cw_Z~i[    Number Theory Seminar at WASEDA Univ.

2005 Nx̓e  (Rg͍u҂ɏĒĂ܂.)

2006 N 2 3 ( )
 uҁF Marc-Hubert Nicole ( JSPS / University of Tokyo ) ^CgF A Geometric Interpretation of Eichler's Basis Problem for Hilbert Modular Forms AuXgNgF Let $E_i, i=1,...,n$ be all the supersingular elliptic curves over $\overline{F_p}$ up to isomorphism. The modules of isogenies $Hom(E_i,E_j)$, equipped with the degree map, are quadratic modules over $\Z$ that give rise to theta series of level $p$. The space of modular forms of weight two for $\Gamma_0(p)$ is spanned by theta series arising from supersingular elliptic curves in this fashion. We generalize this classical result to Hilbert modular forms by showing that for totally real fields $L$ of narrow class number one, the space of Hilbert modular newforms of parallel weight $2$ for $\Gamma_0(p)$, $p$ unramified, is spanned by theta series coming from quadratic modules $\Hom_{\calO_L}(A_i,A_j)$, where $A_i,A_j$ range across all superspecial abelian varieties with real multiplication by $\calO_L$. We also discuss the more delicate case when $p$ is totally ramified in $\calO_L$.
2006 N 1 27 ( )
 uҁF q ( cw ) ^CgF Sg̉~$Z_2$g̊Vsϗʂɂ AuXgNgF $p$$p \equiv 1 \bmod 4$ȂfƂB̂ƂQ̐$a,b$ɂ$p=a^2+b^2$ƏB$\omega=\sqrt{p+a\sqrt{p}}$ƂA$K=Q(\omega)$ƂB $(1)p \equiv 5 \bmod 8$ $(2)p \equiv 1 \bmod 16 2^{\frac{p-1}{4}}\equiv -1, p \equiv 9 \bmod 16 2^{\frac{p-1}{4}}\equiv 1$ 𖞂Ƃ$K$̉~Z_2$g̊V$\lambda-,\mu-$sϗʂ$0$ɂȂ邱ƂؖB 2006 N 1 27 ( )  uҁF ( cw ) ^CgF Integral Points and Rank of Elliptic Curve AuXgNgF K㐔́Af(x,y)K̐WƂRŁAʎ0łȂ̂ƂAebɑ΂Af(x,y)=b̐̌N_f(b)ƂB܂AŒꂽȐC_bƂBN_f(b)>=1̉C_bKꂽȉ~Ȑ̍\B̂ƂA̖lB uK݂̂Ɉˑ萔>0݂āAH(b)\傫AׂĂ̂Rt[Ȑbɑ΂āAN_f(b)<^(rank C_b+1)藧Bv ̖KL̂̂ƂAJ.H.SilvermanɂĉĂB{uł́AQKɑ΂邱̖̉𓚂^B 2006 N 1 20 ( )  uҁF ] Nj ( w ) ^CgF Weilݗ̈ʉƉp AuXgNgF K/kϐ֐̂Ƃ鎞A$f,g \in K$ɑ΂āA$\oveeset{\prod}{v:\text{place of $K/k$}} N_{k(v)/k}(f,g)_v=1$Ƃ̂AÓTIWeilݗłB̌@̌Milnor$K$-QpSuslina玁ɂĈʉĂB {uł́Aʑ􉽊w̐EŌÂmĂGysinʑ̐_􉽓Iȗގ̊{pĂ̌XɈʉB̈ʉꂽ́Aŋ߂̎ƐXpolysymbol̑ݗ쎁̔A[xl̂ɕtMilnor@j_Ƃ[֌WĂAԂ΂̂悤Ȋ֌WɂĂyB 2006 N 1 13 ( )  uҁF J ׎q ( cw ) ^CgF AuXgNgF ϐ͓I Eisenstein ̋ɂ2͓̎I Siegel-Eisenstein Fourier WJ𖾎IɏqׂDāCCɁCъe_ł Laurant WJ̑ꍀD܂Cdł̈ϐ͓I Cohen ^ Eisenstein (ɐRCV) Ƃ̊֌WɂďqׂD 2005 N 12 16 ( )  uҁF r ( cw ) ^CgF p VQ torsion ɂ AuXgNgF ̂悤Ȗl܂F ub̂łȂ΁Aő命d Z_p g p VQ͊V㐔 torsion free ł͂ȂHv d Z_p g 1970 N R. Greenberg ɂČĂA1978 N̘_̒ŏL̖ɓĂ͂܂Ȃ𔭌łȂAƏĂBɐ藧ǂ͍̂Ƃ悭킩ȂAV_ɂʔۂ̈łƎvB {uł́A̎̂ Greenberg \z΁A̋񎟑̂ɑ΂Ė肪Ƃb܂B܂Ap=2 ̏ꍇɈ Greenberg \zւ̉pb܂B( AA Greenberg \zɊւ錋ʂ͊ɗّẅɓiɂʂ̎@pēĂ܂ ) 2005 N 12 2 ( )  uҁF V rY (Bw) ^CgF Finiteness of abelian fundamental groups with restricted ramification AuXgNgF KXL[ɑ΂ăG^[{Q̈鏤ɂ鐔_I{Q𓱓. G^[{Qs핢𓝐Qł̂Ɠl, ̊{Q͗^ꂽq, \܂ނ蕡Gȕ핢𓝐Ă. {uł Abel {Q̗LɊւ 1. 㐔̂̐̐_IXL[, 2. L̏̑l, 3. Ǐ̏̑l, ̏ꍇɕĂ鎖ƕĂȂЉ. 2005 N 11 25 ( )  uҁF (cw) ^CgF [ 2 ̑dΐ֐̒aς 2 ϐ KZ ̐ڑ AuXgNgF 2 ϐdΐ֐̖ KZ ̉ϕCϕƂ̐ڑɂčl@Dɐ[ 2 2 ϐdΐ֐Li_{k_1,k_2}(z_1,z_2) ɂđΉ KZ ̉̔ϕ\̓Iɗ^C(z_1,z_2)=(1,0),(0,1) ɂ̐ڑ肪a Li_{k_1}(z_1) Li_{k_2}(z_2) = Li_{k_1,k_2}(z_1,z_2) + Li_{k_1+k_2}(z_1 z_2) + Li_{k_2,k_1}(z_2,z_1) y 1 ϐ Euler ^]Ɠlł邱ƂD Deligne-[̎ associator ̊􉽊wɂ鑽d[[^l̕Vbt֌WɊւʘ_̗̔ꂩ͓̉Iȉ߂^ĂD 2005 N 11 18 ( ) y w@u z  uҁF Christopher Deninger (Muenster univ.) ^CgF Number theory and foliations AuXgNgF In this lecture we want to explain analogues between certain notions in number theory and corresponding notions in the theory of foliated dynamical systems. In particular we discuss dynamical analogues of the product formula, of the explicit formulas of analytic number theory and of Lichtenbaum's conjectures on vanishing orders and leading coefficients at s=0 of zeta functions of arithmetic schemes. 2005 N 11 11 ( )  uҁF V (qw) ^CgF On non-metacyclic metabelian 2-class field towers over the cyclotomic Z_2-extensions of imaginary quadratic fields AuXgNgF 㐔 k 2-ޑ̓́A̍ős pro-2 gGaloisQ Gal(L~(k)/k) ̌qQɑΉ钆ԑ̗̂ƂĒAlX pro-2 QGaloisQƂČ邱ƂmĂ܂DQ k ̉~ Z_2 g K ɑ΂āA̍ős pro-2 gGaloisQ G~=Gal(L~(K)/K) l@A֘Aۑӂ܂ȂAG~ non-metacyclic metabelian pro-2 QƂȂ k ̖ꂽƂɂĕ񍐂܂D 2005 N 11 4 ( )  uҁF bi (cw) ^CgF 㐔̏ non-abelian p-ޑ̓ɂ AuXgNgF 㐔̏ p-ޑ̓ Galois Q̍\ɂ, abel QƂȂ邩ۂɂĂb܂. 񎟑̂ p=2 ̏ꍇɂ, V--莁ɂ钆Sޑ̂p@. ͂̎@Љ, f̏ꍇɓKpƂ, QƂȂ, ̒œꂽʂɂĘb\ł. 2005 N 10 28 ( )  uҁF ɓ i (ّw) ^CgF A[x̂̃CfAތQɊւ鈽ɂ AuXgNgF$k$2 ̂ƂB$k$̗ސ傤Ǌf$p$łA$pk$ɂĕĂꍇɂ́A$pk$̑fCfA͒PCfAɂ͂ȂȂAƂƂ͂łɒmĂB ́Ǎʂ̗ގǂ܂Ő藧HƂ_ɊւĂ̖iސ$2p$̋ 2 ́A4 A[x̂̏ꍇȂǁjグAɂēꂽƂqׂ\łBiAb錋ʂ̂̂͂łɒmĂ̂m܂Bj 2005 N 10 21 ( )  uҁF  (HƑw) ^CgF ~̂̐̋ K Qɂ AuXgNgF F L Q ̗LA[xg, p f, i 2 ȏ̐Ƃ. uł͎n߂ 2 ̃G^[RzW[QH^2(Spec(O_F[1/p],Z_p(i)) ̈ʐɂčl@. ̃RzW[Q F ɂ̏肷ƊVQ̏ŏ\Ƃo. ȒP̈, F Q 1 p^{m+1} 捪ŶƂ. ̃RzW[Q Teichmuller wW̙p ^j Œa i, j ̋vꍇ ^j p[g͊V\žnɂ Dedekind [[^֐̐̕ɂ镪q̒l p p[gƈv.ċ[̂, ȊO i, j ̋vȂꍇł. uł, L̃RzW[Q̃^j p[g̈ʐ̕\~P (i, j ̋vꍇ), KEXa (i, j ̋vȂꍇ) pė^. (i, j ̋vꍇ̉~Pɂ\͒mĂƎv.) ܂, K Q 2 ̃G^[RzW[ p-adic Chern map ɂ铯^ (Quillen-Lichtenbaum \z) ̉̉, CfAތQ K_0(O_F){tors} p-adic class number formula ̍łƎv K Q̈ʐ^\邱ƂɂĐG. 2005 N 10 14 ( )  uҁF Robert Erdahl (Queen's Univ., Canada) ^CgF Structure Theorem for Voronoi Polytopes for Lattices AuXgNgF It is natural to ask whether the Voronoi polytope for a lattice can be written as the Minkowski sum of simpler Voronoi polytopes. It turns out that this is possible if and only if the corresponding Delaunay tilings for the two simpler Voronoi polytopes are commen-surate. The Minkowski sum of two polytopes is a familiar notion in convexity theory, but the notion of commensurate Delaunay tiling is new and will be explained in the course of the talk. This Structure Theorem is the main result that will be reported, and generalizes an earlier result of S. S. Ryshkov. This Structure Theorem allows an arbitrary Voronoi polytope to be written as a Minkowski sum of simpler irreducible Voronoi polytopes, which are the building blocks and correspond through duality to "edge forms" in Voronoi's theory of lattice types. I will report what is known about the numbers of types of building blocks, or edge forms, and how these numbers of types grow with dimension. The problem of describing these elementary building blocks, and characterizing when they can combine through Minkowski sums to form more complicated Voronoi polytopes is an interesting new problem in geometry of numbers. 2005 N 10 7 ( )  uҁF (w Ȋw) ^CgF ȉ~Ȑp-SelmerQTate-ShafarevichQp-part AuXgNgF 㐔̏̑ȉ~ȐMordell-WeilQ̑傫vZ鎞ɁǍvZ̏QƂȂTate-ShafarevichQ̑傫肷邱Ƃ́Ȁ̕dvȉۑł邪AɂĂ͎̂悤ȂƂ\zĂB uefpƁA㐔Kɑ΂K̑ȉ~ȐŁATate-ShafarevichQp-partAł傫݂Bv p=2,3,5,7,13̎́A͐ƂꂼؖĂB̖̓Ƃ́AQӏЂƂ́A傫SelmerQ̍\ƁAЂƂ́Aǂ̂悤ɂāATate-ShafarevichQołB{uł́A̖ɂĂȂׂIȎbƎv܂B ̓Iɂ͎̂悤Ȗl܂B uf p Ƒ㐔 K ɑ΂āAmodular curve X_0(p)K-L_Ƃɂ̑㐔K̑ȉ~Ȑ̑ŁAp-SelmerQAł傫Ȃ݂̂Bv uXɂ̏󋵂ŁA㐔KŁAȉ~Ȑmodular̎ɁȂȉ~Ȑ̑Tate-ShafarevichQp-partł傫ȂBv 2005 N 8 3 ( ) y w@u z  uҁF Anton Deitmar (Tuebingen Univ., Germany) ^CgF A conjectural Lefschetz formula for arithmetic groups AuXgNgF The Arthur/Selberg trace formula is not well suited for geometric applications as the contributions of different rank subgroups are mixed up in the geometric side of the formula. Remedy is given in the compact case by the dynamical Lefschetz formula. In the non-compact case there is a conjectural analogue in which the spectral contributions are discretised. In the talk I shall present the compact case with geometric and arithmetic applications, such as the prime geodesic theorem and asymptotics of class numbers, and report on the state of the art in the non-compact case. 2005 N 7 15 ( )  uҁF J m ( cmwHw ) ^CgF ϐMahler֐̑㐔_ɂl̒z AuXgNgF z_ɂdvȉۑ̈ƂāAz֐̑㐔_ɂl̐_ǏBɊ֐̉ɊւMahler̗_́AFredholm$\sum_{n=0}^{\infty}z^{r^n}$($r\geq2$͐)Morse-Thue$\prod_{n=0}^{\infty}(1-z^{2^n})$͂߂Ƃ鑽̒z֐̒l̒z𓱂B֐𐶂Mahler̗_͒萫IȒ藝ɂēl㐔IɂȂ֐̗OSɌł_ŋ͂łB {ułMahler̕@瓱钴z̎ЉƋɁAŋ Duverney-ɂēꂽVȒz̏ؖ@̊TBXɔނ̕@p邱Ƃɂē閳ς𒆐SƂ̒zɂĂЉB 2005 N 7 8 ( )  uҁF Y ( cw ) ^CgF LQ̊\̍\@ɂ () AuXgNgF LQ̕\_̎vȖƂ, (1) LQ G ̋ނ肷邱 (2) k ɂ, G (SĂ) k-wW肷邱 (3) G (SĂ) k-^\̓Iɍ\邱 . (1,2) rIeՂɍsꍇł, (3) ͈ʓIɂ͔ɓł. ܂, k 㐔I̂łȂꍇɂ, Q k[G] ̍\̕Gɂ, (3) ͍XɍɂȂ. , ̖L㐔 (ɗL) ̏ꍇɋ̓IɉZ\, _̎ۂɉĂ΂Ηv̂ł. uł, ߔN㐔vZ\tgEFA̔Wӂ, (3) ɂċ̓Iȍl@sďLւ̈ꏕƂ. 2005 N 7 1 ( )  uҁF ( Bw ) ^CgF On Artin symbols of a cyclic polynomial AuXgNgF I̐_IpƂĐ\쐬ۂ̌vZ@␔\̕\@ɂďqׂ. ̗ƂĐIR񑽍$X^3-3sX^2-(3s+3)X-1$. 2005 N 6 24 ( )  uҁF v ( ʑw ) ^CgF Maass spaces of Siegel modular forms of degree 2n and the image of Ikeda lifting AuXgNgF X͏d 2k elliptic cusp forms 玟 2n (4|n,4|n-1) ŏd k+n Siegel cusp forms ւ̒rc lifts ɂĐ鎟 2n ŏd k+n Siegel cusp forms ̕Ԃ Fourier W̊Ԃ̊ȒPȐ֌WƂēt邱Ƃؖ. 2005 N 6 10 ( )  uҁF [ ( cw ) ^CgF CM^ Abelian Surface Mordell Weil rank ɂ AuXgNgF 2005 N 6 3 ( )  uҁF s j ( w ) ^CgF Stickelberger ideals and normal bases of rings of integers (with H. Sumida) AuXgNgF$p$Œ肳ꂽfƂ, 㐔$F(A_p')$𖞂Ƃ,$F$ׂ̂Ă$p$g傪$p$ɊւĐKƂ܂. L$Q$͔Cӂ$p$ł݂̏ƂmĂ܂. Z~i[ł, McCulloh ̎d𓥂܂$(A_p')$g$K=F(\zeta_p)/F$̃KAQɕt"Stickelberger ideal"$K$̃CfAތQ̂Ƃ΂ő, ̏ (1) Stickelberger ideal ̐q, pƂ (2) ͈͂$p$ɂ$p$̂̂̕$(A_p')$𖞂ۂɂďq, (3) ʓIɐ藧ȂƂ\žŏqׂ܂. ȏ̘b͋c_ (L) Ƃ̋ł. 2005 N 5 27 ( )  uҁF O Ɓi cw j ^CgF 㐔wjk AuXgNgF 㐔wɂĂ̐wj̗v_̂ЉDɁC㐔ẘ{藝ɑ΂ Gauss ̑QؖɂāCuLL̐wvւ̔ĴƂĒڂD 2005 N 5 20 ( ) yw@uz  uҁF Jerome William Hoffman (Louisiana State Univ.) ^CgF Topology of Siegel modular varieties AuXgNgF This is a survey of topological properties of the algebraic varieties that are quotients of the Siegel space of degree g. We first review the case g=1, which is the case of modular curves. Then we focus on the case g=2, which are three dimensional varieties. There are still some open questions in this case. We will discuss these aspects: a. Determination of Betti numbers. b. Relation to automorphic forms. c. Zeta functions. Time permitting we will also go into detail in some examples. 2005 N 5 13 ( )  uҁF H ( ) ^CgF Artin-SchreiergGaloisQ\ AuXgNgF WO̗Lׂ̏̂̂̂yтOQg̐̃KAQ\ɂĘb܂. 2005 N 5 6 ( )  uҁF { N@(cw Hw) ^CgF ȉ~Ȑ$2$-iKA\ɂā@ AuXgNgF W$0$̑ k ̃jbN$n$I$f(X) \in k[X]$ɑ΂$y^2 = f(x)$Ƃ$k$̒ȉ~Ȑ$X_f$Ή. ̑Ήʂ$f(X)$̃KAQ𒭂߂邱, ܂͋t$f(X)$̃KAQ̏$X_f\$ ɂǂ̒xf邩ɂ 2,3 ̊ȒPȍl@ƖNs.
2005 N 4 22 ( )
 uҁF j FV ( cw@Hw ) ^CgF 퐔Q̒ȉ~Ȑ̃RrǍQ\ɂ AuXgNgF L̂j̑ȉ~Ȑd̂jL_Qd(j)ɂ(#d(j)̑fĂƂ̌)Weil pairingpĂd(j)̌Q\肷ASY Miller ɂė^ĂD {uł́CL̂jꂽY^2=(X5)^̒ȉ~ȐJacobil i E#i(j)̑fĂ Ei(j)̏Q̐ςƓ^ł ̗𖞂Ă̂ɑ΂ Miller ̃ASYgʂɂĕ񍐂D
2005 N 4 15 ( )
 uҁF r ( cw@Hw ) ^CgF Some non-abelian extensions over Z_p-extensions and p-class field towers AuXgNgF Z_pg̗_̈̑ʂƂāAZ_pgł̐_IΏۂ̍\̍l@Aт̑Ώۂւ Z_p̍pʂ𐶂ݏoĂƂƂB̓x̃Z~i[ł́AZ_pg̔KAg̃KAQւ Z_p̍pAƂƂɂāA܂ōsĂɂĘbƎvB