{"title":"A bound for the exterior product of S-units","authors":"Shabnam Akhtari, Jeffrey D. Vaaler","doi":"10.2140/ant.2024.18.1589","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1589","url":null,"abstract":"<p>We generalize an inequality for the determinant of a real matrix proved by A. Schinzel, to more general exterior products of vectors in Euclidean space. We apply this inequality to the logarithmic embedding of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>S</mi></math>-units contained in a number field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math>. This leads to a bound for the exterior product of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>S</mi></math>-units expressed as a product of heights. Using a volume formula of P. McMullen we show that our inequality is sharp up to a constant that depends only on the rank of the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>S</mi></math>-unit group but not on the field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math>. Our inequality is related to a conjecture of F. Rodriguez Villegas. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"64 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142245251","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Prime values of f(a,b2) and f(a,p2), f quadratic","authors":"Stanley Yao Xiao","doi":"10.2140/ant.2024.18.1619","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1619","url":null,"abstract":"<p>We prove an asymptotic formula for primes of the shape <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>f</mi><mo stretchy=\"false\">(</mo><mi>a</mi><mo>,</mo><msup><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy=\"false\">)</mo></math> with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>a</mi></math>, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>b</mi></math> integers and of the shape <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>f</mi><mo stretchy=\"false\">(</mo><mi>a</mi><mo>,</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy=\"false\">)</mo></math> with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math> prime. Here <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>f</mi></math> is a binary quadratic form with integer coefficients, irreducible over <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℚ</mi></math> and has no local obstructions. This refines the seminal work of Friedlander and Iwaniec on primes of the form <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup>\u0000<mo>+</mo> <msup><mrow><mi>y</mi></mrow><mrow><mn>4</mn></mrow></msup></math> and of Heath-Brown and Li on primes of the form <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup>\u0000<mo>+</mo> <msup><mrow><mi>p</mi></mrow><mrow><mn>4</mn></mrow></msup></math>, as well as earlier work of the author with Lam and Schindler on primes of the form <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>f</mi><mo stretchy=\"false\">(</mo><mi>a</mi><mo>,</mo><mi>p</mi><mo stretchy=\"false\">)</mo></math> with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>f</mi></math> a positive definite form. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"12 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142245246","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Semistable models for some unitary Shimura varieties over ramified primes","authors":"Ioannis Zachos","doi":"10.2140/ant.2024.18.1715","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1715","url":null,"abstract":"<p>We consider Shimura varieties associated to a unitary group of signature <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><mi>n</mi>\u0000<mo>−</mo> <mn>2</mn><mo>,</mo><mn>2</mn><mo stretchy=\"false\">)</mo></math>. We give regular <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-adic integral models for these varieties over odd primes <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math> which ramify in the imaginary quadratic field with level subgroup at <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math> given by the stabilizer of a selfdual lattice in the hermitian space. Our construction is given by an explicit resolution of a corresponding local model. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"197 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142245250","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Application of a polynomial sieve: beyond separation of variables","authors":"Dante Bonolis, Lillian B. Pierce","doi":"10.2140/ant.2024.18.1515","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1515","url":null,"abstract":"<p>Let a polynomial <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>f</mi>\u0000<mo>∈</mo>\u0000<mi>ℤ</mi><mo stretchy=\"false\">[</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mi>…</mi><mo> <!--FUNCTION APPLICATION--></mo><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">]</mo></math> be given. The square sieve can provide an upper bound for the number of integral <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathvariant=\"bold-italic\"><mi>x</mi></mstyle>\u0000<mo>∈</mo> <msup><mrow><mo stretchy=\"false\">[</mo><mo>−</mo><mi>B</mi><mo>,</mo><mi>B</mi><mo stretchy=\"false\">]</mo></mrow><mrow><mi>n</mi></mrow></msup></math> such that <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>f</mi><mo stretchy=\"false\">(</mo><mstyle mathvariant=\"bold-italic\"><mi>x</mi></mstyle><mo stretchy=\"false\">)</mo></math> is a perfect square. Recently this has been generalized substantially: first to a power sieve, counting <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathvariant=\"bold-italic\"><mi>x</mi></mstyle>\u0000<mo>∈</mo> <msup><mrow><mo stretchy=\"false\">[</mo><mo>−</mo><mi>B</mi><mo>,</mo><mi>B</mi><mo stretchy=\"false\">]</mo></mrow><mrow><mi>n</mi></mrow></msup></math> for which <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>f</mi><mo stretchy=\"false\">(</mo><mstyle mathvariant=\"bold-italic\"><mi>x</mi></mstyle><mo stretchy=\"false\">)</mo>\u0000<mo>=</mo> <msup><mrow><mi>y</mi></mrow><mrow><mi>r</mi></mrow></msup></math> is solvable for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>y</mi>\u0000<mo>∈</mo>\u0000<mi>ℤ</mi></math>; then to a polynomial sieve, counting <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathvariant=\"bold-italic\"><mi>x</mi></mstyle>\u0000<mo>∈</mo> <msup><mrow><mo stretchy=\"false\">[</mo><mo>−</mo><mi>B</mi><mo>,</mo><mi>B</mi><mo stretchy=\"false\">]</mo></mrow><mrow><mi>n</mi></mrow></msup></math> for which <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>f</mi><mo stretchy=\"false\">(</mo><mstyle mathvariant=\"bold-italic\"><mi>x</mi></mstyle><mo stretchy=\"false\">)</mo>\u0000<mo>=</mo>\u0000<mi>g</mi><mo stretchy=\"false\">(</mo><mi>y</mi><mo stretchy=\"false\">)</mo></math> is solvable, for a given polynomial <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>g</mi></math>. Formally, a polynomial sieve lemma can encompass the more general problem of counting <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathvariant=\"bold-italic\"><mi>x</mi></mstyle>\u0000<mo>∈</mo> <msup><mrow><mo stretchy=\"false\">[</mo><mo>−</mo><mi>B</mi><mo>,</mo><mi>B</mi><mo stretchy=\"false\">]</mo></mrow><mrow><mi>n</mi></mrow></msup></math> for which <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>F</mi><mo stretchy=\"false\">(</mo><mi>y</mi><mo>,</mo><mstyle mathvariant=\"bold-italic\"><mi>x</mi></mstyle><mo stre","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"12 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142236172","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Unramifiedness of weight 1 Hilbert Hecke algebras","authors":"Shaunak V. Deo, Mladen Dimitrov, Gabor Wiese","doi":"10.2140/ant.2024.18.1465","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1465","url":null,"abstract":"<p>We prove that the Galois pseudorepresentation valued in the mod <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></math> cuspidal Hecke algebra for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> GL</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mn>2</mn><mo stretchy=\"false\">)</mo></math> over a totally real number field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>F</mi></math>, of parallel weight <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>1</mn></math> and level prime to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>, is unramified at any place above <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>. The same is true for the noncuspidal Hecke algebra at places above <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math> whose ramification index is not divisible by <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi><mo>−</mo><mn>1</mn></math>. A novel geometric ingredient, which is also of independent interest, is the construction and study, in the case when <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math> ramifies in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>F</mi></math>, of generalised <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>Θ</mi></math>-operators using Reduzzi and Xiao’s generalised Hasse invariants, including especially an injectivity criterion in terms of minimal weights. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"29 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142236177","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Failure of the local-global principle for isotropy of quadratic forms over function fields","authors":"Asher Auel, V. Suresh","doi":"10.2140/ant.2024.18.1497","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1497","url":null,"abstract":"<p>We prove the failure of the local-global principle, with respect to discrete valuations, for isotropy of quadratic forms in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></math> variables over function fields of transcendence degree <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi>\u0000<mo>≥</mo> <mn>2</mn></math> over an algebraically closed field of characteristic <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo>≠</mo><mn>2</mn></math>. Our construction involves the generalized Kummer varieties considered by Borcea and by Cynk and Hulek as well as new results on the nontriviality of unramified cohomology of products of elliptic curves over discretely valued fields. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"8 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142236173","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Fu Liu, Brian Osserman, Montserrat Teixidor i Bigas, Naizhen Zhang
{"title":"The strong maximal rank conjecture and moduli spaces of curves","authors":"Fu Liu, Brian Osserman, Montserrat Teixidor i Bigas, Naizhen Zhang","doi":"10.2140/ant.2024.18.1403","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1403","url":null,"abstract":"<p>Building on recent work of the authors, we use degenerations to chains of elliptic curves to prove two cases of the Aprodu–Farkas strong maximal rank conjecture, in genus <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn><mn>2</mn></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn><mn>3</mn></math>. This constitutes a major step forward in Farkas’ program to prove that the moduli spaces of curves of genus <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn><mn>2</mn></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn><mn>3</mn></math> are of general type. Our techniques involve a combination of the Eisenbud–Harris theory of limit linear series, and the notion of linked linear series developed by Osserman. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"75 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142236169","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Dan Abramovich, Michael Temkin, Jarosław Włodarczyk
{"title":"Functorial embedded resolution via weighted blowings up","authors":"Dan Abramovich, Michael Temkin, Jarosław Włodarczyk","doi":"10.2140/ant.2024.18.1557","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1557","url":null,"abstract":"<p>We provide a simple procedure for resolving, in characteristic 0, singularities of a variety <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>X</mi></math> embedded in a smooth variety <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>Y</mi> </math> by repeatedly blowing up the worst singularities, in the sense of stack-theoretic weighted blowings up. No history, no exceptional divisors, and no logarithmic structures are necessary to carry this out; the steps are explicit geometric operations requiring no choices; and the resulting algorithm is efficient. </p><p> A similar result was discovered independently by McQuillan (2020). </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"15 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142236174","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Exceptional characters and prime numbers in sparse sets","authors":"Jori Merikoski","doi":"10.2140/ant.2024.18.1305","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1305","url":null,"abstract":"<p>We develop a lower bound sieve for primes under the (unlikely) assumption of infinitely many exceptional characters. Compared with the illusory sieve due to Friedlander and Iwaniec which produces asymptotic formulas, we show that less arithmetic information is required to prove nontrivial lower bounds. As an application of our method, assuming the existence of infinitely many exceptional characters we show that there are infinitely many primes of the form <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup>\u0000<mo>+</mo> <msup><mrow><mi>b</mi></mrow><mrow><mn>8</mn></mrow></msup></math>. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"21 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141315704","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Combining Igusa’s conjectures on exponential sums and monodromy with semicontinuity of the minimal exponent","authors":"Raf Cluckers, Kien Huu Nguyen","doi":"10.2140/ant.2024.18.1275","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1275","url":null,"abstract":"<p>We combine two of Igusa’s conjectures with recent semicontinuity results by Mustaţă and Popa to form a new, natural conjecture about bounds for exponential sums. These bounds have a deceivingly simple and general formulation in terms of degrees and dimensions only. We provide evidence consisting partly of adaptations of already known results about Igusa’s conjecture on exponential sums, but also some new evidence like for all polynomials in up to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>4</mn></math> variables. We show that, in turn, these bounds imply consequences for Igusa’s (strong) monodromy conjecture. The bounds are related to estimates for major arcs appearing in the circle method for local-global principles. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"59 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141315546","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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