Epsilon factors of symplectic type characters in the wild case

By work of John Tate we can associate an epsilon factor with every multiplicative character of a local field. In this paper, we determine the explicit signs of the epsilon factors for symplectic type characters of K × {K^{\times}} , where K / F {K/F} is a wildly ramified quadratic extension of a non-Archimedean local field F of characteristic zero.

A proof of the linear Arithmetic Fundamental Lemma for

Let $K/F$ be an unramified quadratic extension of a non-Archimedean local field. In a previous work [1], we proved a formula for the intersection number on Lubin–Tate spaces. The main result of this article is an algorithm for computation of this formula in certain special cases. As an application, we prove the linear Arithmetic Fundamental Lemma for $ \operatorname {{\mathrm {GL}}}_4$ with the unit element in the spherical Hecke Algebra.

Higher genus theory

In $1801$, Gauss found an explicit description, in the language of binary quadratic forms, for the $2$-torsion of the narrow class group and dual narrow class group of a quadratic number field. This is now known as Gauss’s genus theory. In this paper, we extend Gauss’s work to the setting of multi-quadratic number fields. To this end, we introduce and parametrize the categories of expansion groups and expansion Lie algebras, giving an explicit description for the universal objects of these categories. This description is inspired by the ideas of Smith [ 16] in his recent breakthrough on Goldfeld’s conjecture and the Cohen–Lenstra conjectures. Our main result shows that the maximal unramified multi-quadratic extension $L$ of a multi-quadratic number field $K$ can be reconstructed from the set of generalized governing expansions supported in the set of primes that ramify in $K$. This provides a recursive description for the group $\textrm{Gal}(L/\mathbb{Q})$ and a systematic procedure to construct the field $L$. A special case of our main result gives an upper bound for the size of $\textrm{Cl}^{+}(K)[2]$.

PERIOD IDENTITIES OF CM FORMS ON QUATERNION ALGEBRAS

Waldspurger’s formula gives an identity between the norm of a torus period and an $L$ -function of the twist of an automorphic representation on GL(2). For any two Hecke characters of a fixed quadratic extension, one can consider the two torus periods coming from integrating one character against the automorphic induction of the other. Because the corresponding $L$ -functions agree, (the norms of) these periods—which occur on different quaternion algebras—are closely related. In this paper, we give a direct proof of an explicit identity between the torus periods themselves.

Multiplicity One for Pairs of Prasad–Takloo-Bighash Type

Let ${\textrm{E}}/{\textrm{F}}$ be a quadratic extension of non-archimedean local fields of characteristic different from $2$. Let ${\textrm{A}}$ be an ${\textrm{F}}$-central simple algebra of even dimension so that it contains ${\textrm{E}}$ as a subfield, set ${\textrm{G}}={\textrm{A}}^\times $ and ${\textrm{H}}$ for the centralizer of ${\textrm{E}}^\times $ in ${\textrm{G}}$. Using a Galois descent argument, we prove that all double cosets ${\textrm{H}} g {\textrm{H}}\subset{\textrm{G}}$ are stable under the anti-involution $g\mapsto g^{-1}$, reducing to Guo’s result for ${\textrm{F}}$-split ${\textrm{G}}$ [14], which we extend to fields of positive characteristic different from $2$. We then show, combining global and local results, that ${\textrm{H}}$-distinguished irreducible representations of ${\textrm{G}}$ are self-dual and this implies that $({\textrm{G}},{\textrm{H}})$ is a Gelfand pair $$\begin{equation*}\operatorname{dim}_{\mathbb{C}}(\operatorname{Hom}_{{\textrm{H}}}(\pi,\mathbb{C}))\leq 1\end{equation*}$$for all smooth irreducible representations $\pi $ of ${\textrm{G}}$. Finally we explain how to obtain the multiplicity one statement in the archimedean case using the criteria of Aizenbud and Gourevitch ([1]), and we then show self-duality of irreducible distinguished representations in the archimedean case too.

Subjective value then confidence in human ventromedial prefrontal cortex

Two fundamental goals of decision making are to select actions that maximize rewards while minimizing costs and to have strong confidence in the accuracy of a judgment. Neural signatures of these two forms of value: the subjective value (SV) of choice alternatives and the value of the judgment (confidence), have both been observed in ventromedial prefrontal cortex (vmPFC). However, the relationship between these dual value signals and their relative time courses are unknown. We recorded fMRI while 28 men and women performed a two-phase Ap-Av task with mixed-outcomes of monetary rewards paired with painful shock stimuli. Neural responses were measured during offer valuation (offer phase) and choice valuation (commit phase) and analyzed with respect to observed decision outcomes, model-estimated SV and confidence. During the offer phase, vmPFC tracked SV and decision outcomes, but it not confidence. During the commit phase, vmPFC tracked confidence, computed as the quadratic extension of SV, but it bore no significant relationship with the offer valuation itself, nor the decision. In fact, vmPFC responses from the commit phase were selective for confidence even for rejected offers, wherein confidence and SV were inversely related. Conversely, activation of the cognitive control network, including within lateral prefrontal cortex (lPFC) and dorsal anterior cingulate cortex (dACC) was associated with ambivalence, during both the offer and commit phases. Taken together, our results reveal complementary representations in vmPFC during value-based decision making that temporally dissociate such that offer valuation (SV) emerges before decision valuation (confidence).