Given a graph $G$, the independence complex $I(G)$ is the simplicial complex whose faces are the independent sets of $V(G)$. Let $tilde{b}_i$ denote the $i$-th reduced Betti number of $I(G)$, and let $b(G)$ denote the sum of the $tilde{b}_i(G)$’s. A graph is ternary if it does not contain induced cycles with length divisible by 3. Kalai and Meshulam conjectured that $b(G)le 1$ whenever $G$ is ternary. We prove this conjecture. This extends a recent result proved by Chudnovsky, Scott, Seymour and Spirkl that for any ternary graph $G$, the number of independent sets with even cardinality and the number of independent sets with odd cardinality differ by at most 1.
{"title":"The total Betti number of the independence complex of ternary graphs","authors":"Wentao Zhang, Hehui Wu","doi":"10.4171/jems/1378","DOIUrl":"https://doi.org/10.4171/jems/1378","url":null,"abstract":"Given a graph $G$, the independence complex $I(G)$ is the simplicial complex whose faces are the independent sets of $V(G)$. Let $tilde{b}_i$ denote the $i$-th reduced Betti number of $I(G)$, and let $b(G)$ denote the sum of the $tilde{b}_i(G)$’s. A graph is ternary if it does not contain induced cycles with length divisible by 3. Kalai and Meshulam conjectured that $b(G)le 1$ whenever $G$ is ternary. We prove this conjecture. This extends a recent result proved by Chudnovsky, Scott, Seymour and Spirkl that for any ternary graph $G$, the number of independent sets with even cardinality and the number of independent sets with odd cardinality differ by at most 1.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135393560","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}
We consider the problem of counting the number of varieties in a family over $mathbb{Q}$ with a rational point. We obtain lower bounds for this counting problem for some families over $mathbb{P}^1$, even if the Hasse principle fails. We also obtain sharp results for some multinorm equations and for specialisations of certain Brauer group elements on higher-dimensional projective spaces, where we answer some cases of a question of Serre. Our techniques come from arithmetic geometry and additive combinatorics.
{"title":"Frobenian multiplicative functions and rational points in fibrations","authors":"Daniel Loughran, Lilian Matthiesen","doi":"10.4171/jems/1374","DOIUrl":"https://doi.org/10.4171/jems/1374","url":null,"abstract":"We consider the problem of counting the number of varieties in a family over $mathbb{Q}$ with a rational point. We obtain lower bounds for this counting problem for some families over $mathbb{P}^1$, even if the Hasse principle fails. We also obtain sharp results for some multinorm equations and for specialisations of certain Brauer group elements on higher-dimensional projective spaces, where we answer some cases of a question of Serre. Our techniques come from arithmetic geometry and additive combinatorics.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135353680","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}
Kestutis Cesnavicius, Michael Neururer, Abhishek Saha
The Manin constant $c$ of an elliptic curve $E$ over $mathbb{Q}$ is the nonzero integer that scales the differential $omega_f$ determined by the normalized newform $f$ associated to $E$ into the pullback of a Néron differential under a minimal parametrization $phicolon X_0(N)mathbb{Q} twoheadrightarrow E$. Manin conjectured that $c = pm 1$ for optimal parametrizations, and we prove that in general $c mid deg(phi)$ under a minor assumption at $2$ and $3$ that is not needed for cube-free $N$ or for parametrizations by $X_1(N)mathbb{Q}$. Since $c$ is supported at the additive reduction primes, which need not divide $deg(phi)$, this improves the status of the Manin conjecture for many $E$. Our core result that gives this divisibility is the containment $omega_f in H^0(X_0(N), Omega)$, which we establish by combining automorphic methods with techniques from arithmetic geometry; here the modular curve $X_0(N)$ is considered over $mathbb{Z}$ and $Omega$ is its relative dualizing sheaf over $mathbb{Z}$. We reduce this containment to $p$-adic bounds on denominators of the Fourier expansions of $f$ at all the cusps of $X_0(N)_mathbb{C}$ and then use the recent basic identity for the $p$-adic Whittaker newform to establish stronger bounds in the more general setup of newforms of weight $k$ on $X_0(N)$. To overcome obstacles at $2$ and $3$, we analyze nondihedral supercuspidal representations of $operatorname{GL}_2(mathbb{Q}2)$ and exhibit new cases in which $X_0(N)mathbb{Z}$ has rational singularities.
{"title":"The Manin constant and the modular degree","authors":"Kestutis Cesnavicius, Michael Neururer, Abhishek Saha","doi":"10.4171/jems/1367","DOIUrl":"https://doi.org/10.4171/jems/1367","url":null,"abstract":"The Manin constant $c$ of an elliptic curve $E$ over $mathbb{Q}$ is the nonzero integer that scales the differential $omega_f$ determined by the normalized newform $f$ associated to $E$ into the pullback of a Néron differential under a minimal parametrization $phicolon X_0(N)mathbb{Q} twoheadrightarrow E$. Manin conjectured that $c = pm 1$ for optimal parametrizations, and we prove that in general $c mid deg(phi)$ under a minor assumption at $2$ and $3$ that is not needed for cube-free $N$ or for parametrizations by $X_1(N)mathbb{Q}$. Since $c$ is supported at the additive reduction primes, which need not divide $deg(phi)$, this improves the status of the Manin conjecture for many $E$. Our core result that gives this divisibility is the containment $omega_f in H^0(X_0(N), Omega)$, which we establish by combining automorphic methods with techniques from arithmetic geometry; here the modular curve $X_0(N)$ is considered over $mathbb{Z}$ and $Omega$ is its relative dualizing sheaf over $mathbb{Z}$. We reduce this containment to $p$-adic bounds on denominators of the Fourier expansions of $f$ at all the cusps of $X_0(N)_mathbb{C}$ and then use the recent basic identity for the $p$-adic Whittaker newform to establish stronger bounds in the more general setup of newforms of weight $k$ on $X_0(N)$. To overcome obstacles at $2$ and $3$, we analyze nondihedral supercuspidal representations of $operatorname{GL}_2(mathbb{Q}2)$ and exhibit new cases in which $X_0(N)mathbb{Z}$ has rational singularities.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135488731","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}
We prove the Arithmetic Fundamental Lemma conjecture over a general $p$-adic field with odd residue cardinality $qgeq dim V$. Our strategy is similar to the one used by the second author in his proof of the AFL over $mathbb{Q}_p$, but only requires the modularity of divisor generating series on the Shimura variety (as opposed to its integral model). The resulting increase in flexibility allows us to work over an arbitrary base field. To carry out our strategy, we also generalize results of Howard (2012) on CM cycle intersection and of Ehlen–Sankaran (2018) on Green function comparison from $mathbb{Q}$ to general totally real base fields.
在具有奇残基基数$qgeq dim V$的一般$p$ -进域上证明了算术基本引理猜想。我们的策略类似于第二个作者在$mathbb{Q}_p$上证明AFL时使用的策略，但只需要在Shimura变量上的除数生成级数的模块化(而不是其积分模型)。由此带来的灵活性的增加使我们能够在任意基域上工作。为了实施我们的策略，我们还将Howard(2012)关于CM循环相交的结果和Ehlen-Sankaran(2018)关于Green函数比较的结果从$mathbb{Q}$推广到一般的全实基场。
{"title":"On the Arithmetic Fundamental Lemma conjecture over a general $p$-adic field","authors":"A. Mihatsch, W. Zhang","doi":"10.4171/jems/1375","DOIUrl":"https://doi.org/10.4171/jems/1375","url":null,"abstract":"We prove the Arithmetic Fundamental Lemma conjecture over a general $p$-adic field with odd residue cardinality $qgeq dim V$. Our strategy is similar to the one used by the second author in his proof of the AFL over $mathbb{Q}_p$, but only requires the modularity of divisor generating series on the Shimura variety (as opposed to its integral model). The resulting increase in flexibility allows us to work over an arbitrary base field. To carry out our strategy, we also generalize results of Howard (2012) on CM cycle intersection and of Ehlen–Sankaran (2018) on Green function comparison from $mathbb{Q}$ to general totally real base fields.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134909946","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}
We introduce the volume entropy semi-norm and the systolic volume semi-norm in real homology and show that they satisfy functorial properties similar to the ones of the simplicial volume. Along the way, we also establish a roughly optimal upper bound on the systolic volume of the multiples of any homology class. Finally, we prove that the volume entropy semi-norm, the systolic volume semi-norm and the simplicial volume semi-norm are equivalent in every dimension.
{"title":"Volume entropy semi-norm and systolic volume semi-norm","authors":"I. Babenko, S. Sabourau","doi":"10.4171/jems/1370","DOIUrl":"https://doi.org/10.4171/jems/1370","url":null,"abstract":"We introduce the volume entropy semi-norm and the systolic volume semi-norm in real homology and show that they satisfy functorial properties similar to the ones of the simplicial volume. Along the way, we also establish a roughly optimal upper bound on the systolic volume of the multiples of any homology class. Finally, we prove that the volume entropy semi-norm, the systolic volume semi-norm and the simplicial volume semi-norm are equivalent in every dimension.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2023-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74219226","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":"Deletion-contraction triangles for Hausel–Proudfoot varieties","authors":"Zuszsanna Dancso, M. McBreen, V. Shende","doi":"10.4171/jems/1369","DOIUrl":"https://doi.org/10.4171/jems/1369","url":null,"abstract":"","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2023-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85357840","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":"Statistical reconstruction of the GFF and KT transition","authors":"C. Garban, Avelio Sepúlveda","doi":"10.4171/jems/1288","DOIUrl":"https://doi.org/10.4171/jems/1288","url":null,"abstract":"","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2023-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72663523","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}
P. Balister, B. Bollobás, R. Morris, J. Sahasrabudhe, M. Tiba
{"title":"The structure and number of Erdős covering systems","authors":"P. Balister, B. Bollobás, R. Morris, J. Sahasrabudhe, M. Tiba","doi":"10.4171/jems/1357","DOIUrl":"https://doi.org/10.4171/jems/1357","url":null,"abstract":"","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2023-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82081035","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":"Nonuniqueness in law of stochastic 3D Navier–Stokes equations","authors":"M. Hofmanová, Rongchan Zhu, Xiangchan Zhu","doi":"10.4171/jems/1360","DOIUrl":"https://doi.org/10.4171/jems/1360","url":null,"abstract":"","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2023-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88627010","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":"Elliptic surfaces and intersections of adelic $mathbb{R}$-divisors","authors":"Laura DeMarco, Niki Myrto Mavraki","doi":"10.4171/jems/1340","DOIUrl":"https://doi.org/10.4171/jems/1340","url":null,"abstract":"","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135402488","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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