This paper provides an introduction to Eisenstein measures, a powerful tool for constructing certain $p$-adic $L$-functions. First seen in Serre's realization of $p$-adic Dedekind zeta functions associated to totally real fields, Eisenstein measures provide a way to extend the style of congruences Kummer observed for values of the Riemann zeta function (so-called {em Kummer congruences}) to certain other $L$-functions. In addition to tracing key developments, we discuss some challenges that arise in more general settings, concluding with some that remain open.
{"title":"An introduction to Eisenstein measures","authors":"E. Eischen","doi":"10.5802/jtnb.1178","DOIUrl":"https://doi.org/10.5802/jtnb.1178","url":null,"abstract":"This paper provides an introduction to Eisenstein measures, a powerful tool for constructing certain $p$-adic $L$-functions. First seen in Serre's realization of $p$-adic Dedekind zeta functions associated to totally real fields, Eisenstein measures provide a way to extend the style of congruences Kummer observed for values of the Riemann zeta function (so-called {em Kummer congruences}) to certain other $L$-functions. In addition to tracing key developments, we discuss some challenges that arise in more general settings, concluding with some that remain open.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45615066","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $K$ be a number field and $fin K[X]$ an irreducible monic polynomial with coefficients in $O_K$, the ring of integers of $K$. We aim to enounce an effective criterion, in terms of the Galois group of $f$ over $K$ and a linear recurrence sequence associated to $f$, allowing sometimes to characterize the prime ideals of $O_K$ modulo which $f$ completely splits. If $alpha$ is a root of $f$, this criterion therefore gives a characterization of the prime ideals of $O_K$ which split completely in $K(alpha)$. It does apply if the degree of $f$ is at least $4$ and the Galois group of $f$ is the symmetric group or the alternating group.
{"title":"Idéaux premiers totalement décomposés et sommes de Newton","authors":"D. Bernardi, A. Kraus","doi":"10.5802/jtnb.1213","DOIUrl":"https://doi.org/10.5802/jtnb.1213","url":null,"abstract":"Let $K$ be a number field and $fin K[X]$ an irreducible monic polynomial with coefficients in $O_K$, the ring of integers of $K$. We aim to enounce an effective criterion, in terms of the Galois group of $f$ over $K$ and a linear recurrence sequence associated to $f$, allowing sometimes to characterize the prime ideals of $O_K$ modulo which $f$ completely splits. If $alpha$ is a root of $f$, this criterion therefore gives a characterization of the prime ideals of $O_K$ which split completely in $K(alpha)$. It does apply if the degree of $f$ is at least $4$ and the Galois group of $f$ is the symmetric group or the alternating group.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47062868","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For any noncocompact Fuchsian group $Gamma$, we show that periods of the canonical differential of the third kind associated to residue divisors of cusps are expressed in terms of Rademacher symbols for $Gamma$ - generalizations of periods appearing in the classical theory of modular forms. This result provides a relation between Rademacher symbols and the famous theorem of Manin and Drinfeld. On this basis, we present a straightforward group-theoretic argument to establish both the rationality of Rademacher symbols and the validity of the Manin-Drinfeld theorem for new families of Fuchsian groups and algebraic curves.
{"title":"The Manin–Drinfeld theorem and the rationality of Rademacher symbols","authors":"Claire Burrin","doi":"10.5802/jtnb.1225","DOIUrl":"https://doi.org/10.5802/jtnb.1225","url":null,"abstract":"For any noncocompact Fuchsian group $Gamma$, we show that periods of the canonical differential of the third kind associated to residue divisors of cusps are expressed in terms of Rademacher symbols for $Gamma$ - generalizations of periods appearing in the classical theory of modular forms. This result provides a relation between Rademacher symbols and the famous theorem of Manin and Drinfeld. On this basis, we present a straightforward group-theoretic argument to establish both the rationality of Rademacher symbols and the validity of the Manin-Drinfeld theorem for new families of Fuchsian groups and algebraic curves.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44446114","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The first aim of this note is to fill a gap in the literature by giving a proof of the following refinement of Shafarevich's theorem on solvable Galois groups: Given a global field $k$, a finite set $mathcal{S}$ of primes of $k$, and a finite solvable group $G$, there is a Galois field extension of $k$ of Galois group $G$ in which all primes in $mathcal{S}$ are totally split. To that end, we prove that, given a global field $k$ and a finite set $mathcal{S}$ of primes of $k$, every finite split embedding problem $G rightarrow {rm{Gal}}(L/k)$ over $k$ with nilpotent kernel has a solution ${rm{Gal}}(F/k) rightarrow G$ such that all primes in $mathcal{S}$ are totally split in $F/L$. We then use this to contribute to inverse Galois theory over division rings. Namely, given a finite split embedding problem with nilpotent kernel over a finite field $k$, we fully describe for which automorphisms $sigma$ of $k$ the embedding problem acquires a solution over the skew field of fractions $k(T, sigma)$ of the twisted polynomial ring $k[T, sigma]$.
{"title":"A note on finite embedding problems with nilpotent kernel","authors":"Arno Fehm, Franccois Legrand","doi":"10.5802/jtnb.1215","DOIUrl":"https://doi.org/10.5802/jtnb.1215","url":null,"abstract":"The first aim of this note is to fill a gap in the literature by giving a proof of the following refinement of Shafarevich's theorem on solvable Galois groups: Given a global field $k$, a finite set $mathcal{S}$ of primes of $k$, and a finite solvable group $G$, there is a Galois field extension of $k$ of Galois group $G$ in which all primes in $mathcal{S}$ are totally split. To that end, we prove that, given a global field $k$ and a finite set $mathcal{S}$ of primes of $k$, every finite split embedding problem $G rightarrow {rm{Gal}}(L/k)$ over $k$ with nilpotent kernel has a solution ${rm{Gal}}(F/k) rightarrow G$ such that all primes in $mathcal{S}$ are totally split in $F/L$. We then use this to contribute to inverse Galois theory over division rings. Namely, given a finite split embedding problem with nilpotent kernel over a finite field $k$, we fully describe for which automorphisms $sigma$ of $k$ the embedding problem acquires a solution over the skew field of fractions $k(T, sigma)$ of the twisted polynomial ring $k[T, sigma]$.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46918028","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
These notes expand on the presentation given by the second author at the Iwasawa 2019 conference in Bordeaux of our joint work on the geometry of the $p$-adic eigencurve at a weight one CM form $f$ irregular at $p$, namely its implications in Iwasawa and in Hida theories. Novel features include the determination of -- the Fourier coefficients of the infinitesimal deformations of $f$ along each Hida family containing it in terms of $p$-adic logarithms of algebraic numbers. -- the "mysterious" cross-ratios of the ordinary filtrations of the Hida families containing $f$.
{"title":"A geometric view on Iwasawa theory","authors":"Adel Betina, Mladen Dimitrov","doi":"10.5802/jtnb.1176","DOIUrl":"https://doi.org/10.5802/jtnb.1176","url":null,"abstract":"These notes expand on the presentation given by the second author at the Iwasawa 2019 conference in Bordeaux of our joint work on the geometry of the $p$-adic eigencurve at a weight one CM form $f$ irregular at $p$, namely its implications in Iwasawa and in Hida theories. Novel features include the determination of \u0000-- the Fourier coefficients of the infinitesimal deformations of $f$ along each Hida family containing it in terms of $p$-adic logarithms of algebraic numbers. \u0000-- the \"mysterious\" cross-ratios of the ordinary filtrations of the Hida families containing $f$.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48137262","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this short paper we prove a quantitative version of the Khintchine-Groshev Theorem with congruence conditions. Our argument relies on a classical argument of Schmidt on counting generic lattice points, which in turn relies on a certain variance bound on the space of lattices.
{"title":"Quantitative Diophantine approximation with congruence conditions","authors":"Mahbub Alam, Anish Ghosh, Shucheng Yu","doi":"10.5802/jtnb.1161","DOIUrl":"https://doi.org/10.5802/jtnb.1161","url":null,"abstract":"In this short paper we prove a quantitative version of the Khintchine-Groshev Theorem with congruence conditions. Our argument relies on a classical argument of Schmidt on counting generic lattice points, which in turn relies on a certain variance bound on the space of lattices.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42793247","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 describe an algorithm that, for a smooth connected curve X over a field k, a finite locally constant sheaf A on Xét of abelian groups of torsion invertible in k, computes the first étale cohomology H(Xksep,ét,A) and the first étale cohomology with proper support Hc(Xksep,ét,A) as sets of torsors. The complexity of this algorithm is exponential in nlog , pa(X), and pa(A), where pa(X) is the arithmetic genus of the normal completion of X, pa(A) is the arithmetic genus of the normal completion Y of the smooth curve representing A, and n is the degree of Y over X. The computation in this algorithm is done via the computation of a groupoid scheme classifying the A-torsors with some extra rigidifying data.
{"title":"Computation of étale cohomology on curves in single exponential time","authors":"Jinbi Jin","doi":"10.5802/jtnb.1124","DOIUrl":"https://doi.org/10.5802/jtnb.1124","url":null,"abstract":"In this paper, we describe an algorithm that, for a smooth connected curve X over a field k, a finite locally constant sheaf A on Xét of abelian groups of torsion invertible in k, computes the first étale cohomology H(Xksep,ét,A) and the first étale cohomology with proper support Hc(Xksep,ét,A) as sets of torsors. The complexity of this algorithm is exponential in nlog , pa(X), and pa(A), where pa(X) is the arithmetic genus of the normal completion of X, pa(A) is the arithmetic genus of the normal completion Y of the smooth curve representing A, and n is the degree of Y over X. The computation in this algorithm is done via the computation of a groupoid scheme classifying the A-torsors with some extra rigidifying data.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81432791","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In local class field theory, the Schmid–Witt symbol encodes interesting data about the ramification theory of p-extensi-ons of K and can, for example, be used to compute the higher ramification groups of such extensions. In 1936, Schmid discovered an explicit formula for the Schmid–Witt symbol of Artin–Schreier extensions of local fields. Later, his formula was generalized to Artin–Schreier–Witt extensions, but still over a local field. In this paper we generalize Schmid’s formula to compute the Artin–Schreier–Witt– Parshin symbol for Artin–Schreier–Witt extensions of two-dimensional local fields of positive characteristic.
{"title":"Schmid’s Formula for Higher Local Fields","authors":"M. Schmidt","doi":"10.5802/jtnb.1125","DOIUrl":"https://doi.org/10.5802/jtnb.1125","url":null,"abstract":"In local class field theory, the Schmid–Witt symbol encodes interesting data about the ramification theory of p-extensi-ons of K and can, for example, be used to compute the higher ramification groups of such extensions. In 1936, Schmid discovered an explicit formula for the Schmid–Witt symbol of Artin–Schreier extensions of local fields. Later, his formula was generalized to Artin–Schreier–Witt extensions, but still over a local field. In this paper we generalize Schmid’s formula to compute the Artin–Schreier–Witt– Parshin symbol for Artin–Schreier–Witt extensions of two-dimensional local fields of positive characteristic.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81722640","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. In [Tha04] Thakur defines function field analogs of the classical multiple zeta function, namely, ζ d ( F q [ T ]; s 1 ,...,s d ) and ζ d ( K ; s 1 ,...,s d ), where K is a global function field. Star versions of these functions were further studied by Masri [Mas06]. We prove reduction formulas for these star functions, extend the construction to function field analogs of multiple polylogarithms, and exhibit some formulas for multiple zeta values.
{"title":"Multiple zeta functions and polylogarithms over global function fields","authors":"Debmalya Basak, Nicolas Degré-Pelletier, M. Lalín","doi":"10.5802/jtnb.1128","DOIUrl":"https://doi.org/10.5802/jtnb.1128","url":null,"abstract":". In [Tha04] Thakur defines function field analogs of the classical multiple zeta function, namely, ζ d ( F q [ T ]; s 1 ,...,s d ) and ζ d ( K ; s 1 ,...,s d ), where K is a global function field. Star versions of these functions were further studied by Masri [Mas06]. We prove reduction formulas for these star functions, extend the construction to function field analogs of multiple polylogarithms, and exhibit some formulas for multiple zeta values.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45347795","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 study the field of definition of abelian subvarieties $Bsubset A_{overline{K}}$ for an abelian variety $A$ over a field $K$ of characteristic $0$. We show that, provided that no isotypic component of $A_{overline{K}}$ is simple, there are infinitely many abelian subvarieties of $A_{overline{K}}$ with field of definition $K_A$, the field of definition of the endomorphisms of $A_{overline{K}}$. This result combined with earlier work of R'emond gives an explicit maximum for the minimal degree of a field extension over which an abelian subvariety of $A_{overline{K}}$ is defined with varying $A$ of fixed dimension and $K$ of characteristic $0$.
{"title":"Fields of definition of abelian subvarieties","authors":"S. Philip","doi":"10.5802/jtnb.1214","DOIUrl":"https://doi.org/10.5802/jtnb.1214","url":null,"abstract":"In this paper we study the field of definition of abelian subvarieties $Bsubset A_{overline{K}}$ for an abelian variety $A$ over a field $K$ of characteristic $0$. We show that, provided that no isotypic component of $A_{overline{K}}$ is simple, there are infinitely many abelian subvarieties of $A_{overline{K}}$ with field of definition $K_A$, the field of definition of the endomorphisms of $A_{overline{K}}$. This result combined with earlier work of R'emond gives an explicit maximum for the minimal degree of a field extension over which an abelian subvariety of $A_{overline{K}}$ is defined with varying $A$ of fixed dimension and $K$ of characteristic $0$.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44039208","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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