms of the free module basis supplied by Corollary 3.6, which itself comes from the isomorphism of Theorem 2.3 and the freeness of matrix-index Jacobi forms. The U(p) operator connects to the heat cycle because $\\varphi|U(p)\\equiv 0\\pmod p$ exactly when $L_m^{p-1}(\\varphi)\\equiv\\varphi\\pmod p$; the filtration constraints then force the exact values in Theorem 4.3.","core_discovery":"The paper's central claim is that U(p)-annihilation in Hermitian Jacobi forms is visible in the heat cycle. Concretely, Theorem 4.3 states that under the hypotheses above, the filtration of the $(p+2-k)$-th heat iterate has one of two exact values according as $\\varphi|U(p)$ survives mod $p$. The proof route is a new isomorphism (Theorem 2.3) between the space of Jacobi forms on $\\Gamma_1(O)$ and the space of matrix-index Jacobi forms with index matrix $B=\\mathrm{diag}(m,m)$; this transfers structural facts about freeness and coefficient reduction mod $p$ from matrix-index Jacobi forms. A filtration-jump theorem for the heat operator (Theorem 3.3) then controls how $\\Omega$ changes through the $p-1$ iterates of the heat cycle, and a counting argument with high and low points forces the two displayed values. For Ramanujan-type congruences, the paper proves an equivalence with a congruence between two heat iterates, and derives a non-existence theorem: when $L_m(\\varphi)$ is nonzero mod $p$, $p>k$, and $p\\neq 2k-3$, no such congruence exists. Symmetric Hermitian modular forms of degree 2 inherit the same statements through their Fourier-Jacobi expansion.","pith_inferences":["The index-raising construction used in the examples suggests that a U(p) congruence for index 1 can be transferred to higher indices by twisting with $\\rho\\in O$; this could be systematized into a general index-raising congruence theorem.","If the transferred freeness assumption holds for all indices, the same filtration formula should hold verbatim when $p$ divides $m$; a direct check for $p=5,m=5$ would either close the gap or reveal a genuinely different mod $p$ structure.","The non-existence theorem leaves $p=2k-3$ as the only possible Ramanujan-type regime; the paper's tables suggest a classification of all such congruences at that borderline weight is within reach."],"forward_implications":["For every Hermitian Jacobi form in the stated range, computing $\\Omega(L_m^{p+2-k}(\\varphi))$ decides whether $\\varphi|U(p)$ vanishes mod $p$, so U(p) congruences become checkable from finitely many Fourier coefficients.","The heat cycle of a nonzero form has either one or two low points, and no filtration in the cycle is congruent to 2 mod $p$; these structural constraints will constrain any future mod $p$ theory of Hermitian Jacobi forms.","A Hermitian Jacobi form with $L_m(\\varphi)\\not\\equiv 0\\pmod p$, $p>k$, and $p\\neq 2k-3$ has no Ramanujan-type congruence at any nonzero $b$ mod $p$; the only possible exceptional weight is $k=(p+3)/2$.","For symmetric Hermitian modular forms of degree 2 satisfying a mild nonvanishing Fourier-coefficient condition, the same U(p) dichotomy and the same Ramanujan-type criterion hold, so the Hermitian Jacobi results transfer to Hermitian modular forms."],"supporting_citations":[{"why":"Supplies the freeness and mod p coefficient-reduction theorems for matrix-index Jacobi forms that are transferred to the Hermitian setting through the paper's isomorphism.","marker":"[22]"},{"why":"Gives the heat-operator filtration argument for Jacobi forms that Proposition 3.7 adapts to prove the Hermitian filtration jump.","marker":"[24]"},{"why":"Provides the index-1 Hermitian Jacobi forms mod p results and U(p) congruence characterization that this paper extends to arbitrary index.","marker":"[25]"},{"why":"Supplies the filtration results for modular forms multiplied by E_2, used to control the Eisenstein correction in the heat operator.","marker":"[30]"},{"why":"Provides the Ramanujan-type congruence criterion via the (p-1)/2 heat iterate, which Proposition 4.5 transfers to Hermitian Jacobi forms.","marker":"[7]"},{"why":"Provides the Siegel modular form U(p) filtration argument that the Hermitian modular form version follows.","marker":"[2]"},{"why":"Gives the polynomial ring structure of symmetric Hermitian modular forms modulo p, used in the filtration proof for Hermitian modular forms.","marker":"[14]"},{"why":"Gives the heat/theta operator congruence for Hermitian modular forms modulo p used in Proposition 7.7.","marker":"[15]"},{"why":"Supplies the heat operator lemma for Hermitian Jacobi forms and the index-1 examples used to illustrate the congruences.","marker":"[26]"}],"fun_headline_variants":["Heat cycle filtration reveals U(p) vanishing in Hermitian Jacobi forms","New isomorphism pins down U(p) congruences and Ramanujan-type p-congruences","Filtration jumps decide when U(p) kills Hermitian Jacobi forms mod p","Heat iterates expose U(p) congruences and non-existence of Ramanujan-type","Two-step filtration test for U(p)-annihilation in Hermitian Jacobi forms"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"All the filtration theorems rest on Corollary 3.6: the module of Hermitian Jacobi forms of index $m$ over the ring of elliptic modular forms is free of rank $4m^2$ with a basis of integral forms, so reduction modulo $p$ is unique; this is imported from matrix-index Jacobi forms via the new isomorphism, and the required hypotheses are not verified for indices $m$ divisible by $p$.","fun_headline_variants_meta":{"raw":{"variants":["Heat cycle filtration reveals U(p) vanishing in Hermitian Jacobi forms","New isomorphism pins down U(p) congruences and Ramanujan-type p-congruences","Filtration jumps decide when U(p) kills Hermitian Jacobi forms mod p","Heat iterates expose U(p) congruences and non-existence of Ramanujan-type","Two-step filtration test for U(p)-annihilation in Hermitian Jacobi forms"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00024,"raw_usage":{"total_tokens":1502,"prompt_tokens":911,"completion_tokens":591,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":527,"completion_tokens_details":{"reasoning_tokens":485}},"tokens_in":527,"tokens_out":591,"duration_ms":5240,"temperature":1.0,"reasoning_tokens":485,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:04:56.167566+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $p=5$, $m=5$, $k=4$, and any Hermitian Jacobi form $\\varphi$ that is nonzero mod $5$; compute $\\varphi|U(5)$ mod $5$ and $\\Omega(L_5^3(\\varphi))$ (since $p+2-k=3$). The theorem predicts $\\Omega=10$ if $\\varphi|U(5)$ survives and $\\Omega=6$ if it vanishes; a single computation matching neither value would refute the dichotomy, and independently, checking whether an integral free basis exists for index $5$ directly tests the transferred freeness assumption.","supporting_citations":[{"cited_title":"Raum and O","cited_arxiv_id":null,"evidence_quote":"Supplies the freeness and mod p coefficient-reduction theorems for matrix-index Jacobi forms that are transferred to the Hermitian setting through the paper's isomorphism."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the heat-operator filtration argument for Jacobi forms that Proposition 3.7 adapts to prove the Hermitian filtration jump."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the index-1 Hermitian Jacobi forms mod p results and U(p) congruence characterization that this paper extends to arbitrary index."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the filtration results for modular forms multiplied by E_2, used to control the Eisenstein correction in the heat operator."},{"cited_title":"Dewar and O","cited_arxiv_id":null,"evidence_quote":"Provides the Ramanujan-type congruence criterion via the (p-1)/2 heat iterate, which Proposition 4.5 transfers to Hermitian Jacobi forms."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Siegel modular form U(p) filtration argument that the Hermitian modular form version follows."},{"cited_title":"Kikuta and S","cited_arxiv_id":null,"evidence_quote":"Gives the polynomial ring structure of symmetric Hermitian modular forms modulo p, used in the filtration proof for Hermitian modular forms."},{"cited_title":"Kikuta and S","cited_arxiv_id":null,"evidence_quote":"Gives the heat/theta operator congruence for Hermitian modular forms modulo p used in Proposition 7.7."},{"cited_title":"Senadheera, Hermitian Jacobi forms and congruences , Thesis (Ph.D.)–University of North Texas, 2014, 66 pp","cited_arxiv_id":null,"evidence_quote":"Supplies the heat operator lemma for Hermitian Jacobi forms and the index-1 examples used to illustrate the congruences."}],"review_version":1}