361 research outputs found
L-invariants for cohomological representations of PGL(2) over arbitrary number fields
No full textGehrmann L, Pati MR. L-invariants for cohomological representations of PGL(2) over arbitrary number fields. Forum of Mathematics, Sigma. 2024;12: e71.**Abstract**
Let
be a cuspidal, cohomological automorphic representation of an inner form
G
of
over a number field
F
of arbitrary signature. Further, let
be a prime of
F
such that
G
is split at
and the local component
of
at
is the Steinberg representation. Assuming that the representation is noncritical at
, we construct automorphic
-invariants for the representation
. If the number field
F
is totally real, we show that these automorphic
-invariants agree with the Fontaine–Mazur
-invariant of the associated
p
-adic Galois representation. This generalizes a recent result of Spieß respectively Rosso and the first named author from the case of parallel weight
to arbitrary cohomological weights.
</p
Automorphic ℒ‐invariants for reductive groups
No full textGehrmann L. Automorphic ℒ‐invariants for reductive groups. Journal für die reine und angewandte Mathematik (Crelles Journal). 2021;2021(779):57-103
Shalika models and p-adic L-functions
No full textGehrmann L. Shalika models and p-adic L-functions. Bielefeld: Universitätsbibliothek Bielefeld; 2014
On quaternionic rigid meromorphic cocyles
No full textGehrmann L. On quaternionic rigid meromorphic cocyles. Mathematical Research Letters. 2022;29(5):1429-1444
Leading terms of anticyclotomic Stickelberger elements and -adic periods
No full textBergunde F, Gehrmann L. Leading terms of anticyclotomic Stickelberger elements and -adic periods. Transactions of the American Mathematical Society. 2018;370(9):6297-6329
Gelfand’s Trick for the Spherical Derived Hecke Algebra
No full textGehrmann L. Gelfand’s Trick for the Spherical Derived Hecke Algebra. International Mathematics Research Notices. 2022;2022(18):13841-13856.**Abstract**
Gelfand’s trick shows that the spherical Hecke algebra of a -adic split reductive group is commutative. We adapt this strategy in order to show that the spherical derived Hecke algebra is graded commutative under mild assumptions on the coefficient ring.</p
On the Algebraicity of Polyquadratic Plectic Points
No full textFornea M, Gehrmann L. On the Algebraicity of Polyquadratic Plectic Points. International Mathematics Research Notices. 2023;2023(22):19237-19265.**Abstract**
We establish direct evidence of the arithmetic significance of plectic Stark–Heegner points for elliptic curves of arbitrarily large rank. The main contribution is a proof of the algebraicity of plectic points associated to polyquadratic CM extensions of totally real number fields. Moreover, we relate the non-vanishing of plectic points to analytic and algebraic ranks of elliptic curves.</p
P-adic L-function of GL(2n) via method of p-adic representation.
No full textP -adic L-functions for cohomological cuspidal automorphic representations of GL(2n) were first constructed by Ash and Ginzburg in the case of trivial coefficients. We will discuss a new construction, which works for arbitrary coefficient systems. The construction relies on the representation theory of p-adic groups as well as properties of the cohomology of p-arithmetic groups. This is a generalization of Spiessâ work on the GL(2)-case.
Related references:
L. Gehrmann, On Shalika models and p-adic L-functions, Israel Journal of Mathematics 226 Issue 1, (June 2018), 237â 294
A. Ash and D. Ginzburg, P -adic L-functions for GL(2n), Inventiones mathematicae 116 (1994), 27â 73.
M. Spiess, On special zeros of p-adic L-functions of Hilbert modular forms, Inventiones mathe- maticae 196 (2014), 69â 138Non UBCUnreviewedAuthor affiliation: Universität Duisburg-EssenPostdoctora
Big principal series, -adic families and -invariants
No full textGehrmann L, Rosso G. Big principal series, -adic families and -invariants. Compositio Mathematica. 2022;158(2):409-436.
In earlier work, the first named author generalized the construction of Darmon-style
-invariants to cuspidal automorphic representations of semisimple groups of higher rank, which are cohomological with respect to the trivial coefficient system and Steinberg at a fixed prime. In this paper, assuming that the Archimedean component of the group has discrete series we show that these automorphic
-invariants can be computed in terms of derivatives of Hecke eigenvalues in
-adic families. Our proof is novel even in the case of modular forms, which was established by Bertolini, Darmon and Iovita. The main new technical ingredient is the Koszul resolution of locally analytic principal series representations by Kohlhaase and Schraen. As an application of our results we settle a conjecture of Spieß: we show that automorphic
-invariants of Hilbert modular forms of parallel weight
are independent of the sign character used to define them. Moreover, we show that they are invariant under Jacquet–Langlands transfer and, in fact, equal to the Fontaine–Mazur
-invariant of the associated Galois representation. Under mild assumptions, we also prove the equality of automorphic and Fontaine–Mazur
-invariants for representations of definite unitary groups of arbitrary rank. Finally, we study the case of Bianchi modular forms to show how our methods, given precise results on eigenvarieties, can also work in the absence of discrete series representations.
</p
Plectic Stark–Heegner points
No full textFornea M, Gehrmann L. Plectic Stark–Heegner points. Advances in Mathematics. 2023;414: 108861.We propose a conjectural construction of determinants of global points on modular elliptic curves over arbitrary number fields, generalizing both the p-adic construction of Heegner points via Čerednik–Drinfeld uniformization and the definition of classical Stark–Heegner points. In alignment with Nekovář and Scholl's plectic conjectures, we expect the non-triviality of these plectic Stark–Heegner points to control the Mordell–Weil group of higher rank elliptic curves. We provide some indirect evidence for our conjectures by showing that higher order derivatives of anticyclotomic p-adic L-functions compute plectic invariants
- …
