arXiv subjects
Publications and source records attributed to Dinakar Ramakrishnan.
Half a century ago Manin showed that given a number field $k$ and a rational prime $\ell$, there exists a uniform bound for the order of cyclic $\ell$-power isogenies between two non-CM elliptic curves over $k$. We generalize this to certain $2$-dimensional families of abelian $3$-folds with multiplication by an imaginary quadratic field.
In this paper we calculate the asymptotics of the second moment of the Bessel periods associated to certain holomorphic cuspidal representations $(π, π')$ of $U(2,1) \times U(1,1)$ of regular infinity type (averaged over $π$). Using these, we obtain quantitative non-vanishing results for the Rankin-Selberg central $L$-values $L(1/2, π\times π')$, which are of degree twelve over $\mathbb{Q}$, with concomitant difficulty in applying standard methods, especially since we are in a `conductor dropping' situation. We use the relative trace formula, and the orbital integrals are evaluated rather than compared with others. Besides their intrinsic interest, non-vanishing of these critical values also lead, by known results, to deducing certain associated Selmer groups have rank zero.
Let E be an elliptic curve over a number field F, A the abelian surface E x E, and T_F(A) the F-rational albanese kernel of A, which is a subgroup of the degree zero part of Chow group of zero cycles on A modulo rational equivalence. The first result is that for all but a finite number of primes p where E has ordinary reduction, the image of T_F(A)/p in the Galois cohomology group H^2(F, sym^2(E[p])) is zero; here E[p] denotes as usual the Galois module of p-division points on E. The second result is that for any prime p where E has good ordinary reduction, there is a finite extension K of F, depending on p and E, such that T_K(A)/p is non-zero. Much of this work was joint with Jacob Murre, and the article is dedicated to his memory.
Fix a Dirichlet character $χ$ and a cuspidal GL$(2)$ eigenform $ϕ$ with relatively prime conductors. Then we show that there are infinitely many cusp forms $π$ on GL$(3)$ such that $L(1/2, π\times χ)$ and $L(1/2, π\times ϕ)$ are simultaneously non-zero. We achieve this by use of Jacquet's Relative Trace Formula. We derive an expression of the average over the GL$(3)$ cuspidal spectrum as a sum of a non-zero main term and two subsidiary terms which are forced to be zero for large enough level by use of a suitable test function. This modest article is dedicated to the memory of Harish Chandra, on the occasion of his hundredth birthday.
This Note answers, and generalizes, a question of Kaisa Matomäki. We show that give two cuspidal automorphic representations $π_1$ and $π_2$ of $GL_n$ over a number field $F$ of respective conductors $N_1,$ $N_2,$ every character $χ$ such that $π_1\otimesχ\simeqπ_2$ of conductor $Q,$ satisfies the bound: $Q^n\mid N_1N_2.$ If at every finite place $v,$ $π_{1,v}$ is a discrete series whenever it is ramified, then $Q^n$ divides the least common multiple $[N_1, N_2].$
Let $K/F$ be a finite Galois extension of number fields. It is well known that the Tchebotarev density theorem implies that an irreducible, finitely ramified $p$-adic representation $ρ$ of the absolute Galois group of $K$ is determined (up to equivalence) by the characteristic polynomials of Frobenius elements Fr$_v$ at any set of primes $v$ of $K$ of degree $1$ over $F$. Here we prove an analogue for GL$(n)$, namely that a cuspidal automorphic representation $π$ of GL$(n, {\mathbb A}_K)$ is determined up by the knowledge of its local components at the primes of degree one over $F$. We prove in fact a stronger theorem, stimulated by a question of Michael Rapoport and Wei Zhang, relaxing to an extent the Galois hypothesis. The method uses, besides the Rankin-Selberg theory of L-functions and the Luo-Rudnick-Sarnak bound for the Hecke roots of $π$, certain consequences of class field theory via Galois cohomology. In an earlier paper (\cite{Ra2}) we obtained such a result up to twist equivalence for $K/F$ cyclic of prime degree by using basic Kummer theory. We make use of suitable solvable base changes $π_M$, relative to certain auxiliary succession of abelian extensions $E/F$, with $M$ being an abelian extension of the compositum $EK$, and deduce that $π_M \simeq π'_M$, and then descend this isomorphism to one over $K$. A key ingredient for progress here is the use of global Tate duality and a local-global result arising from class field theory. In fact we prove the main result for {\it isobaric} automorphic representations, which are analogues of {\it semisimple} Galois representations. In the last section we introduce a notion of {\it semi-temperedness}, which is much weaker than temperedness, but allows for the deduction of the main result without any hypothesis whatsoever on $K/F$.
For any elliptic curve $E$ over $k\subset \Bbb R$ with $E({\Bbb C})={\Bbb C}^\times/q^{\Bbb Z}$, $q=e^{2πiz}, \Im(z)>0$, we study the $q$-average $D_{0,q}$, defined on $E({\Bbb C})$, of the function $D_0(z) = \Im(z/(1-z))$. Let $Ω^+(E)$ denote the real period of $E$. We show that there is a rational function $R \in {\Bbb Q}(X_1(N))$ such that for any non-cuspidal real point $s\in X_1(N)$ (which defines an elliptic curve $E(s)$ over $\Bbb R$ together with a point $P(s)$ of order $N$), $πD_{0,q}(P(s))$ equals $Ω^+(E(s))R(s)$. In particular, if $s$ is $\Bbb Q$-rational point of $X_1(N)$, a rare occurrence according to Mazur, $R(s)$ is a rational number.
We prove that various arithmetic quotients of the unit ball in $\mathbb{C}^n$ are Mordellic, in the sense that they have only finitely many rational points over any finitely generated field extension of $\mathbb{Q}$. In the previously known case of compact hyperbolic complex surfaces, we give a new proof using their Albanese in conjunction with some key results of Faltings, but without appealing to the Shafarevich conjecture. In higher dimension, our methods allow us to solve an alternative of Ullmo and Yafaev. Our strongest result uses in addition Rogawski's theory and establishes the Mordellicity of the Baily-Borel compactifications of Picard modular surfaces of some precise levels related to the discriminant of the imaginary quadratic fields.
Suppose $π$, $π'$ are cusp forms on GL$(2)$, not of solvable polyhedral type, such that they have the same symmetric cubes. Then we show that either $π$, $π'$ are twist equivalent, or else a certain degree $36$ $L$-function associated to the pair has a pole at $s=1$. If we further assume that the symmetric fifth power of $π$ is automorphic, then in the latter case, $π$ is icosahedral in a suitable sense, agreeing with the usual notion when there is an associated Galois representation.
Let $F/k$ be a cyclic extension of number fields of prime degree. Let $ρ$ be an irreducible $2$-dimensional representation of Artin type of the absolute Galois group of $F$, and $π$ a cuspidal automorphic representation of GL$_2(\mathbb A_F)$, such that the $L$-functions $L(s,ρ_v)$ and $L(s,π_v)$ agree at all (but finitely many of) the places $v$ of degree one over $k$. We prove in this case that we have the global identity $L(s,ρ)=L(s,π)$, with $ρ_v \leftrightarrow π_v$ being given by the local Langlands correspondence at all $v$. In particular, $π$ is tempered and $L(s,ρ)$ is entire.
It is well known that the Tchebotarev density theorem implies that an irreducible $\ell$-adic representation $ρ$ of the absolute Galois group of a number field $K$ is determined (up to isomorphism) by the characteristic polynomials of Frobenius elements at any set of primes of density 1. In this Note we make some progress on the automorphic side for GL$(n)$ by showing that, given a cyclic extension $K/k$ of number fields of prime degree $p$, a cuspidal automorphic representation $π$ of GL$(n,{\mathbb A}_K)$ is determined up to twist equivalence by the knowledge of its local components at the (density one) set $S_{K/k}$ of primes of $K$ of degree $1$ over $k$, and moreover that $π$ is determined even up to isomorphism if $p=2$. The proof uses the Luo-Rudnick-Sarnak bound for the Hecke roots of $π$, applied to certain Rankin-Selberg $L$-functions of positive type, in conjunction with some Kummer theory and descent along suitable $p$-power extensions arising as nested sequences of cyclic $p^2$-extensions.
Given a holomorphic newform $f$ of weight $k$ and with rational coefficients, a question of Mazur and van Straten asks if there is an associated Calabi-Yau variety $X$ over ${\mathbb Q}$ of dimension $k-1$ such that the $\ell$-adic Galois representation of $f$ occurs in the cohomology of $X$ in degree $k-1$. We provide some explicit examples giving a positive answer, and show moreover that such $X$ come equipped with an involution $τ$ acting by $-1$ on $H^0(X, Ω^{k-1})$. We also raise a general question regarding the regular algebraic, (essentially) selfdual cusp forms $π$ on GL$(n)$ with ${\mathbb Q}$-coefficients, asking for associated Calabi-Yau varieties $X=X_π$ (with an involution $τ$ on each such $X$ such that the quotient variety $X/τ$ is rational) carrying the (conjectural) motive of $π$. We then investigate the compatibility of this with Rankin-Selberg products of modular forms.
Let F be a number field with adele ring A_F, and πan isobaric, algebraic automorphic representation of GL_4(A_F) of a fixed archimedean weight, which is quasi-regular, meaning that at every archimedean place v of F, the 4-dimensional representation σ_v of the Weil group W_{F_v} attached to π_v is multiplicity free. Suppose there is an associated 4-dimensional, Hodge-Tate p-adic representation ρof the absolute Galois group G_F, whose local L-factors agree with those of π(up to a shift) at almost all primes P of F. Then our first result is that the semisimplification of ρdoes not contain any irreducible 2-dimensional Galois representation which is even. The second result is that if πis regular and ρcrystalline, then for sufficiently large p (see the article for a precise statement), the decomposition type of ρis the same as the isobaric type of π. A consequence is that ρis irreducible when πis cuspidal (and regular algebraic), which has also been proved by F. Calegari and T. Gee, in fact with no hypothesis on p. The third and final result, using Taylor's potential modularity theorem, is that given a pair (σ, σ') of odd, 2-dimensional p-adic representations of the same weight and distinct Hodge-Tate types, such that their direct sum is automorphic, the dimension of the G_F-invariants of the tensor product ηof the dual of σwith σ' equals, for large p, the order of pole at s=1 of the L-function of η(with the bad factors removed). This is as predicted by the Tate conjecture when σ, σ' occur in p-adic etale cohmology of smooth projective varieties over F, and it also provides a useful link to a small piece of the work of C. Skinner and E. Urban.
For any number field F, call a cusp form πon GL(2)/F {\it special icosahedral}, or just s-icosahedral for short, if πis not solvable polyhedral, and for a suitable "conjugate" cusp form π' on GL(2)/F, sym^3(π) is isomorphic to sym^3(π'), and the symmetric fifth power L-series of πequals the Rankin-Selberg L-function L(s, sym^2(π') x π) (up to a finite number of Euler factors). Then the point of this Note is to obtain the following result: Let πbe s-icosahedral (of trivial central character). Then π_f is algebraic without local components of Steinberg type, π_\infty is of Galois type, and π_v is tempered everywhere. Moreover, if π' is also of trivial central character, it is s-icosahedral as well, and the field of rationality \Q(π_f) (of π_f) is K:=\Q[\sqrt{5}], with π'_f being the Galois conjugate of π_f under the non-trivial automorphism of K.
In this paper we prove the following conditional result: Let F be a number field, and pi a cusp form on GL(2)/F which is not solvable polyhedral. Assume that all the symmetric powers sym^m(pi) are modular, i.e., define automorphic forms on GL(m+1)/F. If sym^6(pi) is cuspidal, then all the symmetric powers are cuspidal, for all m. Moreover, sym^6(pi) is Eisenteinian iff sym^5(pi) is an abelian twist of the functorial product of pi with the symmetric square of a cusp form pi' on GL(2)/F.
This article is the first part of a two-part work on the Albanese kernel T_F(E x E), for an elliptic curve E over F. The main result furnishes information, for any odd prime p, about the kernel and image of the Galois symbol map from T_F(E \times E)/p to the Galois cohomology group H^2(F, E[p] (x) E[p]), for F a p-adic field and E/F ordinary, without requiring that the p-torsion points are F-rational. A key step is to show that the image is zero when the Galois module E[p] is non-semisimple. The forthcoming second part will deal with global questions.
We exhibit a closed hyperbolic 3-manifold which satisfies a very strong form of Thurston's Virtual Fibration Conjecture. In particular, this manifold has finite covers which fiber over the circle in arbitrarily many ways. More precisely, it has a tower of finite covers where the number of fibered faces of the Thurston norm ball goes to infinity, in fact faster than any power of the logarithm of the degree of the cover, and we give a more precise quantitative lower bound. The example manifold M is arithmetic, and the proof uses detailed number-theoretic information, at the level of the Hecke eigenvalues, to drive a geometric argument based on Fried's dynamical characterization of the fibered faces. The origin of the basic fibration of M over the circle is the modular elliptic curve E=X_0(49), which admits multiplication by the ring of integers of Q[sqrt(-7)]. We first base change the holomorphic differential on E to a cusp form on GL(2) over K=Q[sqrt(-3)], and then transfer over to a quaternion algebra D/K ramified only at the primes above 7; the fundamental group of M is a quotient of the principal congruence subgroup of level 7 of the multiplicative group of a maximal order of D. To analyze the topological properties of M, we use a new practical method for computing the Thurston norm, which is of independent interest. We also give a non-compact finite-volume hyperbolic 3-manifold with the same properties by using a direct topological argument.
Selfdual representations of any group fall into two classes when they are irreducible: those which carry a symmetric bilinear form, and the others which carry an alternating bilinear form. The Langlands correspondence, which matches the irreducible representations σof the Weil group of a local field k of dimension n with the irreducible representations πof the invertible elements of a division algebra D over k of index n, takes selfdual representations to selfdual representations. In this paper we use global methods to study how the Langlands correspondence behaves relative to this distinction among selfdual representations. We prove in particular that for n even, σis symplectic if and only if πis orthogonal. Our results treat more generally the case of GL_m(B), for B a division algebra over k of index r, and n=mr.