Let K be an imaginary quadratic field of discriminant -D_K<0. We introduce a notion of an adelic Maass space S_{k, -k/2}^M for automorphic forms on the quasi-split unitary group U(2,2) associated with K and prove that it is stable under the action of all Hecke operators. When D_K is prime we obtain a Hecke-equivariant descent from S_{k,-k/2}^M to the space of elliptic cusp forms S_{k-1}(D_K, χ_K), where χ_K is the quadratic character of K. For a given ϕ\in S_{k-1}(D_K, χ_K), a prime l >k, we then construct (mod l) congruences between the Maass form corresponding to ϕand hermitian modular forms orthogonal to S_{k,-k/2}^M whenever the l-adic valuation of L^{alg}(\Sym^2 ϕ, k) is positive. This gives a proof of the holomorphic analogue of the unitary version of Harder's conjecture. Finally, we use these congruences to provide evidence for the Bloch-Kato conjecture for the motives \Sym^2 ρ_ϕ(k-3) and \Sym^2 ρ_ϕ(k), where ρ_ϕ denotes the Galois representation attached to ϕ.
In this article, we study the Beilinson-Bloch-Kato conjecture for motives corresponding to the Rankin-Selberg product of conjugate self-dual automorphic representations, within the framework of the Gan-Gross-Prasad conjecture. We show that if the central critical value of the Rankin-Selberg $L$-function does not vanish, then the Bloch-Kato Selmer group with coefficients in a favorable field of the corresponding motive vanishes. We also show that if the class in the Bloch-Kato Selmer group constructed from certain diagonal cycle does not vanish, which is conjecturally equivalent to the nonvanis…
We extend a formula of Duke, Imamōglu, and Tóth (which itself is a generalization of the Katok-Sarnak formula) to prove the Kohnen-Zagier formula for Maass forms for $Γ_0(4)$.
A Shintani-Katok-Sarnak type correspondence for Maass cusp forms of level $N$ is shown to be derived from analytic properties of prehomogeneous zeta functions whose coefficients involve periods of Maass forms.
K. Kato has recently defined and studied heights of mixed motives and proposed some interesting questions. In this paper, we relate the study of heights to the study of Tamagawa numbers of motives. We also partially answer one of Kato's questions about the number of mixed motives of bounded heights in the case of mixed Tate motives with two graded quotients. Finally, we provide a concrete computation with the number of mixed Tate motives with three graded quotients.
We study heights of motives with integral coefficients over number fields introduced by Kato. It is a generalization of the Faltings height of an abelian variety and we establish generalizations of some properties of the Faltings height in our context as conjectured by Kato. This sheds some light on integral structures of motives.
With the method of moments and the mollification method, we study the central $L$-values of GL(2) Maass forms of weight $0$ and level $1$ and establish a positive-proportional nonvanishing result of such values in the aspect of large spectral parameter in short intervals, which is qualitatively optimal in view of Weyl's law. As an application of this result and a formula of Katok--Sarnak, we give a nonvanishing result on the first Fourier coefficients of Maass forms of weight $\frac{1}{2}$ and level $4$ in the Kohnen plus space.
For smooth linear group schemes over $\bbZ$ we give a cohomological interpretation of the local Tamagawa measures as cohomological periods. This is in the spirit of the Tamagawa measures for motives defined by Bloch and Kato. We show that in the case of tori the cohomological and the motivic Tamagawa measures coincide, which reproves the Bloch-Kato conjecture for motives associated to tori.
Let $E$ be an elliptic curve over $\mathbb{Q}$ and $A$ be another elliptic curve over a real quadratic number field. We construct a $\mathbb{Q}$-motive of rank $8$, together with a distinguished class in the associated Bloch-Kato Selmer group, using Hirzebruch-Zagier cycles, that is, graphs of Hirzebruch-Zagier morphisms. We show that, under certain assumptions on $E$ and $A$, the non-vanishing of the central critical value of the (twisted) triple product $L$-function attached to $(E,A)$ implies that the dimension of the associated Bloch-Kato Selmer group of the motive is $0$; and the non-vani…
This is essentially the author's thesis submited to The University of Chicago (May 1997). I prove the validity of Tamagawa number conjecture of Bloch-Kato for certain Hecke characters. I study the exponential map and local Tamagawa number for all odd primes (both ordinary and supersingular), using Kato's explicit reciprocity law for one dimensional Lubin-Tate formal group. I also study p-part of Shafarevich-Tate group for motives associated to Hecke characters, using Rubin's Main Conjecture in Iwasawa theory. Interestingly a congruence property between p-adic periods of elliptic curve and weig…
In this paper we develop a new method to study Iwasawa theory and Eisenstein families for unitary groups $\mathrm{U}(r,s)$ of general signature over a totally real field $F$. As a consequence we prove that for a motive corresponding to a regular algebraic cuspidal automorphic representation $π$ on $\mathrm{U}(r,s)_{/F}$ which is ordinary at $p$, twisted by a Hecke character, if its Selmer group has rank $0$, then the corresponding central $L$-value is nonzero. This generalizes a result of Skinner-Urban in their ICM 2006 report in the special case when $F=\mathbb{Q}$ and the motive is conjugate…
We study the Eisenstein ideal for modular forms of even weight $k>2$ and prime level $N$. We pay special attention to the phenomenon of $\mathit{extra \ reducibility}$: the Eisenstein ideal is strictly larger than the ideal cutting out reducible Galois representations. We prove a modularity theorem for these extra reducible representations. As consequences, we relate the derivative of a Mazur-Tate $L$-function to the rank of the Hecke algebra, generalizing a theorem of Merel, and give a new proof of a special case of an equivariant main conjecture of Kato. In the second half of the paper, we r…
Let $E$ be a modular elliptic curve over a totally real cubic field. We have a cubic-triple product motive over $\mathbb{Q}$ constructed from $E$ through multiplicative induction; it is of rank $8$. We show that, under certain assumptions on $E$, the non-vanishing of the central critical value of the $L$-function attached to the motive implies that the dimension of the associated Bloch-Kato Selmer group is $0$.
The local Tamagawa number conjecure, first formulated by Fontaine and Perrin-Riou, expresses the compatibility of the (global) Tamagawa number conjecture on motivic $L$-functions with the functional equation. The local conjecture was proven for Tate motives over finite unramified extensions $K/\mathbb{Q}_p$ by Bloch and Kato. We use the theory of $(ϕ, Γ_K)$-modules and a reciprocity law due to Cherbonnier and Colmez to provide a new proof in the case of unramified extensions, and to prove the conjecture for the motive $\mathbb{Q}_p(2)$ over certain tamely ramified extensions.
Our first goal in this note is to explain that a weak form of Perrin-Riou's conjecture on the non-triviality of Beilinson-Kato classes follows as an easy consequence of the Iwasawa main conjectures, and deduce its refined versions in the supersingular case from this fact and a variety of Gross-Zagier formulae. Our second goal is to set up a conceptual framework in the context of $Λ$-adic Kolyvagin systems to treat analogues of Perrin-Riou's conjectures for higher motives of higher rank. We apply this general discussion in order to establish a link between Heegner points on a general class of C…
Let $p$ be an odd prime. We prove the cyclotomic Iwasawa Main Conjecture of K.Kato for the motive attached to an eigencuspform $f\in S_{k}(Γ_{0}(N))$ with arbitrary reduction type at $p$ under mild assumptions on the residual Galois representation $\barρ_{f}$. Under the same hypotheses, we also prove the generalized Iwasawa Main Conjecture for $p$-adic families of modular forms. The Iwasawa Main Conjecture for $f$ is deduced by a limit argument involving fundamental lines from a universal Iwasawa Main Conjecture over the universal deformation space of $\barρ_{f}$, which itself follows from the…
In 2006, Fukaya and Kato formulated a general conjecture about $p$-adic $L$-functions for a large class of motives and derived a precise conjectural interpolation formula from the ETNC. We study this conjectural $p$-adic $L$-function in the setting of Hida families and compare it to the $p$-adic $L$-function constructed by Kitagawa. A priori it is not clear whether the interpolation formulas coincide, mainly because of the rather inexplicit definition of periods (via comparison isomorphisms) that appear in the general formula. Using $p$-adic Eichler-Shimura isomorphisms and their interpolation…
Assuming specific instances of two general conjectures in arithmetic algebraic geometry (bijectivity of $p$-adic regulator maps, injectivity of $p$-adic Abel-Jacobi maps), we prove several cases of the $p$-part of the Tamagawa number conjecture ($p$-TNC) of Bloch-Kato and Fontaine-Perrin-Riou for (homological) motives of modular forms of even weight $\geq4$ in analytic rank $1$. More precisely, we prove our results for a large class of newforms $f$ and prime numbers $p$ that are ordinary for $f$ and such that the weight of $f$ is congruent to $2$ modulo $2(p-1)$. Inspired by work of W. Zhang i…
This introductory paper studies a class of real analytic functions on the upper half plane satisfying a certain modular transformation property. They are not eigenfunctions of the Laplacian and are quite distinct from Maass forms. These functions are modular equivariant versions of real and imaginary parts of iterated integrals of holomorphic modular forms, and are modular analogues of single-valued polylogarithms. The coefficients of these functions in a suitable power series expansion are periods. They are related both to mixed motives (iterated extensions of pure motives of classical modula…
The aim of the present paper is to give evidence, largely numerical, in support of the non-commutative main conjecture of Iwasawa theory for the motive of a primitive modular form of weight k>2 over the Galois extension of Q obtained by adjoining to Q all p-power roots of unity, and all p-power roots of a fixed integer m>1. The predictions of the main conjecture are rather intricate in this case because there is more than one critical point, and also there is no canonical choice of periods. Nevertheless, our numerical data agrees perfectly with all aspects of the main conjecture, including Kat…
Answering problems of Manin, we use the critical $L$-values of even weight $k\geq 4$ newforms $f\in S_k(Γ_0(N))$ to define zeta-polynomials $Z_f(s)$ which satisfy the functional equation $Z_f(s)=\pm Z_f(1-s)$, and which obey the Riemann Hypothesis: if $Z_f(ρ)=0$, then $\operatorname{Re}(ρ)=1/2$. The zeros of the $Z_f(s)$ on the critical line in $t$-aspect are distributed in a manner which is somewhat analogous to those of classical zeta-functions. These polynomials are assembled using (signed) Stirling numbers and "weighted moments" of critical values $L$-values. In analogy with Ehrhart polyno…
We define canonical real analytic versions of modular forms of integral weight for the full modular group, generalising real analytic Eisenstein series. They are harmonic Maass waveforms with poles at the cusp, whose Fourier coefficients involve periods and quasi-periods of cusp forms, which are conjecturally transcendental. In particular, we settle the question of finding explicit `weak harmonic lifts' for every eigenform of integral weight $k$ and level one. We show that mock modular forms of integral weight are algebro-geometric and have Fourier coefficients proportional to $n^{1-k}(a'_n + …
