The values of the Dedekind-Rademacher cocycle at real multiplication points

Darmon, Henri; Pozzi, Alice; Vonk, J. Arie

doi:10.48550/arxiv.2103.02490

Cited by 2 publications

(3 citation statements)

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“…We conclude by describing a proof of Conjecture 7.4 in the case that F is a real quadratic field in the beautiful work of Darmon, Pozzi, and Vonk [12]. Their method is purely p-adic (i.e.…”

Section: The Methods Of Darmon-pozzi-vonkmentioning

confidence: 99%

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On the Brumer-Stark Conjecture and Refinements

Dasgupta¹,

Kakde²

2022

Preprint

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We state the Brumer-Stark conjecture and motivate it from two perspectives. Stark's perspective arose in his attempts to generalize the classical Dirichlet class number formula for the leading term of the Dedekind zeta function at s = 1 (equivalently, s = 0). Brumer's perspective arose by generalizing Stickelberger's work regarding the factorization of Gauss sums and the annihilation of class groups of cyclotomic fields. These viewpoints were synthesized by Tate, who stated the Brumer-Stark conjecture in its current form.The conjecture considers a totally real field F and a finite abelian CM extension H/F . It states the existence of p-units in H whose valuations at places above p are related to the special values of the L-functions of the extension H/F at s = 0. Essentially equivalently, the conjecture states that a Stickelberger element associated to H/F annihilates the (appropriately smoothed) class group of H.This conjecture has been refined by many authors in multiple directions. Notably, Kurihara conjectured that the Stickelberger element lies in the Fitting ideal of the Pontryagin dual of the class group, and furthermore conjectured an exact formula for this Fitting ideal. Burns constructed a Selmer group whose Fitting ideal he conjectured to be generated by the Stickelberger element. Atsuta and Kataoka conjectured a formula for the Fitting ideal of the class group, rather than its dual. Rubin stated a higher rank generalization of the Brumer-Stark conjecture. The first author and his collaborators stated an exact p-adic analytic formula for Brumer-Stark units, generalizing conjectures of Gross that give the image of the units under the Artin reciprocity map.We conclude by stating our results toward these various conjectures and summarizing the proofs. In particular, we prove the Brumer-Stark conjecture, Rubin's higher rank version, and Kurihara's conjecture, all "away from 2." We also prove strong partial results toward Gross's conjecture and the exact p-adic analytic formula for Brumer-Stark units. The key technique involved in the proofs is Ribet's method. We demonstrate congruences between Hilbert modular Eisenstein series and cusp forms, and use the associated Galois representations to construct Galois cohomology classes. These cohomology classes are interpreted in terms of Ritter-Weiss modules, from which results on class groups may be deduced.

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“…We conclude by describing a proof of Conjecture 7.4 in the case that F is a real quadratic field in the beautiful work of Darmon, Pozzi, and Vonk [12]. Their method is purely p-adic (i.e.…”

Section: The Methods Of Darmon-pozzi-vonkmentioning

confidence: 99%

“…Our interest in this result is that by enlarging S and taking larger and larger field extensions L/F , one can use (12) to specify all of the p-adic digits of u p . One therefore obtains an exact p-adic analytic formula for u p .…”

Section: Explicit Formula For Brumer-stark Unitsmentioning

confidence: 99%

On the Brumer-Stark Conjecture and Refinements

Dasgupta¹,

Kakde²

2022

Preprint

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“…To obtain similar results to those of Gross, Zagier and Kolyvagin in other settings, for example when is not imaginary, one should first generalize Heegner points. Note however that the algebraicity of Heegner points was provided by the theory of complex multiplication, which is either absent or conjectural in the real setting; there are some intriguing results for special cases and conjectures in this direction though, see [DPV21]. One first step is to phrase the construction of Heegner points differently, by using modular symbols.…”

Section: -Adic Heegner Pointsmentioning

confidence: 99%

p-adic L-functions, p-adic Gross-Zagier formulas and plectic points

Hernández Barrios

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In this work we generalize the construction of p-adic anticyclotomic L-functions associated to an elliptic curve E/F and a quadratic extension K/F, by defining a measure µ_f^p attached to K/F and an automorphic form. In the case of parallel 2, the automorphic form is associated with an elliptic curve E/F. The first main result is a p-adic Gross-Zagier formula: if E has split multiplicative reduction at p and p does not split at K/F, we compute the first derivative of the p-adic L-function by relating it with the conjugate difference of a Darmon point twisted by a character ¿. The proof uses the reciprocity map provided by class field theory as a natural way to interpret conjugate differences of points in E(Kp) as elements in the augmentation ideal for the aluation at the character ¿. This generalizes a result of Bertolini and Darmon. With a similar argument, after discovering the work of Fornea and ehrmann on plectic points, we prove an exceptional zero formula which relates a higher order derivative of In this work we generalize the construction of p-adic anticyclotomic L-functions associated to an elliptic curve E/F and a quadratic extension K/F, by defining a measure µ_f^p attached to K/F and an automorphic form. In the case of parallel 2, the automorphic form is associated with an elliptic curve E/F. The first main result is a p-adic Gross-Zagier formula: if E has split multiplicative reduction at p and p does not split at K/F, we compute the first derivative of the p-adic L-function by relating it with the conjugate difference of a Darmon point twisted by a character ¿. The proof uses the reciprocity map provided by class field theory as a natural way to interpret conjugate differences of points in E(Kp) as elements in the augmentation ideal for the evaluation at the character ¿. This generalizes a result of Bertolini and Darmon. With a similar argument, after discovering the work of Fornea and Gehrmann on plectic points, we prove an exceptional zero formula which relates a higher order derivative of µ_f^S with plectic points. We find an interpolating measure µ_F^p for µ_f^p attached to an interpolating Hida family F for f. Here µ_F^p can be regarded as a two variable p-adic L-function, which now includes the weight as a variable. Then we define the Hida-Rankin p-adic L-function Lp(f^p, ¿, k) as the restriction of µ_F^p to the weight space. Finally, we prove a formula which relates the weight-leading term of Lp(f^p, ¿, k) with plectic points. In short, the leading term is an explicit constant times Euler factors times the logarithm of the trace of a plectic point. This formula is a generalization of a result of Longo, Kimball and Hu, which has been used to prove the rationality of a Darmon point under some hypotheses. En aquesta tesi generalitzem la construcció de funcions L p-àdiques anticiclotòmiques associades a una corba el·líptica E/F i una extensió quadràtica K/F, definint una mesura µ_f^p associada a K/F i una forma automorfa. En el cas de pes paral·lel 2, la forma automorfa s’associa a una corba el·líptica E/F. El primer resultat és una fórmula p-àdica de Gross-Zagier: si E té reducció multiplicativa split a p i p descomposa a K/F, calculem la primera derivada de la funció L p-àdica relacionant-la amb la diferència conjugada d’un punt de Darmon twistat per un caràcter ¿. La demostració utilitza l’aplicació de reciprocitat de la teoria de cossos com una manera natural d’interpretar les diferències conjugades de punts de E(Kp) com elements en l’ideal d’augmentació de l’avaluació en el caràcter ¿. Això generalitza un resultat de Bertolini i Darmon. Amb un argument semblant, després de descobrir el treball de Fornea i Gehrmann sobre els punts plèctics, demostrem una fórmula de zero excepcional que relaciona una derivada d’ordre superior de µ_f^S amb punts plèctics. Trobem una mesura d’interpolació µ_F^p per a µ_f^p associada a la família de Hida F que passa per f. Aquí µ_F^p es pot considerar com una funció L de dues variables, que ara inclou el pes com a variable. Aleshores definim una funció L de Hida-Rankin p-àdica Lp(f^p, ¿, k) com la restricció de µ_F^p a l’espai de pesos. Finalment, demostrem una fórmula que relaciona el terme principal de Lp(f^p, ¿, k) respecte al pes amb punts plèctics. En resum, el terme principal és una constant explícita multiplicada per factors d’Euler i pel logaritme de la traça d’un punt plèctic. Aquesta fórmula és una generalització d’un resultat de Longo, Kimball i Hu, que s’ha utilitzat per demostrar la racionalitat d’un punt de Darmon sota certes hipòtesis.

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The values of the Dedekind-Rademacher cocycle at real multiplication points

Cited by 2 publications

References 12 publications

On the Brumer-Stark Conjecture and Refinements

On the Brumer-Stark Conjecture and Refinements

p-adic L-functions, p-adic Gross-Zagier formulas and plectic points

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