We develop a theory of derived rigid spaces and quasi-coherent sheaves and analytic crystals on them. Amongst other things, we obtain a six-functor formalism for these quasi-coherent sheaves and analytic crystals. We provide evidence that the category of analytic crystals is related to the theory of D-cap-modules introduced by Ardakov--Wadsley.
{"title":"A six-functor formalism for quasi-coherent sheaves and crystals on rigid-analytic varieties","authors":"Arun Soor","doi":"arxiv-2409.07592","DOIUrl":"https://doi.org/arxiv-2409.07592","url":null,"abstract":"We develop a theory of derived rigid spaces and quasi-coherent sheaves and\u0000analytic crystals on them. Amongst other things, we obtain a six-functor\u0000formalism for these quasi-coherent sheaves and analytic crystals. We provide\u0000evidence that the category of analytic crystals is related to the theory of\u0000D-cap-modules introduced by Ardakov--Wadsley.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203802","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}
Kathrin Bringmann, Guoniu Han, Bernhard Heim, Ben Kane
In this paper, we study sign changes of weakly holomorphic modular forms which are given as $eta$-quotients. We give representative examples for forms of negative weight, weight zero, and positive weight.
{"title":"Periodic sign changes for weakly holomorphic $η$-quotients","authors":"Kathrin Bringmann, Guoniu Han, Bernhard Heim, Ben Kane","doi":"arxiv-2409.07164","DOIUrl":"https://doi.org/arxiv-2409.07164","url":null,"abstract":"In this paper, we study sign changes of weakly holomorphic modular forms\u0000which are given as $eta$-quotients. We give representative examples for forms\u0000of negative weight, weight zero, and positive weight.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203824","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}
We present natural conjectural generalizations of the `positivity and integrality of mirror maps' phenomenon, encompassing the mirror maps appearing in the Batyrev--Borisov construction of mirror Calabi--Yau complete intersections in Fano toric varieties as a special case. We find that, given the combinatorial data from which one constructs a mirror pair of Calabi--Yau complete intersections, there are two ways of writing down an associated `mirror map': one which is the `true mirror map', meaning the one which appears in mirror symmetry theorems; and one which is the `naive mirror map'. The two are equal under a certain combinatorial criterion which holds e.g. for the quintic threefold, but not in general. We conjecture (based on substantial computer checks, together with proofs under extra hypotheses) that the naive mirror map always has positive integer coefficients, while the true mirror map always has integer (but not necessarily positive) coefficients. Almost all previous works on the integrality of mirror maps concern the naive mirror map, and in particular, only apply to the true mirror map under the combinatorial criterion mentioned above.
{"title":"On the positivity and integrality of coefficients of mirror maps","authors":"Sophie Bleau, Nick Sheridan","doi":"arxiv-2409.07601","DOIUrl":"https://doi.org/arxiv-2409.07601","url":null,"abstract":"We present natural conjectural generalizations of the `positivity and\u0000integrality of mirror maps' phenomenon, encompassing the mirror maps appearing\u0000in the Batyrev--Borisov construction of mirror Calabi--Yau complete\u0000intersections in Fano toric varieties as a special case. We find that, given\u0000the combinatorial data from which one constructs a mirror pair of Calabi--Yau\u0000complete intersections, there are two ways of writing down an associated\u0000`mirror map': one which is the `true mirror map', meaning the one which appears\u0000in mirror symmetry theorems; and one which is the `naive mirror map'. The two\u0000are equal under a certain combinatorial criterion which holds e.g. for the\u0000quintic threefold, but not in general. We conjecture (based on substantial\u0000computer checks, together with proofs under extra hypotheses) that the naive\u0000mirror map always has positive integer coefficients, while the true mirror map\u0000always has integer (but not necessarily positive) coefficients. Almost all\u0000previous works on the integrality of mirror maps concern the naive mirror map,\u0000and in particular, only apply to the true mirror map under the combinatorial\u0000criterion mentioned above.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203801","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}
Let $p$ be a prime and $c,dinmathbb{Z}$. Sun introduced the determinant $D_p^-(c,d):=det[(i^2+cij+dj^2)^{p-2}]_{13$. In this paper, we confirm three conjectures on $D_p^-(c,d)$ proposed by Zhi-Wei Sun.
{"title":"On some determinant conjectures","authors":"Ze-Hua Zhu, Chen-Kai Ren","doi":"arxiv-2409.07008","DOIUrl":"https://doi.org/arxiv-2409.07008","url":null,"abstract":"Let $p$ be a prime and $c,dinmathbb{Z}$. Sun introduced the determinant\u0000$D_p^-(c,d):=det[(i^2+cij+dj^2)^{p-2}]_{1<i,j<p-1}$ for $p>3$. In this paper,\u0000we confirm three conjectures on $D_p^-(c,d)$ proposed by Zhi-Wei Sun.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203806","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}
We carry out a survey on curves defined over finite fields that are Diophantine stable; that is, with the property that the set of points of the curve is not altered under a proper field extension. First, we derive some general results of such curves and then we analyze several families of curves that happen to be Diophantine stable.
{"title":"Diophantine stability for curves over finite fields","authors":"Francesc Bars, Joan Carles Lario","doi":"arxiv-2409.07086","DOIUrl":"https://doi.org/arxiv-2409.07086","url":null,"abstract":"We carry out a survey on curves defined over finite fields that are\u0000Diophantine stable; that is, with the property that the set of points of the\u0000curve is not altered under a proper field extension. First, we derive some\u0000general results of such curves and then we analyze several families of curves\u0000that happen to be Diophantine stable.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203837","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}
We show that Submonoid Membership is decidable in n-dimensional lamplighter groups $(mathbb{Z}/pmathbb{Z}) wr mathbb{Z}^n$ for any prime $p$ and integer $n$. More generally, we show decidability of Submonoid Membership in semidirect products of the form $mathcal{Y} rtimes mathbb{Z}^n$, where $mathcal{Y}$ is any finitely presented module over the Laurent polynomial ring $mathbb{F}_p[X_1^{pm}, ldots, X_n^{pm}]$. Combined with a result of Shafrir (2024), this gives the first example of a group $G$ and a finite index subgroup $widetilde{G} leq G$, such that Submonoid Membership is decidable in $widetilde{G}$ but undecidable in $G$. To obtain our decidability result, we reduce Submonoid Membership in $mathcal{Y} rtimes mathbb{Z}^n$ to solving S-unit equations over $mathbb{F}_p[X_1^{pm}, ldots, X_n^{pm}]$-modules. We show that the solution set of such equations is effectively $p$-automatic, extending a result of Adamczewski and Bell (2012). As an intermediate result, we also obtain that the solution set of the Knapsack Problem in $mathcal{Y} rtimes mathbb{Z}^n$ is effectively $p$-automatic.
{"title":"Submonoid Membership in n-dimensional lamplighter groups and S-unit equations","authors":"Ruiwen Dong","doi":"arxiv-2409.07077","DOIUrl":"https://doi.org/arxiv-2409.07077","url":null,"abstract":"We show that Submonoid Membership is decidable in n-dimensional lamplighter\u0000groups $(mathbb{Z}/pmathbb{Z}) wr mathbb{Z}^n$ for any prime $p$ and\u0000integer $n$. More generally, we show decidability of Submonoid Membership in\u0000semidirect products of the form $mathcal{Y} rtimes mathbb{Z}^n$, where\u0000$mathcal{Y}$ is any finitely presented module over the Laurent polynomial ring\u0000$mathbb{F}_p[X_1^{pm}, ldots, X_n^{pm}]$. Combined with a result of Shafrir\u0000(2024), this gives the first example of a group $G$ and a finite index subgroup\u0000$widetilde{G} leq G$, such that Submonoid Membership is decidable in\u0000$widetilde{G}$ but undecidable in $G$. To obtain our decidability result, we reduce Submonoid Membership in\u0000$mathcal{Y} rtimes mathbb{Z}^n$ to solving S-unit equations over\u0000$mathbb{F}_p[X_1^{pm}, ldots, X_n^{pm}]$-modules. We show that the solution\u0000set of such equations is effectively $p$-automatic, extending a result of\u0000Adamczewski and Bell (2012). As an intermediate result, we also obtain that the\u0000solution set of the Knapsack Problem in $mathcal{Y} rtimes mathbb{Z}^n$ is\u0000effectively $p$-automatic.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203825","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}
We obfuscate words of a given length in a free monoid on two generators with a simple factorization algorithm (namely $SL_2(mathbb{N})$) to create a public-key encryption scheme. We provide a reference implementation in Python and suggested parameters. The security analysis is between weak and non-existent, left to future work.
{"title":"Public-key encryption from a trapdoor one-way embedding of $SL_2(mathbb{N}$)","authors":"Robert Hines","doi":"arxiv-2409.07616","DOIUrl":"https://doi.org/arxiv-2409.07616","url":null,"abstract":"We obfuscate words of a given length in a free monoid on two generators with\u0000a simple factorization algorithm (namely $SL_2(mathbb{N})$) to create a\u0000public-key encryption scheme. We provide a reference implementation in Python\u0000and suggested parameters. The security analysis is between weak and\u0000non-existent, left to future work.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"76 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203803","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}
We consider two questions of Ruzsa on how the minimum size of an additive basis $B$ of a given set $A$ depends on the domain of $B$. To state these questions, for an abelian group $G$ and $A subseteq D subseteq G$ we write $ell_D(A) colon =min { |B|: B subseteq D, A subseteq B+B }$. Ruzsa asked how much larger can $ell_{mathbb{Z}}(A)$ be than $ell_{mathbb{Q}}(A)$ for $Asubsetmathbb{Z}$, and how much larger can $ell_{mathbb{N}}(A)$ be than $ell_{mathbb{Z}}(A)$ for $Asubsetmathbb{N}$. For the first question we show that if $ell_{mathbb{Q}}(A) = n$ then $ell_{mathbb{Z}}(A) le 2n$, and that this is tight up to an additive error of at most $O(sqrt{n})$. For the second question, we show that if $ell_{mathbb{Z}}(A) = n$ then $ell_{mathbb{N}}(A) le O(nlog n)$, and this is tight up to the constant factor. We also consider these questions for higher order bases. Our proofs use some ideas that are unexpected in this context, including linear algebra and Diophantine approximation.
{"title":"Additive Bases: Change of Domain","authors":"Boris Bukh, Peter van Hintum, Peter Keevash","doi":"arxiv-2409.07442","DOIUrl":"https://doi.org/arxiv-2409.07442","url":null,"abstract":"We consider two questions of Ruzsa on how the minimum size of an additive\u0000basis $B$ of a given set $A$ depends on the domain of $B$. To state these\u0000questions, for an abelian group $G$ and $A subseteq D subseteq G$ we write\u0000$ell_D(A) colon =min { |B|: B subseteq D, A subseteq B+B }$. Ruzsa\u0000asked how much larger can $ell_{mathbb{Z}}(A)$ be than $ell_{mathbb{Q}}(A)$\u0000for $Asubsetmathbb{Z}$, and how much larger can $ell_{mathbb{N}}(A)$ be\u0000than $ell_{mathbb{Z}}(A)$ for $Asubsetmathbb{N}$. For the first question we\u0000show that if $ell_{mathbb{Q}}(A) = n$ then $ell_{mathbb{Z}}(A) le 2n$, and\u0000that this is tight up to an additive error of at most $O(sqrt{n})$. For the\u0000second question, we show that if $ell_{mathbb{Z}}(A) = n$ then\u0000$ell_{mathbb{N}}(A) le O(nlog n)$, and this is tight up to the constant\u0000factor. We also consider these questions for higher order bases. Our proofs use\u0000some ideas that are unexpected in this context, including linear algebra and\u0000Diophantine approximation.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203804","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}
A 1971 conjecture of Graham (later repeated by ErdH{o}s and Graham) asserts that every set $A subseteq mathbb{F}_p setminus {0}$ has an ordering whose partial sums are all distinct. We prove this conjecture for sets of size $|A| leqslant e^{(log p)^{1/4}}$; our result improves the previous bound of $log p/log log p$. One ingredient in our argument is a structure theorem involving dissociated sets, which may be of independent interest.
{"title":"Graham's rearrangement conjecture beyond the rectification barrier","authors":"Benjamin Bedert, Noah Kravitz","doi":"arxiv-2409.07403","DOIUrl":"https://doi.org/arxiv-2409.07403","url":null,"abstract":"A 1971 conjecture of Graham (later repeated by ErdH{o}s and Graham) asserts\u0000that every set $A subseteq mathbb{F}_p setminus {0}$ has an ordering whose\u0000partial sums are all distinct. We prove this conjecture for sets of size $|A|\u0000leqslant e^{(log p)^{1/4}}$; our result improves the previous bound of $log\u0000p/log log p$. One ingredient in our argument is a structure theorem involving\u0000dissociated sets, which may be of independent interest.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"110 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203823","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}
Using combinations of weight-1 and weight-2 of Kronecker-Eisenstein series to construct currents in the distributional de Rham complex of a squared elliptic curve, we find a simple explicit formula for the type II $(text{GL}_2, text{GL}_2)$ theta lift without smoothing, analogous to the classical formula of Siegel for periods of Eisenstein series. For $K$ a CM field, the same technique applies without change to obtain an analogous formula for the $(text{GL}_2(K),K^times)$ theta correspondence.
{"title":"Explicit formula for the $(text{GL}_2, text{GL}_2)$ theta lift via Bruhat decomposition","authors":"Peter Xu","doi":"arxiv-2409.06940","DOIUrl":"https://doi.org/arxiv-2409.06940","url":null,"abstract":"Using combinations of weight-1 and weight-2 of Kronecker-Eisenstein series to\u0000construct currents in the distributional de Rham complex of a squared elliptic\u0000curve, we find a simple explicit formula for the type II $(text{GL}_2,\u0000text{GL}_2)$ theta lift without smoothing, analogous to the classical formula\u0000of Siegel for periods of Eisenstein series. For $K$ a CM field, the same\u0000technique applies without change to obtain an analogous formula for the\u0000$(text{GL}_2(K),K^times)$ theta correspondence.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203807","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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