We establish a comparison isomorphism between prismatic cohomology and derived de Rham cohomology respecting various structures, such as their Frobenius actions and filtrations. As an application, when $X$ is a proper smooth formal scheme over $mathcal O_K$ with $K$ being a $p$-adic field, we improve Breuil--Caruso's theory on comparison between torsion crystalline cohomology and torsion 'etale cohomology.
{"title":"Comparison of prismatic cohomology and derived de Rham cohomology","authors":"Shizhang Li, Tong Liu","doi":"10.4171/jems/1377","DOIUrl":"https://doi.org/10.4171/jems/1377","url":null,"abstract":"We establish a comparison isomorphism between prismatic cohomology and derived de Rham cohomology respecting various structures, such as their Frobenius actions and filtrations. As an application, when $X$ is a proper smooth formal scheme over $mathcal O_K$ with $K$ being a $p$-adic field, we improve Breuil--Caruso's theory on comparison between torsion crystalline cohomology and torsion 'etale cohomology.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"43 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135217936","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}
Braverman and Kazhdan proposed a conjecture, later refined by Ngo and broadened to the framework of spherical varieties by Sakellaridis, that asserts that affine spherical varieties admit Schwartz spaces, Fourier transforms, and Poisson summation formulae. The first author in joint work with B.~Liu and later the first two authors proved these conjectures for certain spherical varieties $Y$ built out of triples of quadratic spaces. However, in these works the Fourier transform was only proven to exist. In the present paper we give, for the first time, an explicit formula for the Fourier transform on $Y.$ We also prove that it is unitary in the nonarchimedean case. As preparation for this result, we give explicit formulae for Fourier transforms on Braverman-Kazhdan spaces attached to maximal parabolic subgroups of split, simple, simply connected groups. These Fourier transforms are of independent interest, for example, from the point of view of analytic number theory.
{"title":"Harmonic analysis on certain spherical varieties","authors":"Jayce R. Getz, Chun-Hsien Hsu, Spencer Leslie","doi":"10.4171/jems/1381","DOIUrl":"https://doi.org/10.4171/jems/1381","url":null,"abstract":"Braverman and Kazhdan proposed a conjecture, later refined by Ngo and broadened to the framework of spherical varieties by Sakellaridis, that asserts that affine spherical varieties admit Schwartz spaces, Fourier transforms, and Poisson summation formulae. The first author in joint work with B.~Liu and later the first two authors proved these conjectures for certain spherical varieties $Y$ built out of triples of quadratic spaces. However, in these works the Fourier transform was only proven to exist. In the present paper we give, for the first time, an explicit formula for the Fourier transform on $Y.$ We also prove that it is unitary in the nonarchimedean case. As preparation for this result, we give explicit formulae for Fourier transforms on Braverman-Kazhdan spaces attached to maximal parabolic subgroups of split, simple, simply connected groups. These Fourier transforms are of independent interest, for example, from the point of view of analytic number theory.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135323065","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}
Motivated by the search for methods to establish strong minimality of certain low order algebraic differential equations, a measure of how far a finite rank stationary type is from being minimal is introduced and studied: The degree of nonminimality is the minimum number of realisations of the type required to witness a nonalgebraic forking extension. Conditional on the truth of a conjecture of Borovik and Cherlin on the generic multiple-transitivity of homogeneous spaces definable in the stable theory being considered, it is shown that the nonminimality degree is bounded by the $U$-rank plus 2. The Borovik–Cherlin conjecture itself is verified for algebraic and meromorphic group actions, and a bound of $U$-rank plus 1 is then deduced unconditionally for differentially closed fields and compact complex manifolds. An application is given regarding transcendence of solutions to algebraic differential equations.
{"title":"Bounding nonminimality and a conjecture of Borovik–Cherlin","authors":"James Freitag, Rahim Moosa","doi":"10.4171/jems/1384","DOIUrl":"https://doi.org/10.4171/jems/1384","url":null,"abstract":"Motivated by the search for methods to establish strong minimality of certain low order algebraic differential equations, a measure of how far a finite rank stationary type is from being minimal is introduced and studied: The degree of nonminimality is the minimum number of realisations of the type required to witness a nonalgebraic forking extension. Conditional on the truth of a conjecture of Borovik and Cherlin on the generic multiple-transitivity of homogeneous spaces definable in the stable theory being considered, it is shown that the nonminimality degree is bounded by the $U$-rank plus 2. The Borovik–Cherlin conjecture itself is verified for algebraic and meromorphic group actions, and a bound of $U$-rank plus 1 is then deduced unconditionally for differentially closed fields and compact complex manifolds. An application is given regarding transcendence of solutions to algebraic differential equations.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"62 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136057755","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 interpret the log-Brunn–Minkowski conjecture of Böröczky–Lutwak–Yang–Zhang as a spectral problem in centro-affine differential geometry. In particular, we show that the Hilbert--Brunn--Minkowski operator coincides with the centro-affine Laplacian, thus obtaining a new avenue for tackling the conjecture using insights from affine differential geometry. As every strongly convex hypersurface in $mathbb{R}^n$ is a centro-affine unit sphere, it has constant centro-affine Ricci curvature equal to $n-2$, in stark contrast to the standard weighted Ricci curvature of the associated metric-measure space, which will in general be negative. In particular, we may use the classical argument of Lichnerowicz and a centro-affine Bochner formula to give a new proof of the Brunn–Minkowski inequality. For origin-symmetric convex bodies enjoying fairly generous curvature pinching bounds (improving with dimension), we are able to show global uniqueness in the $L^p$- and log-Minkowski problems, as well as the corresponding global $L^p$- and log-Minkowski conjectured inequalities. As a consequence, we resolve the isomorphic version of the log-Minkowski problem: for any origin-symmetric convex body $bar K$ in $mathbb{R}^n$, there exists an origin-symmetric convex body $K$ with $bar K subset K subset 8 bar K$ such that $K$ satisfies the log-Minkowski conjectured inequality, and such that $K$ is uniquely determined by its cone-volume measure $V_K$. If $bar K$ is not extremely far from a Euclidean ball to begin with, an analogous isometric result, where $8$ is replaced by $1+varepsilon$, is obtained as well.
我们将Böröczky-Lutwak-Yang-Zhang的log-Brunn-Minkowski猜想解释为中心仿射微分几何中的一个谱问题。特别是，我们证明了Hilbert- Brunn- Minkowski算子与中心仿射拉普拉斯算子重合，从而获得了利用仿射微分几何的见解来解决猜想的新途径。由于$mathbb{R}^n$中的每个强凸超曲面都是中心仿射单位球，因此它具有恒定的中心仿射Ricci曲率，等于$n-2$，与相关度量度量空间的标准加权Ricci曲率形成鲜明对比，后者通常为负。特别地，我们可以利用Lichnerowicz的经典论证和中心仿射Bochner公式给出布伦-闵可夫斯基不等式的新证明。对于原点对称凸体具有相当大的曲率挤压界(随维数的增加而提高)，我们能够在$L^p$ -和log-Minkowski问题中显示全局唯一性，以及相应的全局$L^p$ -和log-Minkowski猜想不等式。因此，我们解决了对数-闵可夫斯基问题的同构版本:对于$mathbb{R}^n$中的任意原点对称凸体$bar K$，存在一个具有$bar K subset K subset 8 bar K$的原点对称凸体$K$，使得$K$满足对数-闵可夫斯基猜想不等式，并且使得$K$是由其锥体积测度$V_K$唯一决定的。如果$bar K$一开始离欧几里得球不远，也可以得到类似的等距结果，其中$8$用$1+varepsilon$代替。
{"title":"Centro-affine differential geometry and the log-Minkowski problem","authors":"Emanuel Milman","doi":"10.4171/jems/1386","DOIUrl":"https://doi.org/10.4171/jems/1386","url":null,"abstract":"We interpret the log-Brunn–Minkowski conjecture of Böröczky–Lutwak–Yang–Zhang as a spectral problem in centro-affine differential geometry. In particular, we show that the Hilbert--Brunn--Minkowski operator coincides with the centro-affine Laplacian, thus obtaining a new avenue for tackling the conjecture using insights from affine differential geometry. As every strongly convex hypersurface in $mathbb{R}^n$ is a centro-affine unit sphere, it has constant centro-affine Ricci curvature equal to $n-2$, in stark contrast to the standard weighted Ricci curvature of the associated metric-measure space, which will in general be negative. In particular, we may use the classical argument of Lichnerowicz and a centro-affine Bochner formula to give a new proof of the Brunn–Minkowski inequality. For origin-symmetric convex bodies enjoying fairly generous curvature pinching bounds (improving with dimension), we are able to show global uniqueness in the $L^p$- and log-Minkowski problems, as well as the corresponding global $L^p$- and log-Minkowski conjectured inequalities. As a consequence, we resolve the isomorphic version of the log-Minkowski problem: for any origin-symmetric convex body $bar K$ in $mathbb{R}^n$, there exists an origin-symmetric convex body $K$ with $bar K subset K subset 8 bar K$ such that $K$ satisfies the log-Minkowski conjectured inequality, and such that $K$ is uniquely determined by its cone-volume measure $V_K$. If $bar K$ is not extremely far from a Euclidean ball to begin with, an analogous isometric result, where $8$ is replaced by $1+varepsilon$, is obtained as well.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136254446","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}
In this paper we consider nonlinearly elastic, frame-indifferent, and singularly perturbed two-well models for materials undergoing solid-solid phase transitions in any space dimensions, and we perform a simultaneous passage to sharp-interface and small-strain limits. Sequences of deformations with equibounded energies are decomposed via suitable Caccioppoli partitions into the sum of piecewise constant rigid movements and suitably rescaled displacements. These converge to limiting partitions, deformations, and displacements, respectively. Whereas limiting deformations are simple laminates whose gradients only attain two values, the limiting displacements belong to the class of special functions with bounded variation (SBV). The latter feature elastic contributions measuring the distance to simple laminates, as well as jumps associated to two consecutive phase transitions having vanishing distance, and thus undetected by the limiting deformations. By $Gamma$- convergence we identify an effective limiting model given by the sum of a quadratic linearized elastic energy in terms of displacements along with two surface terms. The first one is proportional to the total length of interfaces created by jumps in the gradient of the limiting deformation. The second one is proportional to twice the total length of interfaces created by jumps in the limiting displacement, as well as by the boundaries of limiting partitions. A main tool of our analysis is a novel two-well rigidity estimate which has been derived in [Calc. Var. Partial Differential Equations 59, art. 44 (2020)] for a model with anisotropic second-order perturbation.
{"title":"Two-well linearization for solid-solid phase transitions","authors":"Elisa Davoli, Manuel Friedrich","doi":"10.4171/jems/1385","DOIUrl":"https://doi.org/10.4171/jems/1385","url":null,"abstract":"In this paper we consider nonlinearly elastic, frame-indifferent, and singularly perturbed two-well models for materials undergoing solid-solid phase transitions in any space dimensions, and we perform a simultaneous passage to sharp-interface and small-strain limits. Sequences of deformations with equibounded energies are decomposed via suitable Caccioppoli partitions into the sum of piecewise constant rigid movements and suitably rescaled displacements. These converge to limiting partitions, deformations, and displacements, respectively. Whereas limiting deformations are simple laminates whose gradients only attain two values, the limiting displacements belong to the class of special functions with bounded variation (SBV). The latter feature elastic contributions measuring the distance to simple laminates, as well as jumps associated to two consecutive phase transitions having vanishing distance, and thus undetected by the limiting deformations. By $Gamma$- convergence we identify an effective limiting model given by the sum of a quadratic linearized elastic energy in terms of displacements along with two surface terms. The first one is proportional to the total length of interfaces created by jumps in the gradient of the limiting deformation. The second one is proportional to twice the total length of interfaces created by jumps in the limiting displacement, as well as by the boundaries of limiting partitions. A main tool of our analysis is a novel two-well rigidity estimate which has been derived in [Calc. Var. Partial Differential Equations 59, art. 44 (2020)] for a model with anisotropic second-order perturbation.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136254444","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}
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":"43 1","pages":"0"},"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":"52 1","pages":"0"},"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":"18 1","pages":"0"},"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":"36 1","pages":"0"},"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":"1 1","pages":""},"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}
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