In this paper, for positive integers $H$ and $k leq n$, we obtain some estimates on the cardinality of the set of monic integer polynomials of degree $n$ and height bounded by $H$ with exactly $k$ roots of maximal modulus. These include lower and upper bounds in terms of $H$ for fixed $k$ and $n$. We also count reducible and irreducible polynomials in that set separately. Our results imply, for instance, that the number of monic integer irreducible polynomials of degree $n$ and height at most $H$ whose all $n$ roots have equal moduli is approximately $2H$ for odd $n$, while for even $n$ there are more than $H^{n/8}$ of such polynomials.
{"title":"Counting integer polynomials with several roots of maximal modulus","authors":"Artūras Dubickas, Min Sha","doi":"arxiv-2409.08625","DOIUrl":"https://doi.org/arxiv-2409.08625","url":null,"abstract":"In this paper, for positive integers $H$ and $k leq n$, we obtain some\u0000estimates on the cardinality of the set of monic integer polynomials of degree\u0000$n$ and height bounded by $H$ with exactly $k$ roots of maximal modulus. These\u0000include lower and upper bounds in terms of $H$ for fixed $k$ and $n$. We also\u0000count reducible and irreducible polynomials in that set separately. Our results\u0000imply, for instance, that the number of monic integer irreducible polynomials\u0000of degree $n$ and height at most $H$ whose all $n$ roots have equal moduli is\u0000approximately $2H$ for odd $n$, while for even $n$ there are more than\u0000$H^{n/8}$ of such polynomials.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259922","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Yudong Liu, Chenglong Ma, Xiecheng Nie, Xiaoyu Qu, Yupeng Wang
Let $C$ be an algebraically closed perfectoid field over $Qp$ with the ring of integer $calO_C$ and the infinitesimal thickening $Ainf$. Let $frakX$ be a smooth formal scheme over $calO_C$ with a fixed smooth lifting $wtx$ over $Ainf$. Let $X$ be the generic fiber of $frakX$ and $wtX$ be its lifting over $BdRp$ induced by $wtx$. Let $MIC_r(wtX)^{{rm H-small}}$ and $rLrS_r(X,BBdRp)^{{rm H-small}}$ be the $v$-stacks of rank-$r$ Hitchin-small integrable connections on $wtX_{et}$ and $BBdRp$-local systems on $X_{v}$, respectively. In this paper, we establish an equivalence between this two stacks by introducing a new period sheaf with connection $(calObB_{dR,pd}^+,rd)$ on $X_{v}$.
{"title":"A stacky $p$-adic Riemann--Hilbert correspondence on Hitchin-small locus","authors":"Yudong Liu, Chenglong Ma, Xiecheng Nie, Xiaoyu Qu, Yupeng Wang","doi":"arxiv-2409.08785","DOIUrl":"https://doi.org/arxiv-2409.08785","url":null,"abstract":"Let $C$ be an algebraically closed perfectoid field over $Qp$ with the ring\u0000of integer $calO_C$ and the infinitesimal thickening $Ainf$. Let $frakX$ be\u0000a smooth formal scheme over $calO_C$ with a fixed smooth lifting $wtx$ over\u0000$Ainf$. Let $X$ be the generic fiber of $frakX$ and $wtX$ be its lifting\u0000over $BdRp$ induced by $wtx$. Let $MIC_r(wtX)^{{rm H-small}}$ and\u0000$rLrS_r(X,BBdRp)^{{rm H-small}}$ be the $v$-stacks of rank-$r$\u0000Hitchin-small integrable connections on $wtX_{et}$ and $BBdRp$-local systems\u0000on $X_{v}$, respectively. In this paper, we establish an equivalence between\u0000this two stacks by introducing a new period sheaf with connection\u0000$(calObB_{dR,pd}^+,rd)$ on $X_{v}$.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"214 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259977","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Sourabhashis Das, Ertan Elma, Wentang Kuo, Yu-Ru Liu
Let $k geq 1$ be a natural number and $f in mathbb{F}_q[t]$ be a monic polynomial. Let $omega_k(f)$ denote the number of distinct monic irreducible factors of $f$ with multiplicity $k$. We obtain asymptotic estimates for the first and the second moments of $omega_k(f)$ with $k geq 1$. Moreover, we prove that the function $omega_1(f)$ has normal order $log (text{deg}(f))$ and also satisfies the ErdH{o}s-Kac Theorem. Finally, we prove that the functions $omega_k(f)$ with $k geq 2$ do not have normal order.
{"title":"On the number of irreducible factors with a given multiplicity in function fields","authors":"Sourabhashis Das, Ertan Elma, Wentang Kuo, Yu-Ru Liu","doi":"arxiv-2409.08559","DOIUrl":"https://doi.org/arxiv-2409.08559","url":null,"abstract":"Let $k geq 1$ be a natural number and $f in mathbb{F}_q[t]$ be a monic\u0000polynomial. Let $omega_k(f)$ denote the number of distinct monic irreducible\u0000factors of $f$ with multiplicity $k$. We obtain asymptotic estimates for the\u0000first and the second moments of $omega_k(f)$ with $k geq 1$. Moreover, we\u0000prove that the function $omega_1(f)$ has normal order $log (text{deg}(f))$\u0000and also satisfies the ErdH{o}s-Kac Theorem. Finally, we prove that the\u0000functions $omega_k(f)$ with $k geq 2$ do not have normal order.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"84 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259923","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Lattices have many significant applications in cryptography. In 2021, the $p$-adic signature scheme and public-key encryption cryptosystem were introduced. They are based on the Longest Vector Problem (LVP) and the Closest Vector Problem (CVP) in $p$-adic lattices. These problems are considered to be challenging and there are no known deterministic polynomial time algorithms to solve them. In this paper, we improve the LVP algorithm in local fields. The modified LVP algorithm is a deterministic polynomial time algorithm when the field is totally ramified and $p$ is a polynomial in the rank of the input lattice. We utilize this algorithm to attack the above schemes so that we are able to forge a valid signature of any message and decrypt any ciphertext. Although these schemes are broken, this work does not mean that $p$-adic lattices are not suitable in constructing cryptographic primitives. We propose some possible modifications to avoid our attack at the end of this paper.
{"title":"An Attack on $p$-adic Lattice Public-key Cryptosystems and Signature Schemes","authors":"Chi Zhang","doi":"arxiv-2409.08774","DOIUrl":"https://doi.org/arxiv-2409.08774","url":null,"abstract":"Lattices have many significant applications in cryptography. In 2021, the\u0000$p$-adic signature scheme and public-key encryption cryptosystem were\u0000introduced. They are based on the Longest Vector Problem (LVP) and the Closest\u0000Vector Problem (CVP) in $p$-adic lattices. These problems are considered to be\u0000challenging and there are no known deterministic polynomial time algorithms to\u0000solve them. In this paper, we improve the LVP algorithm in local fields. The\u0000modified LVP algorithm is a deterministic polynomial time algorithm when the\u0000field is totally ramified and $p$ is a polynomial in the rank of the input\u0000lattice. We utilize this algorithm to attack the above schemes so that we are\u0000able to forge a valid signature of any message and decrypt any ciphertext.\u0000Although these schemes are broken, this work does not mean that $p$-adic\u0000lattices are not suitable in constructing cryptographic primitives. We propose\u0000some possible modifications to avoid our attack at the end of this paper.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142269318","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Lewis Combes, John W. Jones, Jennifer Paulhus, David Roe, Manami Roy, Sam Schiavone
A database of abstract groups has been added to the L-functions and Modular Forms Database (LMFDB), available at https://www.lmfdb.org/Groups/Abstract/. We discuss the functionality of the database and what makes it distinct from other available databases of abstract groups. We describe solutions to mathematical problems we encountered while creating the database, as well as connections between the abstract groups database and other collections of objects in the LMFDB.
L 函数和模块形式数据库（LMFDB）中增加了一个抽象群数据库，可在 https://www.lmfdb.org/Groups/Abstract/ 上查阅。我们讨论了该数据库的功能，以及它与其他现有抽象群数据库的不同之处。我们描述了创建数据库时遇到的数学问题的解决方案，以及抽象群数据库与 LMFDB 中其他对象集合之间的连接。
{"title":"Creating a dynamic database of finite groups","authors":"Lewis Combes, John W. Jones, Jennifer Paulhus, David Roe, Manami Roy, Sam Schiavone","doi":"arxiv-2409.09189","DOIUrl":"https://doi.org/arxiv-2409.09189","url":null,"abstract":"A database of abstract groups has been added to the L-functions and Modular\u0000Forms Database (LMFDB), available at https://www.lmfdb.org/Groups/Abstract/. We\u0000discuss the functionality of the database and what makes it distinct from other\u0000available databases of abstract groups. We describe solutions to mathematical\u0000problems we encountered while creating the database, as well as connections\u0000between the abstract groups database and other collections of objects in the\u0000LMFDB.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259917","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jonas Bergström, Valentijn Karemaker, Stefano Marseglia
We give a categorical description of all abelian varieties with commutative endomorphism ring over a finite field with $q=p^a$ elements in a fixed isogeny class in terms of pairs consisting of a fractional $mathbb Z[pi,q/pi]$-ideal and a fractional $Wotimes_{mathbb Z_p} mathbb Z_p[pi,q/pi]$-ideal, with $pi$ the Frobenius endomorphism and $W$ the ring of integers in an unramified extension of $mathbb Q_p$ of degree $a$. The latter ideal should be compatible at $p$ with the former and stable under the action of a semilinear Frobenius (and Verschiebung) operator; it will be the Dieudonn'e module of the corresponding abelian variety. Using this categorical description we create effective algorithms to compute isomorphism classes of these objects and we produce many new examples exhibiting exotic patterns.
{"title":"Abelian varieties over finite fields with commutative endomorphism algebra: theory and algorithms","authors":"Jonas Bergström, Valentijn Karemaker, Stefano Marseglia","doi":"arxiv-2409.08865","DOIUrl":"https://doi.org/arxiv-2409.08865","url":null,"abstract":"We give a categorical description of all abelian varieties with commutative\u0000endomorphism ring over a finite field with $q=p^a$ elements in a fixed isogeny\u0000class in terms of pairs consisting of a fractional $mathbb Z[pi,q/pi]$-ideal\u0000and a fractional $Wotimes_{mathbb Z_p} mathbb Z_p[pi,q/pi]$-ideal, with\u0000$pi$ the Frobenius endomorphism and $W$ the ring of integers in an unramified\u0000extension of $mathbb Q_p$ of degree $a$. The latter ideal should be compatible\u0000at $p$ with the former and stable under the action of a semilinear Frobenius\u0000(and Verschiebung) operator; it will be the Dieudonn'e module of the\u0000corresponding abelian variety. Using this categorical description we create\u0000effective algorithms to compute isomorphism classes of these objects and we\u0000produce many new examples exhibiting exotic patterns.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"214 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259919","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In projective dimension growth results, one bounds the number of rational points of height at most $H$ on an irreducible hypersurface in $mathbb P^n$ of degree $d>3$ by $C(n)d^2 H^{n-1}(log H)^{M(n)}$, where the quadratic dependence in $d$ has been recently obtained by Binyamini, Cluckers and Kato in 2024 [1]. For these bounds, it was already shown by Castryck, Cluckers, Dittmann and Nguyen in 2020 [3] that one cannot do better than a linear dependence in $d$. In this paper we show that, for the mentioned projective dimension growth bounds, the quadratic dependence in $d$ is eventually tight when $n$ grows. More precisely the upper bounds cannot be better than $c(n)d^{2-2/n} H^{n-1}$ in general. Note that for affine dimension growth (for affine hypersurfaces of degree $d$, satisfying some extra conditions), the dependence on $d$ is also quadratic by [1], which is already known to be optimal by [3]. Our projective case thus complements the picture of tightness for dimension growth bounds for hypersurfaces.
{"title":"Eventual tightness of projective dimension growth bounds: quadratic in the degree","authors":"Raf Cluckers, Itay Glazer","doi":"arxiv-2409.08776","DOIUrl":"https://doi.org/arxiv-2409.08776","url":null,"abstract":"In projective dimension growth results, one bounds the number of rational\u0000points of height at most $H$ on an irreducible hypersurface in $mathbb P^n$ of\u0000degree $d>3$ by $C(n)d^2 H^{n-1}(log H)^{M(n)}$, where the quadratic\u0000dependence in $d$ has been recently obtained by Binyamini, Cluckers and Kato in\u00002024 [1]. For these bounds, it was already shown by Castryck, Cluckers,\u0000Dittmann and Nguyen in 2020 [3] that one cannot do better than a linear\u0000dependence in $d$. In this paper we show that, for the mentioned projective\u0000dimension growth bounds, the quadratic dependence in $d$ is eventually tight\u0000when $n$ grows. More precisely the upper bounds cannot be better than\u0000$c(n)d^{2-2/n} H^{n-1}$ in general. Note that for affine dimension growth (for\u0000affine hypersurfaces of degree $d$, satisfying some extra conditions), the\u0000dependence on $d$ is also quadratic by [1], which is already known to be\u0000optimal by [3]. Our projective case thus complements the picture of tightness\u0000for dimension growth bounds for hypersurfaces.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259920","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Kâzım Büyükboduk, Daniele Casazza, Aprameyo Pal, Carlos de Vera-Piquero
Our main objective in this paper (which is expository for the most part) is to study the necessary steps to prove a factorization formula for a certain triple product $p$-adic $L$-function guided by the Artin formalism. The key ingredients are: a) the explicit reciprocity laws governing the relationship of diagonal cycles and generalized Heegner cycles to $p$-adic $L$-functions; b) a careful comparison of Chow--Heegner points and twisted Heegner points in Hida families, via formulae of Gross--Zagier type.
{"title":"On the Artin formalism for triple product $p$-adic $L$-functions: Chow--Heegner points vs. Heegner points","authors":"Kâzım Büyükboduk, Daniele Casazza, Aprameyo Pal, Carlos de Vera-Piquero","doi":"arxiv-2409.08645","DOIUrl":"https://doi.org/arxiv-2409.08645","url":null,"abstract":"Our main objective in this paper (which is expository for the most part) is\u0000to study the necessary steps to prove a factorization formula for a certain\u0000triple product $p$-adic $L$-function guided by the Artin formalism. The key\u0000ingredients are: a) the explicit reciprocity laws governing the relationship of\u0000diagonal cycles and generalized Heegner cycles to $p$-adic $L$-functions; b) a\u0000careful comparison of Chow--Heegner points and twisted Heegner points in Hida\u0000families, via formulae of Gross--Zagier type.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259921","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In 1996, Merel showed there exists a function $Bcolon mathbb{Z}^+rightarrow mathbb{Z}^+$ such that for any elliptic curve $E/F$ defined over a number field of degree $d$, one has the torsion group bound $# E(F)[textrm{tors}]leq B(d)$. Based on subsequent work, it is conjectured that one can choose $B$ to be polynomial in the degree $d$. In this paper, we show that such bounds exist for torsion from the family $mathcal{I}_{mathbb{Q}}$ of elliptic curves which are geometrically isogenous to at least one rational elliptic curve. More precisely, we show that for each $epsilon>0$ there exists $c_epsilon>0$ such that for any elliptic curve $E/Fin mathcal{I}_{mathbb{Q}}$, one has [ # E(F)[textrm{tors}]leq c_epsiloncdot [F:mathbb{Q}]^{5+epsilon}. ] This generalizes prior work of Clark and Pollack, as well as work of the second author in the case of rational geometric isogeny classes.
{"title":"Uniform polynomial bounds on torsion from rational geometric isogeny classes","authors":"Abbey Bourdon, Tyler Genao","doi":"arxiv-2409.08214","DOIUrl":"https://doi.org/arxiv-2409.08214","url":null,"abstract":"In 1996, Merel showed there exists a function $Bcolon\u0000mathbb{Z}^+rightarrow mathbb{Z}^+$ such that for any elliptic curve $E/F$\u0000defined over a number field of degree $d$, one has the torsion group bound $#\u0000E(F)[textrm{tors}]leq B(d)$. Based on subsequent work, it is conjectured that\u0000one can choose $B$ to be polynomial in the degree $d$. In this paper, we show\u0000that such bounds exist for torsion from the family $mathcal{I}_{mathbb{Q}}$\u0000of elliptic curves which are geometrically isogenous to at least one rational\u0000elliptic curve. More precisely, we show that for each $epsilon>0$ there exists\u0000$c_epsilon>0$ such that for any elliptic curve $E/Fin\u0000mathcal{I}_{mathbb{Q}}$, one has [ # E(F)[textrm{tors}]leq\u0000c_epsiloncdot [F:mathbb{Q}]^{5+epsilon}. ] This generalizes prior work of\u0000Clark and Pollack, as well as work of the second author in the case of rational\u0000geometric isogeny classes.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203796","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we consider sums of three generalized $m$-gonal numbers whose parameters are restricted to integers with a bounded number of prime divisors. With some restrictions on $m$ modulo $30$, we show that a density one set of integers is represented as such a sum, where the parameters are restricted to have at most 6361 prime factors. Moreover, if the squarefree part of $f_m(n)$ is sufficiently large, then $n$ is represented as such a sum, where $f_m(n)$ is a natural linear function in $n$.
{"title":"Universal sums of generalized polygonal numbers of almost prime \"length\"","authors":"Soumyarup Banerjee, Ben Kane, Daejun Kim","doi":"arxiv-2409.07895","DOIUrl":"https://doi.org/arxiv-2409.07895","url":null,"abstract":"In this paper, we consider sums of three generalized $m$-gonal numbers whose\u0000parameters are restricted to integers with a bounded number of prime divisors.\u0000With some restrictions on $m$ modulo $30$, we show that a density one set of\u0000integers is represented as such a sum, where the parameters are restricted to\u0000have at most 6361 prime factors. Moreover, if the squarefree part of $f_m(n)$\u0000is sufficiently large, then $n$ is represented as such a sum, where $f_m(n)$ is\u0000a natural linear function in $n$.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203799","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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