Prime Power Level Research Articles

On the rational cuspidal divisor class groups of Drinfeld modular curves X0(𝔭r)

Let [Formula: see text] be the rational cuspidal divisor class group of the Drinfeld modular curve [Formula: see text] for a prime power level [Formula: see text]. We relate the rational cuspidal divisors of degree 0 on [Formula: see text] with [Formula: see text]-quotients, where [Formula: see text] is the Drinfeld discriminant function. As a result, we are able to determine explicitly the structure of [Formula: see text] for arbitrary prime [Formula: see text] and [Formula: see text].

Rational torsion of generalized Jacobians of modular and Drinfeld modular curves

Abstract We consider the generalized Jacobian J ~ {\widetilde{J}} of the modular curve X 0 ⁢ ( N ) {X_{0}(N)} of level N with respect to a reduced divisor consisting of all cusps. Supposing N is square free, we explicitly determine the structure of the ℚ {\mathbb{Q}} -rational torsion points on J ~ {\widetilde{J}} up to 6-primary torsion. The result depicts a fuller picture than [18] where the case of prime power level was studied. We also obtain an analogous result for Drinfeld modular curves. Our proof relies on similar results for classical Jacobians due to Ohta, Papikian and the first author. We also discuss the Hecke action on J ~ {\widetilde{J}} and its Eisenstein property.

The pro-l outer Galois actions associated to modular curves of prime power level

The pro-l outer Galois actions associated to modular curves of prime power level

The first moment of twisted Hecke $L$-functions with unbounded shifts

We compute the first moment of twisted Hecke $L$-functions of weight $2$ and prime power level going to infinity, uniformly in the conductor of the twist and in the vertical shift.

Modular curves of prime-power level with infinitely many rational points

For each open subgroup $G$ of ${\rm GL}_2(\hat{\mathbb{Z}})$ containing $-I$ with full determinant, let $X_G/\mathbb{Q}$ denote the modular curve that loosely parametrizes elliptic curves whose Galois representation, which arises from the Galois action on its torsion points, has image contained in $G$. Up to conjugacy, we determine a complete list of the $248$ such groups $G$ of prime power level for which $X_G(\mathbb{Q})$ is infinite. For each $G$, we also construct explicit maps from each $X_G$ to the $j$-line. This list consists of $220$ modular curves of genus $0$ and $28$ modular curves of genus $1$. For each prime $\ell$, these results provide an explicit classification of the possible images of the $\ell$-adic Galois representations arising from elliptic curves over $\mathbb{Q}$ that is complete except for a finite set of exceptional $j$-invariants.

Non-vanishing of automorphic L-functions of prime power level

We prove that at the minimum $$25\%$$ of L-values associated to holomorphic newforms of fixed even integral weight and large prime power level do not vanish at the critical point.

Values of automorphic L-functions with prime power level at the point $$s=1$$ s = 1

We discuss in this work the distributions of values of L(1, f), where f is a primitive cusp form whose level is a prime power. We prove the upper bound part of the Montgomery–Vaughan’s first conjecture and give a weaker version of the lower bound part for automorphic L-functions in this case. We establish an unweighted trace formula in aspect of prime power level in our proof.

A uniform asymptotic formula for the second moment of primitive L-functions on the critical line

We prove an asymptotic formula for the second moment of primitive L-functions of even weight and prime power level. The error term is estimated uniformly in all parameters: level, weight, shift, and twist.

The shifted fourth moment of automorphic $L$-functions of prime power level

We prove the asymptotic formula for the fourth moment of automorphic $L$-functions of level $p^{\nu}$, where $p$ is a fixed prime number and $\nu \rightarrow \infty$. This paper is a continuation of work by Rouymi, who computed asymptotics of the first three moments at prime power level, and a generalization of results obtained for prime level by Duke, Friedlander & Iwaniec and Kowalski, Michel & Vanderkam.

Stable averages of central values of Rankin–Selberg L -functions: Some new variants

As shown by Michel and Ramakrishnan (2007) and later generalized by Feigon and Whitehouse (2008), there are “stable” formulas for the average central L -value of the Rankin–Selberg convolutions of some holomorphic forms of fixed even weight and large level against a fixed imaginary quadratic theta series. We obtain exact finite formulas for twisted first moments of Rankin–Selberg L -values in much greater generality and prove analogous “stable” formulas when one considers either arbitrary modular twists of large prime power level or real dihedral twists of odd type associated to a Hecke character of mixed signature.

Effective Siegel's theorem for modular curves

We prove that integral points can be effectively determined on all but finitely many modular curves, and on all but one modular curve of prime power level.